Point Process Calculus in Time and Space [1 ed.] 9783030627522, 9783030627539

This book provides an introduction to the theory and applications of point processes, both in time and in space. Present

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Point Process Calculus in Time and Space [1 ed.]
 9783030627522, 9783030627539

Table of contents :
Preface
Contents
Chapter 1 Generalities
1.1 Point Processes as Random Measures
1.2 Campbell’s Formula and Moment Measures
1.3 The Distribution of a Point Process
1.4 Convergence in Distribution and Variation
1.5 Cluster Point Processes
1.6 The Stieltjes–Lebesgue Calculus
1.7 Exercises
Chapter 2 Poisson Processes on the Line
2.1 Counting Process and Interval Sequence
2.2 The Smoothing Formula
2.3 Poisson Martingales and Stochastic Integrals
2.4 Watanabe’s Characterization
2.5 HMCs and Stochastic Differential Equations
2.6 HMCs and Time-Scaled HPPs
2.7 Exercises
Chapter 3 Spatial Poisson Processes
3.1 Sampling a Poisson Process
3.2 The Covariance and Exponential Formulas
3.3 Marked Spatial Poisson Processes
3.4 Operations on Poisson Processes
3.5 Change of Probability
3.6 Exact Sampling of Cluster Point Processes
3.7 The Boolean Model
3.8 Exercises
Chapter 4 Renewal and Regenerative Processes
4.1 Renewal Point processes
4.2 The Renewal Theorem
4.3 Blackwell’s Theorem and its Refinements
4.4 Regenerative Processes
4.5 Multivariate Renewal Equations
4.6 Semi-Markov Processes
4.7 Exercises
Chapter 5 Point Processes with a Stochastic Intensity
5.1 The Smoothing Formulas
5.2 Regenerative Form of the Intensity Kernel
5.3 Martingales as Stochastic Integrals
5.4 Time Scaling
5.5 Continuous Change of Probability
5.6 Extension of the Theory of Stochastic Intensity
5.7 Grigelionis’ Representation
5.8 Origin and Motivation of the Martingale Approach
5.9 Exercises
Chapter 6 Exvisible Intensity of Finite Point Processes
6.1 The Janossy Density
6.2 The Spatial Smoothing formula
6.3 Exvisibility and Predictability
6.4 Finite Markov Point Processes
6.5 Spatial Birth-and-Death Point Processes
6.6 An Alternative Model
6.7 Exercises
Chapter 7 Palm Probability on the Line
7.1 Stationary Point Processes
7.2 A First Look at Palm Probability
7.3 Palm Theory on the Line: Basic Formulas
7.4 From Palm Probability to Stationary Probability
7.5 Local Interpretation of Palm Probability
7.6 The Cross-ergodic Theorem
7.7 Palm Probability and Stochastic Intensity
7.8 Exercises
Chapter 8 Palm Probability in Space
8.1 The Voronoi Cell and the Inversion Formula
8.2 The Local Interpretation
8.3 Ergodicity
8.4 The Mecke Measure
8.5 The Reduced Mecke Measure
8.6 Exercises
Chapter 9 The Power Spectral Measure
9.1 The Covariance Measure
9.2 The Bartlett Spectral Measure
9.3 Long-range Dependence
9.4 Transformations of the Spectral Measure
9.5 Exercises
Chapter 10 The Information Content of Point Processes
10.1 Filtering
10.2 Separation of Detection and Filtering
10.3 Hiding Information in a Point Process
10.4 Noisy Points
10.5 Random Sampling
10.6 Exercises
Chapter 11 Point Processes in Queueing
11.1 A Review of Markovian Queueing Theory
11.2 Poisson Systems
11.3 The G/G/1/∞ Queue
11.4 PASTA, Little, etc.
11.5 Selected Applications
11.6 Exercises
Chapter 12 Hawkes Point Processes
12.1 As a Branching Point Process
12.2 Rates of Extinction and of Installation
12.3 The Bartlett Spectrum of the Hawkes Process
12.4 Exact Sampling of Hawkes Processes
12.5 Branching Point Processes Without Ancestor
12.6 Kerstan Point Processes
12.7 Exercises
Appendix
A.1 Measurability and Measure
A.2 Stochastic Processes
A.3 Martingales
A.4 Internal History of a Marked Point Process
A.5 Local vs. Global Absolute Continuity
Bibliography
Index

Citation preview

Probability Theory and Stochastic Modelling 98

Pierre Brémaud

Point Process Calculus in Time and Space An Introduction with Applications

Probability Theory and Stochastic Modelling Volume 98

Editors-in-Chief Peter W. Glynn, Stanford, CA, USA Andreas E. Kyprianou, Bath, UK Yves Le Jan, Orsay, France Advisory Editors Søren Asmussen, Aarhus, Denmark Martin Hairer, Coventry, UK Peter Jagers, Gothenburg, Sweden Ioannis Karatzas, New York, NY, USA Frank P. Kelly, Cambridge, UK Bernt Øksendal, Oslo, Norway George Papanicolaou, Stanford, CA, USA Etienne Pardoux, Marseille, France Edwin Perkins, Vancouver, Canada Halil Mete Soner, Zürich, Switzerland

The Probability Theory and Stochastic Modelling series is a merger and continuation of Springer’s two well established series Stochastic Modelling and Applied Probability and Probability and Its Applications. It publishes research monographs that make a significant contribution to probability theory or an applications domain in which advanced probability methods are fundamental. Books in this series are expected to follow rigorous mathematical standards, while also displaying the expository quality necessary to make them useful and accessible to advanced students as well as researchers. The series covers all aspects of modern probability theory including • • • • • •

Gaussian processes Markov processes Random Fields, point processes and random sets Random matrices Statistical mechanics and random media Stochastic analysis

as well as applications that include (but are not restricted to): • Branching processes and other models of population growth • Communications and processing networks • Computational methods in probability and stochastic processes, including simulation • Genetics and other stochastic models in biology and the life sciences • Information theory, signal processing, and image synthesis • Mathematical economics and finance • Statistical methods (e.g. empirical processes, MCMC) • Statistics for stochastic processes • Stochastic control • Stochastic models in operations research and stochastic optimization • Stochastic models in the physical sciences

More information about this series at http://www.springer.com/series/13205

Pierre Brémaud

Point Process Calculus in Time and Space An Introduction with Applications

123

Pierre Brémaud Paris, France

ISSN 2199-3130 ISSN 2199-3149 (electronic) Probability Theory and Stochastic Modelling ISBN 978-3-030-62752-2 ISBN 978-3-030-62753-9 (eBook) https://doi.org/10.1007/978-3-030-62753-9 Mathematics Subject Classification: 60-xx, 90-xx, 93-xx, 94-xx © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

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 A7 E = Rm M/ B7 ν Bb Q7 i?2 7Q`K ν(C) = C λ(x)dx 7Q` bQK2 MQM@M2;iBp2 K2bm`#H2 7mM+iBQM λ : Rm → R- i?2 SQBbbQM T`Q+2bb N Bb bB/ iQ /KBi i?2 BMi2MbBiv 7mM+iBQM λ(x)X A7 BM //BiBQM λ(x) ≡ λ- N Bb +HH2/  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb U?TTV QM Rm rBi? BMi2MbBiv Q` `i2 λX A7 λ ≡ 1- Bi Bb bQK2iBK2b +HH2/  biM/`/ SQBbbQM T`Q+2bbX *Qt S`Q+2bb2b .Qm#Hv biQ+?biB+ SQBbbQM T`Q+2bb2b- Q` *Qt T`Q+2bb2b-j `2 +QMbi`m+i2/ BM irQ bi2TbX 6B`bi QM2 /`rb  `M/QK BMi2MbBiv K2bm`2 T`Q+2bb- i?i Bb  HQ+HHv }@ MBi2 `M/QK K2bm`2 ν QM i?2 K2bm`#H2 bT+2 (E, E)- M/ ?pBM; /QM2 bQ- QM2 ;2M2`i2b  SQBbbQM T`Q+2bb N rBi? i?2 BMi2MbBiv K2bm`2 νX 6Q`KHHv, .2}MBiBQM RXRXRy G2i ν #2  HQ+HHv }MBi2 `M/QK K2bm`2 QM EX G2i G #2  σ@}2H/ +QMiBMBM; i?2 σ@}2H/ F ν ;2M2`i2/ #v νX  TQBMi T`Q+2bb N QM E bm+? i?i- ;Bp2M G- N Bb  SQBbbQM T`Q+2bb QM E rBi? i?2 BMi2MbBiv K2bm`2 ν Bb +HH2/  /Qm#Hv biQ+?biB+ SQBbbQM T`Q+2bb rBi? `2bT2+i iQ G rBi? i?2 +QM/BiBQMH BMi2MbBiv K2bm`2 ν- Q`  *Qt T`Q+2bb /B`2+i2/ #v i?2 `M/QK K2bm`2 νX A7 E = Rm M/ ν(dx) = λ(x) dx 7Q` bQK2 MQM@M2;iBp2 HQ+HHv BMi2;`#H2 biQ+?biB+ T`Q+2bb {λ(x)}x∈Rm - i?2 TQBMi T`Q+2bb N Bb +HH2/  /Qm#Hv biQ+?b@ iB+ SQBbbQM T`Q+2bb rBi? `2bT2+i iQ G rBi? i?2 U+QM/BiBQMHV BMi2MbBiv 7mM+iBQM {λ(x)}x∈Rm X A7 KQ`2Qp2` λ(x) ≡ Λ-  MQM@M2;iBp2 `M/QK p`B#H2- N Bb +HH2/  KBt2/ SQBbbQM T`Q+2bb Q7 BMi2MbBiv ΛX _2M2rH SQBMi S`Q+2bb G2i {Sn }n≥1 #2 M BB/ b2[m2M+2 Q7 TQbBiBp2 `M/QK p`B#H2b rBi? +QKKQM +mKm@ HiBp2 /Bbi`B#miBQM 7mM+iBQM F X h?2 TQBMi T`Q+2bb QM R+ /2}M2/ #v Bib b2[m2M+2 Q7 TQBMib n  Tn = Sk (n ≥ 1) k=1

Bb +HH2/  `2M2rH TQBMi T`Q+2bbX am+? TQBMi T`Q+2bb2b rBHH #2 bim/B2/ BM *?Ti2` 9X SQBMi S`Q+2bb2b QM :`B/b .2}MBiBQM RXRXRR  ;`B/ QM Rm Um ≥ 1V Bb  +QHH2+iBQM Q7 TQBMib Q7 i?2 ivT2 {(n1 T1 , . . . , nm Tm ) ; n1 , . . . , nm ∈ Z} , r?2`2 T1 , . . . , Tm `2 TQbBiBp2 `2H MmK#2`bX j

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*>Sh1_ RX :1L1_GAhA1a

1tKTH2 RXRXRk, _M/QK SQBMib QM i?2 :`B/X G2i {Xn1 ,n2 }n1 ,n2 ∈Z #2  /Qm#Hv BM/2t2/ b2[m2M+2 Q7 {0, 1}@pHm2/ `M/QK p`B#H2bX  TQBMi T`Q+2bb QM i?2 ;`B/ URXRV UrBi? m = 2V Bb +QMbi`m+i2/ 7`QK i?Bb b2[m2M+2 #v TH+BM;  TQBMi i (n1 T1 , n2 T2 ) B7 M/ QMHv B7 Xn1 ,n2 = 1X G2i pn1 ,n2 := P (Xn1 ,n2 = 1)X h?2 +QM/BiBQM  pn1 ,n2 < ∞ n1 ,n2 ∈Z

Bb bm{+B2Mi 7Q` i?Bb TQBMi T`Q+2bb iQ #2 }MBi2 Ui?i Bb- iQ ?p2 HKQbi@bm`2Hv  }MBi2 MmK#2` Q7 TQBMibV bBM+2 i?2 bmK BM i?2 H27i@?M/ bB/2 2[mHb i?2 p2`;2 iQiH MmK#2` Q7 TQBMibX AM i?2 +b2 i?i {Xn1 ,n2 }n1 ,n2 ∈Z Bb M BM/2T2M/2Mi b2[m2M+2- Bi Bb HbQ M2+2bb`v U#v i?2 "Q`2HĜ*Mi2HHB H2KK 7Q` BM/2T2M/2Mi b2[m2M+2bVX A7 Xn1 ,n2 = 1 7Q` HH n1 , n2 ∈ Z- i?2 +Q``2bTQM/BM; U/2i2`KBMBbiB+V TQBMi T`Q+2bb Bb +HH2/  +QKTH2i2 ;`B/ UQM R2 VX P7 +Qm`b2 bBKBH` /2}MBiBQMb `2 pBH#H2 7Q` i?2 ;`B/ QM Rd M/ Mv d ≥ 1X 1tKTH2 RXRXRj,  aiiBQM`v :`B/X  +QKTH2i2 ;`B/ QM R2 Bb MQi  biiBQM@ `v TQBMi T`Q+2bbX >Qr2p2`- i?2 `M/QKHv b?B7i2/ p2`bBQM Q7 Bi(n1 T1 + V1 , n2 T2 + V2 )

(n1 , n2 ∈ Z) ,

r?2`2 V1 M/ V2 `2 BM/2T2M/2Mi `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, T1 ) M/ [0, T2 ) `2bT2+iBp2Hv- Bb biiBQM`vX h?Bb +M #2 T`Qp2/ /B`2+iHv Q` #v mbBM; i?2 GTH+2 7mM+iBQMH U1tKTH2 RXjXReVX .2}MBiBQM RXRXR9 G2i N #2  `M/QK K2bm`2 QM (E, E)X h?2 b2i 7mM+iBQM C → ν(C) := E [N (C)]

(C ∈ E)

mMB[m2Hv /2}M2b  K2bm`2 QM (E, E) +HH2/ i?2 BMi2MbBiv K2bm`2 Q7 N X .2}MBiBQM RXRXR8 AM i?2 +b2 Q7 M HX+X/X#X bT+2 E-  HQ+HHv }MBi2 TQBMi T`Q+2bb N Bb +HH2/  }`bi@Q`/2` TQBMi T`Q+2bb B7 E [N (C)] < ∞ 7Q` HH `2HiBp2Hv +QKT+i b2ib C ∈ EX J`F2/ SQBMi S`Q+2bb2b hQ 2+? TQBMi Q7  TQBMi T`Q+2bb Kv #2 ii+?2/ M ii`B#mi2- 7Q` BMbiM+2 i?2 ?2B;?i Q7 i?2 i`22 ;`QrBM; i?2`2 Q` i?2 `2[mB`2/ b2`pB+2 iBK2 Q7  +mbiQK2` T`2@ b2MiBM; ?BKb2H7 i  b2`pB+2 /2bF i i?Bb iBK2X h?Bb ii`B#mi2 Bb `2T`2b2Mi2/ #v  `M/QK 2H2K2Mi- +HH2/ i?2 K`F Q7 i?2 TQBMiX h?Bb K`F +M #2 Q7  p2`v ;2M2`H Mim`2- `M;BM; 7`QK  MmK#2` iQ  7mM+iBQM Q`  TQBMi T`Q+2bb- b rBHH i?2 +b2 BM i?2 /2}MBiBQM Q7 SHK T`Q##BHBiv U*?Ti2` dVX .2}MBiBQM RXRXRe G2i N M/ {Xn }n∈N #2 b BM G2KK RXRX8X G2i (K, K) #2 bQK2 K2bm`#H2 bT+2 M/ H2i {Zn }n∈N #2  `M/QK b2[m2M+2 rBi? pHm2b BM KX h?2 b2[m2M+2 {(Xn , Zn )}n∈N Bb +HH2/  K`F2/ TQBMi T`Q+2bb QM E rBi? K`Fb BM KX h?2 b2[m2M+2 {Zn }n∈N Bb i?2 K`F b2[m2M+2 M/ N Bb i?2 #b2 TQBMi T`Q+2bbX

RXRX SPALh S_P*1aa1a a _L.PJ J1al_1a

d

E

L

C

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6B;m`2 RXRXkX  K`F2/ TQBMi T`Q+2bbX >2`2 N (C × L) = 2X .2}M2  TQBMi T`Q+2bb N UBM i?2 b2Mb2 Q7 .2}MBiBQM RXRXkV QM E × K #v  N (C × L) := 1C (Xn )1L (Zn ) (C ∈ B(E) , L ∈ K) . URXkV n∈N

LQi2 i?i bBM+2 Δ ∈ / C- i?2 TQBMib Xn ∈ {Δ} /Q MQi TT2` BM i?2 bmK #Qp2 UdzTQBMib i BM}MBiv `2 2t+Hm/2/ǴVX q2 b?HH Q++bBQMHHv mb2 i?2 MQiiBQM NZ BMbi2/ Q7 N X h?2 7QHHQrBM; T?`b2b `2 i?2M +QMbB/2`2/ 2[mBpH2Mi, dzi?2 K`F2/ TQBMi T`Q+2bb {(Xn , Zn )}n∈N Ǵdzi?2 K`F2/ TQBMi T`Q+2bb N Ǵdzi?2 K`F2/ TQBMi T`Q+2bb NZ Ǵ- M/ dzi?2 K`F2/ TQBMi T`Q+2bb (N, Z)ǴX UAM i?2 Hbi irQ MQiiBQMb- Z Bb  bvK#QHB+ `2T`2b2MiiBQM Q7 i?2 b2[m2M+2 Q7 K`FbXV .2}MBiBQM RXRXRd A7 BM //BiBQM i?2 b2[m2M+2 {Zn }n∈N Bb BB/ M/ BM/2T2M/2Mi Q7 N - i?2M N Bb +HH2/  K`F2/ TQBMi T`Q+2bb rBi? BM/2T2M/2Mi BB/ K`FbX h?2 /2}MBiBQM Q7 i?2 BMi2MbBiv K2bm`2 Q7  K`F2/ TQBMi T`Q+2bb Bb bBKBH` iQ i?i Q7 M mMK`F2/ TQBMi T`Q+2bb U.2}MBiBQM RXRXR9VX .2}MBiBQM RXRXR3 h?2 BMi2MbBiv K2bm`2 Q7 i?2 K`F2/ TQBMi T`Q+2bb NZ := {(Xn , Zn )}n∈N Q7 .2}MBiBQM RXRXRe Bb i?2 K2bm`2 νZ mMB[m2Hv /2}M2/ QM (E × K, E ⊗ K) #v    1D ((Xn , Zn )) (D ∈ E ⊗ K) . νZ (D) := E n∈Z

1tKTH2 RXRXRN, AMi2MbBiv J2bm`2 Q7  J`F2/ SQBMi S`Q+2bb rBi? AM/2T2M/2Mi BB/ J`FbX h?2 BMi2MbBiv K2bm`2 Q7 i?2 K`F2/ TQBMi T`Q+2bb Q7 .2}MBiBQM RXRXRd Bb i?2 T`Q/m+i K2bm`2 ν × QZ - r?2`2 ν Bb i?2 BMi2MbBiv K2bm`2 Q7 i?2 #b2 TQBMi T`Q+2bb U1t2`+Bb2 RXdX9VX G2i E M/ K #2 HX+X/X#X bT+2b rBi? i?2B` "Q`2H σ@}2H/b B(E) M/ K := B(K)  := E × K Bb M HX+X/X#X bT+2 r?Qb2 "Q`2H σ@}2H/ B(E)  Bb i?2 `2bT2+iBp2HvX h?2M E T`Q/m+i B(E) ⊗ KX

3

*>Sh1_ RX :1L1_GAhA1a

.2}MBiBQM RXRXky h?2 +MQMB+H bT+2 Q7 HQ+HHv }MBi2 TQBMi T`Q+2bb2b QM E rBi? p (E)  Q7 HQ+HHv K`Fb BM K Bb- #v /2}MBiBQM- i?2 +MQMB+H bT+2 Mp (E × K) := M  }MBi2 TQBMi K2bm`2b QM EX  TQBMi K2bm`2 μ  ∈ Mp (E × K) Bb `2T`2b2Mi2/ #v  b2[m2M+2 { xn }n∈N = {(xn , zn )}n∈N Q7 TQBMib BM E × KX ++Q`/BM; iQ i?2 mbmH i2`KBMQHQ;v- μ := {xn }n∈N Bb i?2 #b2 TQBMi K2bm`2 M/ i?2 b2[m2M+2 {zn }n∈N Bb i?2 b2[m2M+2 Q7 K`Fb- zn ∈ K #2BM; i?2 K`F Q7 i?2 #b2 TQBMi xn ∈ EX

RXk *KT#2HHǶb 6Q`KmH M/ JQK2Mi J2bm`2b SQBMi T`Q+2bb UbiQ+?biB+V BMi2;`Hb `2 BMi2;`Hb rBi? `2bT2+i iQ  `M/QK TQBMi K2bm`2X aBM+2 i?Bb `M/QK K2bm`2 Bb BM i?Bb T`iB+mH` +b2  +QmMiBM; K2bm`2i?2b2 BMi2;`Hb `2 BM 7+i bmKbX G2i (E, E) #2 M `#Bi``v K2bm`#H2 bT+2 M/ N  HQ+HHv }MBi2 TQBMi T`Q+2bb Q`  TQBMi T`Q+2bb BM i?2 b2Mb2 Q7 .2}MBiBQM RXRXkX G2i μ #2  K2bm`2 QM i?2 K2bm`#H2 bT+2 (E, E)  M/ H2i ϕ : (E, E) → (R, B(R)) #2  K2bm`#H2 7mM+iBQM 7Q` r?B+? i?2 BMi2;`H E ϕ dμ Bb r2HH /2}M2/X h?Bb BMi2;`H rBHH HbQ #2 /2MQi2/ #v μ(ϕ)X q?2M ϕ : E → R Bb  K2bm`#H2 7mM+iBQM M/ N Bb  TQBMi T`Q+2bb- i?2 7QHHQrBM; MQiiBQMb `2T`2b2Mi i?2 bK2 Ki?2KiB+H Q#D2+i UB7 i?2 Hii2` Bb r2HH /2}M2/V,   ϕ(Xn ) , ϕ(x)N (dx) , N (ϕ) . n∈N

E

h?2 Hbi irQ MQiiBQMb TTHv iQ Mv `M/QK K2bm`2X AM i?2 }`bi MQiiBQM- r?B+? +QM+2`Mb QMHv TQBMi T`Q+2bb2b- r2 mb2 i?2 +QMp2MiBQM i?i i?2 bmK 2ti2M/b QMHv iQ i?Qb2 BM/B+2b n bm+? i?i Xn ∈ E- 2t+Hm/BM; i?2 TQBMib dzi BM}MBivǴ UBM 7+iϕ(Δ) Bb MQi /2}M2/VX AM i?2 bBimiBQM Q7 .2}MBiBQM RXRXRe- Q#b2`p2 i?i   ϕ(x, z)N (dx × dz) = ϕ(Xn , Zn ), E×K

n∈N

rBi? i?2 bK2 +QMp2MiBQM b i?2 QM2 Dmbi ;`22/ mTQM +QM+2`MBM; TQBMib i BM}MBivX h?2Q`2K RXkXR G2i N #2  `M/QK K2bm`2 QM i?2 K2bm`#H2 bT+2 (E, E) rBi? BMi2MbBiv K2bm`2 νX h?2M- 7Q` HH K2bm`#H2 7mM+iBQMb ϕ : E → R r?B+? `2 2Bi?2` MQM@M2;iBp2 Q` BM L1R (ν)- i?2 BMi2;`H N (ϕ) Bb r2HH /2}M2/ UTQbbB#Hv BM}MBi2 r?2M ϕ Bb QMHv bbmK2/ iQ #2 MQM@M2;iBp2V M/ E [N (ϕ)] = ν(ϕ) . AM T`iB+mH`- N (ϕ) Bb XbX }MBi2 B7 ϕ ∈ L1R (ν)X

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h?2 7Q`KmH URXjV Bb FMQrM b i?2 U}`biV *KT#2HH 7Q`KmHX9 S`QQ7X 6B`bi- bmTTQb2 i?i ϕ Bb  bBKTH2 MQM@M2;iBp2 K2bm`#H2 7mM+iBQM- i?i Bb- Q7 i?2 7Q`K L  α h 1 Ch , h=1

r?2`2 L ∈ N- αh ∈ R+ M/ C1 , . . . , CL `2 /BbDQBMi K2bm`#H2 bm#b2ib Q7 EX h?2M  L  L   E[N (ϕ)] = E ah N (Ch ) = ah ν(Ch ) = ν(ϕ) . h=1

h=1

LQr H2i ϕ #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM M/ {ϕn }n∈N  MQM@/2+`2bBM; b2[m2M+2 Q7 bBKTH2 MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb rBi? HBKBi ϕX G2iiBM; n i2M/ iQ ∞ BM E[N (ϕn )] = ν(ϕn ) vB2H/b i?2 MMQmM+2/ `2bmHi- #v KQMQiQM2 +QMp2`;2M+2X AM i?2 +b2 r?2`2 ϕ ∈ L1R (ν)- bBM+2 E [N (ϕ± )] = ν(ϕ± ) < ∞- i?2 `M/QK p`B#H2b N (ϕ± ) `2 P @XbX }MBi2- M/ i?2`27Q`2 N (ϕ) = N (ϕ+ ) − N (ϕ− ) Bb r2HH /2}M2/ M/ }MBi2- M/ E[N (ϕ)] = E[N (ϕ+ )] − E[N (ϕ− )] = ν(ϕ+ ) − ν(ϕ− ) = ν(ϕ) .  1tKTH2 RXkXk, *KT#2HHǰb 6Q`KmH 7Q` J`F2/ SQBMi S`Q+2bb2b rBi? AM/2T2M/2Mi BB/ J`FbX *QMbB/2` i?2 K`F2/ TQBMi T`Q+2bb Q7 .2}MBiBQM RXRXRd- i?i Bb-  K`F2/ TQBMi T`Q+2bb N QM E rBi? BM/2T2M/2Mi BB/ K`Fb {Zn }n∈N iFBM; i?2B` pHm2b BM bQK2 K2bm`#H2 bT+2 (K, K) M/ Q7 +QKKQM /Bbi`B#miBQM QZ X Aib BMi2MbBiv K2bm`2 Bb νZ (dx × dz) = ν(dx)QZ (dz) U1tKTH2 RXRXRNVX *KT@ #2HHǶb i?2Q`2K i?2M `2/b b 7QHHQrbX A7 i?2 K2bm`#H2 7mM+iBQM ϕ : Rm × K → R Bb 2Bi?2` MQM@M2;iBp2 Q` BM L1R (ν × QZ )- i?2M i?2 bmK  ϕ(Xn , Zn ) n∈N

Bb P @XbX r2HH /2}M2/ UTQbbB#Hv BM}MBi2 B7 ϕ Bb QMHv bbmK2/ MQM@M2;iBp2V M/     E ϕ(Xn , Zn ) = E [ϕ(x, Z)] ν(dx) , n∈N

E

r?2`2 Z Bb Mv K@pHm2/ `M/QK p`B#H2 rBi? /Bbi`B#miBQM QZ X h?2 7QHHQrBM; pi` Q7 *KT#2HHǶb 7Q`KmH Bb bQK2iBK2b M22/2/ Ub22 i?2 M2ti i?2Q`2K 7Q` BMbiM+2V, 9

(*KT#2HH- RNRy)X

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*Q`QHH`v RXkXj G2i N #2  TQBMi T`Q+2bb QM i?2 K2bm`#H2 bT+2 (E, E) rBi? BMi2MbBiv K2bm`2 ν- /2}M2/ QM i?2 T`Q##BHBiv bT+2 (Ω, F, P )X G2i G #2  bm#@σ@ }2H/ Q7 F- BM/2T2M/2Mi Q7 F N X h?2M- 7Q` HH K2bm`#H2 7mM+iBQMb ϕ : (E × Ω, E ⊗ G) → (R, B(R)) r?B+? `2 2Bi?2` MQM@M2;iBp2 Q` BM L1R (ν × P )- i?2 BMi2;`H  ϕ(x, ω) N (ω, dx) N (ϕ)(ω) := E

Bb XbX r2HH /2}M2/ UTQbbB#Hv BM}MBi2 r?2M ϕ Bb QMHv bbmK2/ MQM@M2;iBp2V M/ E [N (ϕ)] = ν(E [ϕ]). AM T`iB+mH`- B7 ϕ ∈

L1R (ν

× P )- N (ϕ) Bb XbX }MBi2X

S`QQ7X h?2 bb2`iBQM Bb i`m2 7Q` ϕ(x, ω) = u(x)v(ω) , r?2`2 u : E → R+ Bb E@K2bm`#H2 M/ v : Ω → R+ Bb G@K2bm`#H2- bBM+2 BM i?Bb +b2E [N (ϕ)] = E[v]E [N (u)] = E[v]ν(u) = ν(E[v]u) = ν(E [ϕ]) . h?2 `2bi Bb  bi`B;?i7Q`r`/ TTHB+iBQM Q7 .vMFBMǶb 7mM+iBQMH i?2Q`2K Uh?2Q`2K XRXeVX  _2+HH i?i  K2bm`2 μ QM i?2 K2bm`#H2 bT+2 (E, E) Bb +HH2/ MQM@iQKB+Q` /Bzmb2- B7 μ({a}) = 0 7Q` Mv bBM;H2iQM {a}X h?2Q`2K RXkX9 hrQ BM/2T2M/2Mi TQBMi T`Q+2bb2b N1 M/ N2 QM i?2 K2bm`#H2 bT+2 (E, E) rBi? `2bT2+iBp2 BMi2MbBiv K2bm`2b ν1 M/ ν2 ?p2 XbX MQ TQBMib BM +QKKQM B7 M/ QMHv B7  ν1 ({x}) ν2 (dx) = 0 , M/ i?2M Q7 +Qm`b2

URX9V

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ν2 ({x}) ν1 (dx) = 0X

h?Bb Bb i?2 +b2 7Q` BMbiM+2 B7 QM2 Q7 i?2K ?b  MQM@iQKB+ BMi2MbBiv K2bm`2X S`QQ7X "v *Q`QHH`v RXkXj   N1 ({x}) N2 (dx) = ν1 ({x}) ν2 (dx) . E E

E

()

  h?2`27Q`2- B7 E ν1 ({x}) ν2 (dx) = 0- i?2M E E N1 ({x}) N2 (dx) = 0 M/ BM T`iB+mH` E N1 ({x}) N2 (dx) = 0- P @XbX- r?B+? BKTHB2b i?i N1 M/ N2 ?p2 P @XbX  MQ +QKKQM TQBMibX HbQ- B7 N1 M/ N2 ?p2 XbX MQ +QKKQM TQBMibP ( E N1 ({x}) N2 (dx) = 0) = 1 M/ BM T`iB+mH` E E N1 ({x}) N2 (dx) = 0i?i Bb- #v *Q`QHH`v RXkXj- E ν1 ({x}) ν2 (dx) = 0X 

RXkX *JS"1GGǶa 6P_JlG L. JPJ1Lh J1al_1a

RR

JQK2Mi J2bm`2b _2+HH i?i  TQBMi T`Q+2bb N QM i?2 HX+X/X#X bT+2 E bm+? i?i E [N (C)] < ∞ 7Q` HH `2HiBp2Hv +QKT+i b2ib C ⊂ E Bb +HH2/  }`bi@Q`/2` TQBMi T`Q+2bb M/ i?i i?2 K2bm`2 ν QM (E, B(E)) /2}M2/ #v ν(C) = E [N (C)] Bb i?2M  HQ+HHv }MBi2 K2bm`2- i?2 BMi2MbBiv K2bm`2- HbQ +HH2/ i?2 }`bi KQK2Mi K2bm`2 Q7 N X h?2 TQBMi T`Q+2bb N Bb +HH2/  b2+QM/@Q`/2` TQBMi T`Q+2bb B7 7Q` HH `2HiBp2Hv +QKT+i b2ib C ⊂ E E N (C)2 < ∞ . URX8V AM T`iB+mH`-  b2+QM/@Q`/2` TQBMi T`Q+2bb Bb HbQ  }`bi@Q`/2` TQBMi T`Q+2bbX 6Q`  b2+QM/@Q`/2` TQBMi T`Q+2bb N - i?2 b2+QM/ KQK2Mi K2bm`2 M2 QM E × E Bb r2HH M/ mMB[m2Hv /2}M2/ #v i?2 7Q`KmH M2 (A × B) = E [N (A) N (B)]

(A, B ∈ B(E))

M/ i?Bb K2bm`2 Bb HQ+HHv }MBi2 U1t2`+Bb2 RXdXRRVX .2MQi2 #v N 2 = N × N i?2 TQBMi T`Q+2bb QM E × E r?Qb2 TQBMib `2 i?2 Q`/2`2/ TB`b (Xn , Xk ) Un, k ∈ NVX P#b2`pBM; i?i (N 2 )(A × B) = N (A)N (B)r2 b22 i?i M2 Bb i?2 BMi2MbBiv K2bm`2 Q7 N × N M/ i?2`27Q`2- #v *KT#2HHǶb i?2Q`2K- 7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb g : E × E → R     g(Xn , Xk ) = g(t, s) M2 (dt × ds) , E E

n∈N k∈N

E

M/ i?Bb 7Q`KmH Bb HbQ i`m2 7Q` Mv K2bm`#H2 7mM+iBQM g : E × E → C bm+? i?i   |g(t, s)| M2 (dt × ds) < ∞ . E

E

h?2 #Qp2 /2}MBiBQMb M/ `2bmHib 2ti2M/ Mim`HHv iQ ?B;?2` KQK2Mi K2@ bm`2bX G2i n ≥ 1 #2  TQbBiBp2 BMi2;2`X  bBKTH2 HQ+HHv }MBi2 N QM E bm+? i?i E [N (C)n ] < ∞ 7Q` HH `2HiBp2Hv +QKT+i b2ib C ∈ B(E) Bb +HH2/ M n@i? Q`/2` TQBMi T`Q+2bbX h?2 K2bm`2 Mn QM E n /2}M2/ #v Mn (A1 × · · · × An ) := E [N (A1 ) · · · N (An )] Bb i?2 n@i? Q`/2` KQK2Mi K2bm`2X h?Bb HQ+HHv }MBi2 K2bm`2 Bb i?2 BMi2MbBiv K2@ bm`2 Q7 i?2 TQBMi T`Q+2bb N (n) r?Qb2 TQBMib `2 i?2 Q`/2`2/ n@imTH2b (Xi1 , . . . , Xin ) Ui1 , . . . , in ∈ NVX h?2`27Q`2- #v *KT#2HHǶb i?2Q`2K- 7Q` HH MQM@M2;iBp2 K2bm`@ #H2 7mM+iBQMb g : E n → R       E ··· g(Xi1 , . . . , Xin ) = · · · g(x1 , . . . , xn ) Mn (dx1 × · · · × dxn ) . i1 ∈N

in ∈N

E

E

P7 +Qm`b2 M1 = ν- i?2 BMi2MbBiv K2bm`2X G2i N #2 M n@i? Q`/2` TQBMi T`Q+2bb QM EX *QMbB/2` MQr i?2 TQBMi T`Q+2bb N !(n) r?Qb2 TQBMib `2 i?2 Q`/2`2/ n@imTH2b (Xi1 , . . . , Xin ) Q7 /BbiBM+i TQBMibX h?2 HQ+HHv }MBi2 K2bm`2 Mn! QM E n /2}M2/ #v

Rk

*>Sh1_ RX :1L1_GAhA1a Mn! (A1 × · · · × An ) = E N !(n) (A1 × · · · × An ) ,

+HH2/ i?2 n@i? 7+iQ`BH KQK2Mi K2bm`2- Bb i?2 BMi2MbBiv K2bm`2 Q7 i?2 TQBMi T`Q+2bb N !(n) X h?2`27Q`2- #v *KT#2HHǶb i?2Q`2K- 7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb g : E n → R =     E g(Xi1 , . . . , Xin ) = · · · g(x1 , . . . , xn ) Mn! (dx1 × · · · × dxn ) , URXeV E

i1 ,...,in ∈N

E

r?2`2 i?2 bvK#QH = bB;MB}2b i?i i?2 bmK Bb `2bi`B+i2/ iQ i?2 Q`/2`2/ n@imTH2b i1 , . . . , in ∈ N Q7 TB`rBb2 /BbiBM+i BMi2;2`bX P7 +Qm`b2 M1! = M1 = ν Bb i?2 BMi2MbBiv K2bm`2 Q7 N X HbQ MQi2 i?i Mn! (An ) = E [N (A)(N (A) − 1) · · · (N (A) − n + 1)] , r?B+? 2tTHBMb i?2 TT2HHiBQM dz7+iQ`BHǴ K2bm`2X h?2Q`2K RXkX8 6Q`  b2+QM/ Q`/2` TQBMi T`Q+2bb N iQ #2 bBKTH2 Bi Bb M2+2bb`v M/ bm{+B2Mi i?i i?2 M2! @K2bm`2 Q7 i?2 /B;QMH Q7 E × E #2 MmHHX S`QQ7X h?Bb 7QHHQrb 7`QK i?2 2[mHBiv   1{x1 =x2 } M2! (dx1 × dx2 ) = E E

E



=  i1 ,i2 ∈N

 1{Xi1 =Xi2 } . 

1tKTH2 RXkXe, R@+HQb2 SB`b- iF2 RX  TB` (x, y) Q7 TQBMib Q7 Rd Bb +HH2/ R@+HQb2 B7 i?2 1m+HB/2M /BbiM+2 ||x − y|| Bb H2bb i?M Q` 2[mH iQ RX h?2 MmK#2` Q7 R@+HQb2 TB`b BM i?2 b[m`2 [0, 1] × [0, 1] Q7  bBKTH2 b2+QM/@Q`/2` TQBMi T`Q+2bb N QM R2 2[mHb 1  1[0,1]2 (Xn )1[0,1]2 (Xk )1{||Xn −Xk ||≤R} , 2 n k;k=n M/ i?2`27Q`2 Bib K2M 2[mHb- #v URXeV  1 1{||x−y||≤R} M2! (dx × dy) . 2 [0,1]2 [0,1]2

q2 MQr HBbi  72r `2HiBQMb i?i `2 2Bi?2` Q#pBQmb Q` 2bBHv /2`Bp2/X A7 A1 , . . . , An `2 KmimHHv /BbDQBMi Mn! (A1 × · · · × An ) = Mn (A1 × · · · × An ) . 6Q` n = 2-

M2! (A1 × A2 ) = M2 (A1 × A2 ) − ν(A1 ∩ A2 ) .

RXkX *JS"1GGǶa 6P_JlG L. JPJ1Lh J1al_1a

Rj

_2K`F RXkXd JQ`2 rBHH #2 bB/ QM i?2 b2+QM/ KQK2Mi K2bm`2b BM *?Ti2` N /2pQi2/ iQ i?2 +QKTmiiBQM Q7 TQr2` bT2+i`H K2bm`2b Q7 rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb2bX 1tKTH2 RXkX3, JQK2Mi K2bm`2b Q7 i?2 SQBbbQM T`Q+2bb "v i?2 BM/2@ T2M/2M+2 T`QT2`iv Q7 SQBbbQM T`Q+2bb2b- r2 ?p2 i?2 7QHHQrBM; `2bmHib +QM+2`MBM; Bib KQK2Mi K2bm`2b U1t2`+Bb2 RXdXRRVX M2 (A1 × A2 ) = ν(A1 )ν(A2 ) + ν(A1 ∩ A2 ) M/ M2! (A1 × A2 ) = ν(A1 )ν(A2 ) . A7 N Bb M ?TT QM Rm Q7 BMi2MbBiv λ- i?2 K2bm`2 Mn! /KBib  /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 n ;Bp2M #v ρ!n (x1 , . . . , xn ) = λn .

1tKTH2 RXkXN, R@+HQb2 SB`b- hF2 kX "v bT2+BHBxBM; i?2 `2bmHi Q7 1tKTH2 RXkXe- r2 Q#iBM i?i 7Q`  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R2 Q7 BMi2MbBiv λi?2 p2`;2 MmK#2` Q7 R@+HQb2 TB`b BM i?2 b[m`2 [0, 1] × [0, 1] Bb   1 2 λ 1{||x−y||≤R} dx dy . 2 [0,1]2 [0,1]2

h?2 *QBM+B/2M+2 6mM+iBQM A7 i?2 n@i? 7+iQ`BH K2bm`2 /KBib  /2MbBiv ρ!n (x1 , . . . , xn ) rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 n - i?Bb /2MbBiv Bb bQK2iBK2b +HH2/ i?2 Un@i?V +QBM+B/2M+2 7mM+@ iBQM BM pB2r Q7 i?2 7QHHQrBM; BM7Q`KH BMi2`T`2iiBQM, dzi?2 T`Q##BHBiv i?i i?2`2 Bb QM2 TQBMi BM 2+? QM2 Q7 i?2 KmimHHv /BbDQBMi BM}MBi2bBKH #Qt2b dx1 - Ę- dxn Bb T`QTQ`iBQMH iQ ρ!n (x1 , . . . , xn )ǴX h?2 BMi2`2bi Q7 /2}MBM;  TQBMi T`Q+2bb #v K2Mb Q7 Bib +QBM+B/2M+2 7mM+iBQM `2bB/2b BM Bib `2/#BHBiv BM i2`Kb Q7 ii`+iBQM Q` `2TmHbBQMX h?2`2 2tBbi H`;2 +Hbb2b Q7 TQBMi T`Q+2bb2b r?B+? `2 /2}M2/ pB i?2B` +Q@ BM+B/2M+2 7mM+iBQMbX h?2 KBM i?2Q`2iB+H T`Q#H2K bbQ+Bi2/ rBi? i?Bb ivT2 Q7 /2b+`BTiBQM Bb iQ T`Qp2 i?2 2tBbi2M+2 M/ mMB+Biv Q7  TQBMi T`Q+2bb /Bbi`B#miBQM rBi? i?2 ;Bp2M +M/B/i2 +QBM+B/2M+2 7mM+iBQMX h?Bb T`Q#H2K ?b #22M iQ  H`;2 2ti2Mi `2bQHp2/ 7Q` i?2 /2i2`KBMMiH M/ T2`KM2MiH TQBMi T`Q+2bb2bX 1tKTH2 RXkXRy, .2i2`KBMMiH M/ T2`KM2MiH TQBMi T`Q+2bb2bX G2i K : E × E → C #2  K2bm`#H2 7mM+iBQM M/ m #2  HQ+HHv }MBi2 K2bm`2 QM EX h?2 TQBMi T`Q+2bb N QM E Bb +HH2/  /2i2`KBMMiH TQBMi T`Q+2bb rBi? #+F;`QmM/ K2bm`2 m M/ F2`M2H K B7 7Q` HH k ≥ 1

R9

*>Sh1_ RX :1L1_GAhA1a M!k (dx1 · · · dxn ) = det{K(xi , xj )}1≤i,j≤k m(dx1 ) · · · m(dxk ) .

h?2 7mM+iBQM ρ!n (x1 , . . . , xk ) := det{K(xi , xj )}1≤i,j≤k Bb +HH2/ i?2 k@i? 7+iQ`BH KQK2Mi /2MbBiv rBi? `2bT2+i iQ m⊗k X AM T`iB+mH`- i?2 BMi2MbBiv K2bm`2 Q7 N Bb  ν(dx) = K(x, x) m(dx). E

h?2 /2i2`KBMMiH TQBMi T`Q+2bb2b `2 bBKTH2 TQBMi T`Q+2bb2b U1t2`+Bb2 RXdXNVX M BMi2`2biBM; bi#BHBiv T`QT2`iv Q7 i?2B` /Bbi`B#miBQM Bb i?i i?BMMBM;  /2i2`KBMMiH TQBMi T`Q+2bb `2bmHib BM  /2i2`KBMMiH TQBMi T`Q+2bb U1t2`+Bb2 RXdX3VX .2i2`KB@ MMiH TQBMi T`Q+2bb2b `2 `2TmHbBp2 #2+mb2 i?2 /2i2`KBMMi TT`Q+?2b x2`Q r?2M irQ `Qrb Q` +QHmKMb `2 +HQb2 iQ #2BM; HBM2`Hv BM/2T2M/2MiX AM i?2 2H2K2Mi`v T`iB+H2 +QMi2ti- i?Bb MiB@#mM+?BM; 2z2+i Bb +?`+i2`BbiB+ Q7 i?2 72`KBQMb U2H2+@ i`QMbVX h?2 /2}MBiBQM Q7 T2`KM2MiH TQBMi T`Q+2bb2b Bb 7Q`KHHv i?2 bK2 b i?i Q7 /2i2`KBMMiH TQBMi T`Q+2bb2b- 2t+2Ti i?i BMbi2/ Q7 i?2 /2i2`KBMMi QM2 mb2b i?2 T2`KM2Mi Q7 i?2 Ki`B+2b {K(xi , xj )}1≤i,j≤k X8 AM +QMi`bi rBi? i?2 /2i2`KBMMiH TQBMi T`Q+2bb2b- i?2 T2`KM2MiH TQBMi T`Q+2bb2b `2 ii`+iBp2- M/ i?Bb T`QT2`iv Bb +?`+i2`BbiB+ Q7 i?2 #QbQMb UT?QiQMbVX Hi?Qm;? i?2b2 T`Q+2bb2b `2 Q7 ;`2i BMi2`2bi BM bT2+B}+ `2b- BM+Hm/BM; S?vbB+b-e i?2B` MHvbBb Bb mbmHHv BM MHviB+H i2`Kbd M/ i?2`27Q`2 /Q2b MQi [mBi2 }i rBi? i?2 KQ`2 T`Q##BHBbiB+ TT`Q+? Q7 i?Bb #QQF M/ Bib HBKBi2/ ;QH +2Mi2`2/ `QmM/ biQ+?biB+ +H+mHmbX

RXj h?2 .Bbi`B#miBQM Q7  SQBMi S`Q+2bb .2}MBiBQM RXjXR G2i E #2 M HX+X/X#X bT+2 rBi? "Q`2H σ@}2H/ B(E) M/ H2i N #2  `M/QK K2bm`2 QM (E, B(E)) /2}M2/ QM i?2 T`Q##BHBiv bT+2 (Ω, F, P )X h?2 T`Q##BHBiv PN := P ◦ N −1 QM (M (E), M(E)) Bb +HH2/ i?2 /Bbi`B#miBQM Q7 N X 1tT2+iiBQM rBi? `2bT2+i iQ PN rBHH #2 /2MQi2/ #v EPN X  72r 2tKTH2b rBHH 7KBHB`Bx2 i?2 `2/2` rBi? i?2 MQiiBQMX 6Q` C ∈ B(E) M/ a ∈ RPN ({μ ; μ(C) ≥ a}) = P (N (C) ≥ a) . 6Q`  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM f : E → R     μ; f (x) μ(dx) ≤ b =P f (x) N (dx) ≤ b . PN E 8

E

_2+HH i?i i?2 T2`KM2Mi Q7  Ki`Bt Bb Q#iBM2/ 7`QK i?2 7Q`KmH ;BpBM; i?2 /2i2`KBMMi #v +?M;BM; i?2 − bB;Mb iQ + bB;MbX e (J++?B- RNdR- RNdR# M/ RNd8)- ("2M`/ M/ J++?B- RNdj)X d (a?B`B M/ hF?b?B- kyyj)X

RXjX h>1 .Aah_A"lhAPL P6  SPALh S_P*1aa

R8

6Q`  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM g : M (E) → R+ EPN [g] = E [g(N )] Q`- 2[mBpH2MiHv

 g(μ) PN (dμ) =

M (E)

g(N (ω)) P (dω) . Ω

h?2 M2ti /2}MBiBQM ;Bp2b i?2 T`2+Bb2 K2MBM; Q7 i?2 2tT`2bbBQM dzbKTHBM; Ui?2 /Bbi`B#miBQM Q7V  TQBMi T`Q+2bbǴX .2}MBiBQM RXjXk aKTHBM;  /Bbi`B#miBQM P QM (Mp (E), Mp (E)) Bb iQ +QMbi`m+i  dzbKTH2Ǵ N (ω) Q7 bQK2 TQBMi T`Q+2bb rBi? i?2 /Bbi`B#miBQM PN = PX JQ`2 ;2M2`HHv- bKTHBM;  T`Q##BHBiv /Bbi`B#miBQM Q QM  K2bm`#H2 bT+2 (U, U) +QMbBbib- #v /2}MBiBQM- BM ;2M2`iBM;  `M/QK 2H2K2Mi X(ω) rBi? pHm2b BM (U, U ) r?Qb2 /Bbi`B#miBQM Bb QX AM Q`/2` iQ ;2M2`i2 X Bi Bb bmTTQb2/ i?i QM2 ?b i /BbTQbBiBQM Mv MmK#2` Q7 +QTB2b Q7 dz2bBHv ;2M2`i2/Ǵ `M/QK 2H2K2Mib- bm+? b 7Q` BMbiM+2- BB/ `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM  mMBi BMi2`pH Q` SQBbbQM p`B#H2bX AKTHB+Bi BM i?2 /2}MBiBQM Q7 bKTHBM; Bb i?2 M2+2bbBiv i?i i?2 ;2M2`iBQM Q7 X b?QmH/ `2[mB`2 QMHv  }MBi2 UmbmHHv `M/QKV MmK#2` Q7 QT2`iBQMbX aKTHBM; Bb Q7i2M +HH2/ 2t+i bKTHBM; BM Q`/2` iQ BMbBbi QM i?2 /Bz2`2M+2 rBi? TT`QtBKi2 bKTHBM;- r?2`2 i?2 bKTH2 Q#iBM2/ ?b  /Bbi`B#miBQM dz+HQb2Ǵ iQ i?2 i`;2i /Bbi`B#miBQM- r?2`2 i?2 +HQb2M2bb Bb mbmHHv K2bm`2/ BM i2`Kb Q7 i?2 p`BiBQM /BbiM+2X h?2 bKTHBM; Bbbm2 rBHH #2 2tKBM2/ i p`BQmb TH+2b BM i?Bb #QQFX h?2 /Bbi`B#miBQM Q7  TQBMi T`Q+2bb Kv #2 +?`+i2`Bx2/ BM i H2bi i?`22 rvb, #v Bib }MBi2@/BK2MbBQMH /Bbi`B#miBQMb- #v Bib GTH+2 7mM+iBQMH M/ #v Bib pQB/@ M+2 T`Q##BHBiv 7mM+iBQMX "27Q`2 T`Q+22/BM; iQ i?2 +Q``2bTQM/BM; +?`+i2`BxiBQMb 72r T`2HBKBM`v `2bmHib `2 M22/2/X 6B`bi- r2 b?Qr i?i i?2 +QHH2+iBQM Q7 b2ib C ∈ B(E) 7Q` r?B+? r2 ?p2 iQ +?2+F i?i N (C) Bb  `M/QK p`B#H2 BM Q`/2` 7Q` i?2 KTTBM; N : Ω → M (E) iQ #2 K2bm`#H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(E) M/ M(E) +M #2 +QMbB/2`#Hv bKHH2` i?M B(E)X am+?  +QHH2+iBQM Bb /2b+`B#2/ BM i?2 7QHHQrBM; /2}MBiBQM, .2}MBiBQM RXjXj .2}M2 E0 iQ #2 Mv U}t2/V +QHH2+iBQM Q7 `2HiBp2Hv +QKT+i bm#@ b2ib Q7 E bm+? i?i UBV E0 Bb  π@bvbi2K Ui?i Bb, Bb bi#H2 #v }MBi2 BMi2`b2+iBQMbVUBBV σ(E0 ) = B(E)- M/ UBBBV i?2`2 2tBbib  b2[m2M+2 {En }n≥1 BM E0 bm+? i?i 2Bi?2` UV, 7Q`Kb  T`iBiBQM Q7 E- Q` U#V, BM+`2b2b iQ EX

Re

*>Sh1_ RX :1L1_GAhA1a

1tKTH2 RXjX9, h?2 `2+iM;H2bX G2i E := Rd X h?2 +Hbb E0 Q7 b2ib Q7 i?2 7Q`K d  I= (aj , bj ] ([aj , bj ] ⊆ R) j=1

biBb}2b i?2 `2[mB`2K2Mib Q7 .2}MBiBQM RXjXjX

h?2Q`2K RXjX8 X 6Q` i?2 KTTBM; N : Ω → M (E) U`2bTX Mp (E)V iQ #2 K2bm`#H2- Bi bm{+2b i?i N (C) : Ω → R+ #2  `M/QK p`B#H2 7Q` HH C ∈ E0 X "X

  M(E) = σ {μ ; μ(C) ∈ A} ; C ∈ E0 , A ∈ B(R+ ) .

()

AM T`iB+mH`- H2iiBM;   I := ∩kj=1 {μ ; μ(Cj ) ∈ Aj } ; Cj ∈ E0 , Aj ∈ B(R+ ) (1 ≤ j ≤ k) , k ∈ N+ , U π@bvbi2KV- r2 ?p2 i?i σ(I) = M(E)X S`QQ7X X h?2 T`QQ7 Bb ;Bp2M 7Q` i?2 +b2 BBBU#V Q7 .2}MBiBQM RXjXj- i?2 Qi?2` +b2 #2BM; bBKBH`X 6Bt nX h?2 +QHH2+iBQM G := {C ∈ B(E) ; ω → N (ω, C ∩ Kn ) Bb K2bm`#H2 7`QK (Ω, F) iQ (R+ , B(R+ ))} Bb  d@bvbi2KX Uh?2 7+i i?i  K2bm`2 μ ∈ M (E) Tmib }MBi2 Kbb QM  `2HiBp2Hv +QKT+i b2i- ?2`2 Kn - THvb  `QH2 BM i?2 T`QQ7 Q7 bi#BHBiv Q7 G mM/2` T`QT2` /Bz2`2M+2XV aBM+2 G ⊃ F0 - r2 ?p2 i?i G ⊃ σ(E0 ) = B(E)X h?2`27Q`2 N (C ∩ Kn ) Bb  `M/QK p`B#H2 7Q` HH C ∈ B(E) M/ HH n ≥ 1X 6BMHHv- 7Q` HH C ∈ B(E)N (C) Bb  `M/QK p`B#H2- #2BM; i?2 HBKBi b n ↑ ∞ Q7 i?2 `M/QK p`B#H2b N (C ∩ Kn )X "X AM - iF2 Ω = M (E) M/ H2i N #2 i?2 B/2MiBiv KTTBM; QM M (E)X "v /2}MBiBQM- M(E) = σ{{μ ; μ(C) ∈ A}, C ∈ B(E), A ∈ B(R+ )}X h?2 +QM+HmbBQM Q7 T`i  BKTHB2b i?i r2 +M `2TH+2 BM i?2 Hbi B/2MiBiv C ∈ B(E) #v C ∈ E0 X P#pBQmbHv- i?2 `B;?i@?M/ bB/2 Q7 UV Bb 2[mH iQ σ(I)X 

_2K`F RXjXe h?2 bK2 `2bmHi TTHB2b rBi? i?2 bK2 T`QQ7 r?2M `2TH+BM; M (E) #v Mp (E)X AM i?Bb +b2- bBM+2  TQBMi K2bm`2 iF2b Bib pHm2b BM N ∪ {∞}- H2iiBM; i?Bb iBK2   I := ∩kj=1 {μ ; μ(Cj ) = nj } ; Cj ∈ E0 , nj ∈ N+ , (1 ≤ j ≤ k) , k ∈ N+ U π@bvbi2KV- r2 ?p2 i?i σ(I) = Mp (E)X

RXjX h>1 .Aah_A"lhAPL P6  SPALh S_P*1aa

Rd

6BMBi2@/BK2MbBQMH .Bbi`B#miBQMb h?2Q`2K RXjXd G2i E0 #2 b BM .2}MBiBQM RXjXjX h?2 /Bbi`B#miBQM PN Q7  `M/QK K2bm`2 N QM (E, B(E)) Bb +QKTH2i2Hv +?`+i2`Bx2/ #v i?2 /Bbi`B#miBQMb Q7 i?2 p2+iQ`b (N (C1 ), . . . , N (Cm )) 7Q` HH BMi2;2`b m ≥ 1 M/ HH C1 , . . . , Cm ∈ E0 X S`QQ7X q2 ?p2 b22M BM h?2Q`2K RXjX8 i?i i?2 +Hbb I Q7 bm#b2ib Q7 M (E) Q7 i?2 7Q`K {μ; μ(C1 ) ∈ A1 , . . . , μ(Ck ) ∈ Ak } r?2`2 Cj ∈ E0 M/ Aj ∈ B(E) U1 ≤ j ≤ kV Bb bi#H2 #v }MBi2 BMi2`b2+iBQM M/ ;2M2`i2b M(E)X h?2`27Q`2- irQ T`Q##BHBiv K2bm`2b ;`22BM; QM I- ;`22 QM σ(I) Uh?2Q`2K XRXdVX M/- b r2 Dmbi `2+HH2/- σ(I) = M(E)X  .2}MBiBQM RXjX3 h?2 }/B U}MBi2@/BK2MbBQMHV /Bbi`B#miBQM Q7  TQBMi T`Q+2bb N Bb- #v /2}MBiBQM- i?2 +QHH2+iBQM Q7 i?2 T`Q##BHBiv /Bbi`B#miBQMb Q7 i?2 p2+iQ`b (N (C1 ), . . . , N (Cm )) (m ∈ N+ - C1 , . . . , Cm ∈ B(E))X h?2Q`2K RXjXd i2HHb mb BM T`iB+mH` i?i i?2 }/B /Bbi`B#miBQM `2bi`B+i2/ iQ i?2 +QHH2+iBQM Q7 b2ib I /2}M2/ BM h?2Q`2K RXjXd +?`+i2`Bx2b i?2 /Bbi`B#miBQM Q7  HQ+HHv }MBi2 TQBMi T`Q+2bbX h?2 GTH+2 6mM+iBQMH .2}MBiBQM RXjXN G2i N #2  `M/QK K2bm`2 QM i?2 HX+X#X/X bT+2 EX h?2 GTH+2 7mM+iBQMH Q7 N Bb i?2 KTTBM; LN bbQ+BiBM; rBi?  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM ϕ : E → R+ i?2 MQM@M2;iBp2 `2H MmK#2` LN (ϕ) := E e−N (ϕ) . AM i2`Kb Q7 i?2 /Bbi`B#miBQM PN Q7 N   LN (ϕ) = e− E ϕ(x) μ(dx) PN (dμ) . M (E)

1tKTH2 RXjXRy, GTH+2 6mM+iBQMH Q7 i?2 "BMQKBH SQBMi S`Q+2bbX h?Bb TQBMi T`Q+2bb QM Rm ?b  U}MBi2V MmK#2` T Q7 TQBMib- r?2`2 T Bb  #BMQKBH `M/QK p`B#H2 Q7 bBx2 n M/ T`K2i2` p ∈ (0, 1), P (T = k) =

  n k p (1 − p)n−k k

(0 ≤ k ≤ n) .

:Bp2M T = k- i?2 k TQBMib `2 HQ+i2/ BM/2T2M/2MiHv Q7 QM2 MQi?2` QM Rm +@ +Q`/BM; iQ i?2 bK2 T`Q##BHBiv /Bbi`B#miBQM QX Ai Bb bBKTH2 B7 M/ QMHv B7 Q Bb MQM@iQKB+X

R3

*>Sh1_ RX :1L1_GAhA1a  T  LN (ϕ) = E e− j=1 ϕ(Xj )  n   k   − j=1 ϕ(Xj ) =E 1{T =k} e k=1

=

n 

 k  P (T = k)E e− j=1 ϕ(Xj )

k=1

=

n 

k P (T = k)E e−ϕ(X1 )

k=1

=

n    n k=1

k pk (1 − p)n−k E e−ϕ(X1 )

k



k pE e−ϕ(X1 ) = (1 − p)n k 1 − p k=1  n   −ϕ(X1 ) n −ϕ(x) = 1+p e = 1−p e Q(dx) . n    n

E

1tKTH2 RXjXRR, GTH+2 6mM+iBQMH Q7  SQBbbQM S`Q+2bbX MiB+BTiBM;  `2bmHi Q7 *?Ti2` j Uh?2Q`2K jXkXk i?2`2BMV- i?2 GTH+2 7mM+iBQMH Q7  SQBbbQM T`Q+2bb QM E rBi? BMi2MbBiv K2bm`2 ν Bb 



LN (ϕ) = exp

e

−ϕ(x)



− 1 ν(dx)

 .

E

1tKTH2 RXjXRk, GTH+2 6mM+iBQMH Q7  *Qt S`Q+2bbX G2i N #2  *Qt T`Q+2bb QM E rBi? /B`2+iBM; K2bm`2 M Q7 GTH+2 i`Mb7Q`K LM X q2 +QKTmi2 Bib GTH+2 i`Mb7Q`K,    LN (ϕ) = E e−N (ϕ) = E e− E ϕ(x) N (dx)      = E E e− E ϕ(x) N (dx) | F M  

= E exp (e−ϕ(x) − 1) M (dx) . E

h?2`27Q`2 LN (ϕ) = LM (1 − e−ϕ ) .

RXjX h>1 .Aah_A"lhAPL P6  SPALh S_P*1aa

RN

1tKTH2 RXjXRj, GTH+2 6mM+iBQMH Q7  *QMi`+i2/ SQBMi S`Q+2bbX G2i N #2  bBKTH2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N M/ H2i α > 0X .2}M2 i?2 dz+QMi`+i2/Ǵ3 TQBMi T`Q+2bb Nc,α /2}M2/ #v Bib b2[m2M+2 Q7 TQBMib {αXn }n∈N X Aib GTH+2 7mM+iBQMH Bb     LNc,α (ϕ) = E exp − ϕ(αXn ) = LN (ϕ(α ·)) , n∈N

r?2`2 α · /2MQi2b i?2 7mM+iBQM x → αxX h?2 GTH+2 7mM+iBQMH THvb 7Q` TQBMi T`Q+2bb2b  `QH2 MHQ;Qmb iQ i?i Q7 i?2 mbmH GTH+2 i`Mb7Q`K 7Q` `M/QK p2+iQ`bX 6Q` BMbiM+2, h?2Q`2K RXjXR9 h?2 GTH+2 7mM+iBQMH Q7  HQ+HHv }MBi2 `M/QK K2bm`2 N QM i?2 HX+X#X/X bT+2 E +?`+i2`Bx2b Bib /Bbi`B#miBQMX S`QQ7X Ai bm{+2b iQ b?Qr i?i i?2 GTH+2 7mM+iBQMH Q7  TQBMi T`Q+2bb N +?`+@ i2`Bx2b Bib }MBi2@/BK2MbBQMH /Bbi`B#miBQMX 6Q` i?Bb- Dmbi Q#b2`p2 i?i 7Q` HH K ≥ 1 M/ HH /BbDQBMi K2bm`#H2 b2ib C1 - Ę- CK BM B(E)- i?2 GTH+2 i`Mb7Q`K Q7 i?2 p2+iQ` (N (C1 ), . . . , N (CK ))- i?i Bb- i?2 7mM+iBQM −t1 N (C1 )−···−t N (C ) K K (t1 , . . . , tK ) ∈ RK , + → E e −N (ϕ) Bb Q7 i?2 7Q`K E e r?2`2 ϕ = t1 1C1 + · · · + tK 1CK X  h?2 T`QQ7 Q7 i?2 M2ti `2bmHi Bb H27i 7Q` i?2 `2/2`X *Q`QHH`v RXjXR8  TQBMi T`Q+2bb N QM Rm Bb biiBQM`v B7 M/ QMHv B7 Bib GTH+2 i`Mb7Q`K LN Bb bm+? i?i LN (ϕ) = LN (Sa ϕ) 7Q` HH MQM@M2;iBp2 7mM+iBQMb ϕ 7`QK Rm iQ R M/ HH a ∈ Rm - r?2`2 Sa ϕ Bb i?2 7mM+iBQM /2}M2/ #v Sa ϕ : t → ϕ(t − a)X 1tKTH2 RXjXRe,  aiiBQM`v :`B/- hF2 kX AM Q`/2` iQ T`Qp2 i?2 bi@ iBQM`Biv Q7 i?2 b?B7i2/ ;`B/ Q7 1tKTH2 RXRXRj- Bi bm{+2b iQ b?Qr i?i 7Q` Mv MQM@M2;iBp2 7mM+iBQM ϕ 7`QK R2 iQ R- i?2 [mMiBiv  E e n,m∈Z ϕ(nT1 +V1 +α,nT2 +V2 +β) Bb BM/2T2M/2Mi Q7 α- β ∈ RX h?Bb [mMiBiv 2[mHb   T1  T2  e n,m∈Z ϕ(nT1 +v1 +α,nT2 +v2 +β) dv2 dv1 . 0

0

h?2 +QM+HmbBQM i?2M 7QHHQrb 7`QK i?2 b?B7i@BMp`BM+2 Q7 i?2 G2#2b;m2 K2bm`2X h?2 T`QQ7 Q7 i?2 M2ti `2bmHi Bb H27i b M 2t2`+Bb2X 3

P7 +Qm`b2- B7 α > 1- Bi Bb BM 7+i /BHi2/XXX

ky

*>Sh1_ RX :1L1_GAhA1a

h?2Q`2K RXjXRd G2i Ni (i ∈ I) #2  +QHH2+iBQM Q7 TQBMi T`Q+2bb2b QM i?2 HX+X/X#X bT+2 E- r?2`2 I Bb M `#Bi``v BM/2t b2iX A7 7Q` Mv }MBi2 bm#b2i J ⊆ I- Mv +QHH2+iBQM ϕi (i ∈ J) Q7 MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb 7`QK E iQ R   −Ni (ϕi ) E e− i∈J Ni (ϕi ) = , E e

URXdV

i∈J

i?2M Ni (i ∈ I) Bb M BM/2T2M/2Mi 7KBHv Q7 TQBMi T`Q+2bb2bX h?BMM2/ SQBMi S`Q+2bb2b G2i N #2  bBKTH2 TQBMi T`Q+2bb QM i?2 HX+X/X#X bT+2 E rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Zn }n∈N #2 M BB/ b2[m2M+2 Q7 BM/2T2M/2Mi K`Fb Q7 N - 2+? Zn iFBM; Bib pHm2b BM {0, 1}- rBi? i?2 T`Q##BHBiv p ∈ (0, 1) 7Q` i?2 pHm2 1X h?2 TQBMi T`Q+2bb Nthin,p /2}M2/ #v Nthin,p (C) :=



1C (Xn )Zn

n∈N

Bb +HH2/ i?2 p@i?BMMBM; Q7 N X 1+? TQBMi Q7 N Bb `2iBM2/ BM Nthin,p rBi? T`Q##BHBiv p- BM/2T2M/2MiHv Q7 2p2`vi?BM; 2Hb2X 1tKTH2 RXjXR3, GTH+2 6mM+iBQMH Q7 h?BMM2/ SQBMi S`Q+2bb2bX q2 +QKTmi2 i?2 GTH+2 7mM+iBQMH Q7 i?2 i?BMM2/ TQBMi T`Q+2bb,     LNthin,p (ϕ) = E exp − ϕ(Xn )Zn  =E

n∈N





exp (−ϕ(Xn )Zn )

n∈N

  =E E  =E

 exp (−ϕ(Xn )Zn ) |F

N

n∈N





n∈N



n∈N

=E





= E exp



E exp (−ϕ(Xn )Zn ) |F

N



 

{p exp(−ϕ(Xn )) + (1 − p)} 



 log (p exp(−ϕ(Xn )) + (1 − p))

n∈N

   = LN − log pe−ϕ(·) + 1 − p .

h?2 BMi2`K2/B`v `2bmHi Q#iBM2/ BM i?2 T2MmHiBKi2 HBM2 Q7 i?2 +H+mHiBQM Q7 i?2 T`2pBQmb 2tKTH2 rBHH #2 `2+Q`/2/ 7Q` 7mim`2 `272`2M+2 BM i?2 7Q`K,

RXjX h>1 .Aah_A"lhAPL P6  SPALh S_P*1aa  LNthin,p (ϕ) = E exp

Rm

  − log 1 − p(1 − e−ϕ(x) ) N (dx)

kR 

.

URX3V

h?2 p@i?BMMBM; QT2`iBQM Dp U0 ≤ p ≤ 1V +ib QM T`Q##BHBiv /Bbi`B#miBQMb QM Mp (E), Mp (E), B7 P Bb bm+?  TQBMi T`Q+2bb /Bbi`B#miBQM- Dp P Bb i?2 /Bbi`B#miBQM Q7 i?2 TQBMi T`Q+2bb Q#iBM2/ 7`QK  TQBMi T`Q+2bb rBi? /Bbi`B#miBQM P #v p@i?BMMBM;X h?2Q`2K RXjXRN h?2 QT2`iBQM Dp Bb QM2@iQ@QM2X S`QQ7X q2 ?p2 iQ b?Qr i?i i?2`2 +MMQi #2 irQ /Bbi`B#miBQMb P M/ P  bm+? i?i Dp P = Dp P  X AM/22/- B7 7Q` HH MQM@M2;iBp2 7mM+iBQMb ϕ      LN − log pe−ϕ(·) + 1 − p = LN  − log pe−ϕ(·) + 1 − p i?2M LN (f ) = LN  (f ) 7Q` HH MQM@M2;iBp2 7mM+iBQMb f X hQ b22 i?Bb- Bi bm{+2b iQ Q#b2`p2 i?i 7Q` Mv a ≥ 0 i?2`2 2tBbib QMHv QM2 b ≥ 0 bm+? i?i   e−a = 1 − p 1 − e−b () M/- +QMp2`b2Hv- 7Q` Mv b ≥ 0 i?2`2 2tBbib QMHv QM2 a ≥ 0 biBb7vBM; UVX



h?2 `2bmHi Q7 1tKTH2 RXjXR3 +M #2 bHB;?iHv 2ti2M/2/X .2}MBiBQM RXjXky G2i p : E → [0, 1] #2  K2bm`#H2 7mM+iBQMX G2i N #2  bBKTH2 TQBMi T`Q+2bb QM i?2 HX+X/X#X bT+2 E rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Un }n∈N #2 M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X h?2 TQBMi T`Q+2bb Nthin,p(·) /2}M2/ #v  Nthin,p(·) (C) := 1C (Xn )1{Un ≤p(Xn )} n∈N

Bb +HH2/ i?2 p(·)@i?BMMBM; Q7 N X h?2Q`2K RXjXkR h?2 GTH+2 7mM+iBQMH Q7 Nthin,p(·) Bb ;Bp2M #v i?2 7Q`KmH    LN (ϕ) = LN − log p(·)e−ϕ(·) + 1 − p(·) . h?2 T`QQ7 Bb bBKBH` iQ i?2 QM2 ;Bp2M BM 1tKTH2 RXjXR3X Ai Bb `2[mB`2/ BM 1t2`+Bb2 RXdXRjX 1tKTH2 RXjXkk, h?BMMBM; i?2 SQBbbQM S`Q+2bbX A7 N Bb  SQBbbQM T`Q+2bb rBi? i?2 HQ+HHv BMi2;`#H2 BMi2MbBiv K2bm`2 ν

−ψ(x)

−1)ν(dx) , LNthin,p(·) (ϕ) = LN (ψ) = e− E (e   −ϕ(x) −ψ(x) r?2`2 ψ(x) := − log p(x)e + 1 − p(x) X h?2`27Q`2 e = p(x)e−ϕ(x) + 1 − p(x) = p(e−ϕ(x) − 1) + 1 M/ }MHHv 

−ϕ(x) −1 p(x)ν(dx) ) LNthin,p(·) (ϕ) = e− E (e .

h?2`27Q`2- p(·)@i?BMMBM;  SQBbbQM T`Q+2bb Q7 BMi2MbBiv K2bm`2 ν(·) `2bmHib BM  SQBbbQM T`Q+2bb Q7 BMi2MbBiv K2bm`2 νp(·) (dx) := p(x)ν(dx)X

kk

*>Sh1_ RX :1L1_GAhA1a

h?2 S`Q##BHBiv :2M2`iBM; 6mM+iBQMH G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM Rm X G2i V (Rm ) #2 i?2 +QHH2+iBQM Q7 K2bm`#H2 7mM+iBQMb h : Rm → R pMBb?BM; QmibB/2  #QmM/2/ b2i M/ bm+? i?i 0 ≤ h(x) ≤ 1 7Q` HH x ∈ Rm X h?2 T`Q##BHBiv ;2M2`iBM; 7mM+iBQMH Q7 N Bb i?2 KTTBM; GN : V (Rm ) → R /2}M2/ #v    GN (h) := LN (log h) = E e Rm log h(x) N (dx) . 1[mBpH2MiHv-

 GN (h) = E



URXNV

 h(Xn ) ,

n∈N

r?2`2 Bi Bb mM/2`biQQ/ i?i i?2 bmK 2ti2M/b iQ HH TQBMib Xn BM i?2 bmTTQ`i Q7 h M/ i?i Bi Bb 2[mH iQ 1 B7 i?2`2 Bb MQ TQBMi BM i?2 bmTTQ`i  Q7 hX h?2 /2}MBiBQM Q7 GN (h) m 2ti2M/b iQ Mv 7mM+iBQM h : R → R bm+? i?i | log h(x)|ν(dx) < ∞- bBM+2 + Rm  i?2 BMi2;`H Rm log h(x) N (dx) BM URXNV Bb i?2M r2HH /2}M2/ M/ }MBi2 Uh?2Q`2K RXkXRVX h?2 pQB/M+2 S`Q##BHBiv 6mM+iBQM >2`2 Bb v2i MQi?2` +?`+i2`BxiBQM Q7 i?2 /Bbi`B#miBQM Q7  TQBMi T`Q+2bb i?i Bb bQK2iBK2b mb27mHX .2}MBiBQM RXjXkj G2i N #2  TQBMi T`Q+2bb QM i?2 K2bm`#H2 bT+2 (E, E)X h?2 7mM+iBQM v : E → [0, 1] /2}M2/ #v v(B) = P (N (B) = 0)

(B ∈ E)

Bb +HH2/ i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM Q7 N X v(B) Bb i?2`27Q`2 i?2 T`Q##BHBiv i?i N dzpQB/bǴ i?2 b2i BX 1tKTH2 RXjXk9, pQB/M+2 S`Q##BHBiv 6mM+iBQM Q7  SQBbbQM S`Q@ +2bbX 6Q`  SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2bm`2 ν- v(B) = exp(−λν(B))X

h?2Q`2K RXjXk8 G2i (E, d) #2  +QKTH2i2 b2T`#H2 K2i`B+ bT+2X G2i E #2 i?2 "Q`2H σ@}2H/ QM EX h?2 /Bbi`B#miBQM Q7  bBKTH2 TQBMi T`Q+2bb N QM (E, E) Bb +?`+i2`Bx2/ #v Bib pQB/M+2 T`Q##BHBiv 7mM+iBQMX

_2K`F RXjXke LQi2 i?i i?2 bbmKTiBQM Q7 bBKTHB+Biv Bb M2+2bb`v 7Q` i?2 #Qp2 `2bmHiX 6Q` BMbiM+2- /Qm#HBM; i?2 KmHiBTHB+Biv Q7 i?2 TQBMib Q7  ;Bp2M TQBMi T`Q+2bb /Q2b MQi Hi2` i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQMX

RXjX h>1 .Aah_A"lhAPL P6  SPALh S_P*1aa

kj

S`QQ7X h?Bb `2bmHiN bvb i?i B7 irQ bBKTH2 TQBMi T`Q+2bb2b ?p2 i?2 bK2 pQB/M+2 T`Q##BHBiv 7mM+iBQM- i?2v ?p2 i?2 bK2 /Bbi`B#miBQMX hQ b?Qr i?Bb- Bi bm{+2b iQ b?Qr i?i i?2 }MBi2@/BK2MbBQMH /Bbi`B#miBQMb Q7  bBKTH2 TQBMi T`Q+2bb N +M #2 Q#iBM2/ 7`QK Bib pQB/M+2 T`Q##BHBiv 7mM+iBQM HQM2X 6Q` i?Bb- Bi Bb 2MQm;? iQ T`Qp2 i?i 7Q` HH BMi2;2`b k ≥ 1- HH K2bm`#H2 b2ib A1 , . . . , Ak M/ HH BMi2;2`b n 1 , . . . , nk P (N (A1 ) ≤ n1 , . . . , N (Ak ) ≤ nk ) +M #2 2tT`2bb2/ BM i2`Kb Q7 v HQM2X h?Bb Bb /QM2 BM 7Qm` bi2TbX ai2T RX 6Q` Mv "Q`2H b2ib A1 , . . . , Ak , B- r2 ?p2 P (N (A1 ) > 0, . . . , N (Ak ) > 0, N (B) = 0) = P (N (A1 ) > 0, . . . , N (Ak−1 ) > 0, N (B) = 0) − P (N (A1 ) > 0, . . . , N (Ak−1 ) > 0, N (Ak ∪ B) = 0) , M/ 7Q` k = 1P (N (A1 ) > 0, N (B) = 0) = P (N (B) = 0) − P (N (B ∪ A1 ) = 0) = v(B) − v(B ∪ A1 ) . h?Bb b?Qrb i?i P (N (A1 ) > 0, . . . , N (Ak ) > 0, N (B) = 0) +M #2 `2+m`bBp2Hv +QKTmi2/ 7`QK i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM vX n Q7 M2bi2/ T`iBiBQMb ai2T kX amTTQb2 i?i r2 ?p2  b2[m2M+2 Kn = {Kn,i }ki=1 Q7 E bm+? i?i 7Q` Mv /BbiBM+i x- y ∈ E- i?2`2 2tBbib M n bm+? i?i x M/ y #2HQM; iQ irQ /BbiBM+i b2ib Q7 i?2 T`iBiBQM Kn UBM Qi?2` rQ`/b- i?2 b2[m2M+2 Q7 T`iBiBQMb {Kn }n≥0 2p2MimHHv b2T`i2b i?2 TQBMib Q7 EVX G2i 7Q` n ≥ 1 M/ A ∈ B(E)-

Hn (A) :=

kn 

H(A ∩ Kn,i ) ,

i=0 n X aBM+2 i?2 b2[m2M+2 r?2`2 H(C) := 1{N (C)>0} UC ∈ EVX G2i Kn ∩A := {A ∩ Kn,i }ki=0 Q7 T`iBiBQMb {Kn ∩A}n≥1 Q7 A 2p2MimHHv b2T`i2b i?2 TQBMib Q7 A- M/ bBM+2 Hn (A) +QmMib i?2 MmK#2` Q7 b2ib Q7 Kn ∩ A i?i +QMiBM i H2bi QM2 TQBMi Q7 i?2 TQBMi T`Q+2bb- r2 ?p2 BM pB2r Q7 i?2 bbmK2/ bBKTHB+Biv Q7 i?2 TQBMi T`Q+2bb

lim Hn (A) = N (A) ,

n↑∞

XbX

ai2T jX h?2 T`Q##BHBiv P (Hn (A) = l) +M #2 2tT`2bb2/ BM i2`Kb Q7 i?2 pQB/@ M+2 T`Q##BHBiv 7mM+iBQM v HQM2 bBM+2 UrBi? An,i = A ∩ Kn,i V  P (H(An,0 ) = i0 , . . . , H(An,kn ) = ikn ) P (Hn (A) = l) = i0 ,...,ik ∈{0,1} n k Σ n ij =l j=1

M/ 7Q` i0 , . . . , ikn ∈ {0, 1} N LK2/ 7i2` _ûMvB- r?Q BMi`Q/m+2/ i?2 MQiBQM M/ TTHB2/ Bi iQ SQBbbQM T`Q+2bb2b (_ûMvBRNed)X 6Q` i?2 ;2M2`H +b2, (JƺM+?- RNdR)X

k9

*>Sh1_ RX :1L1_GAhA1a P (H(An,0 ) = i0 , . . . , H(An,kn ) = ikn ) = P (∩l;il =1 {N (An,l ) > 0} ∩ {N (∪m;im =0 An,m ) = 0}) ,

 [mMiBiv r?B+? +M #2 2tT`2bb2/ BM i2`Kb Q7 i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM v HQM2- b r2 br BM ai2T RX JQ`2 ;2M2`HHv- 7Q` HH l1 , . . . , lk ∈ NP (Hn (A1 ) = l1 , . . . , Hn (Ak ) = lk ) Bb 2tT`2bbB#H2 BM i2`Kb Q7 i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM- M/ i?2 bK2 Bb i`m2 Q7 P (Hn (A1 ) ≤ n1 , . . . , Hn (Ak ) ≤ nk ) (n1 , . . . , nk ∈ N) . ai2T 9X 6BMHHv- Q#b2`p2 i?i {Hn (A1 ) ≤ n1 , . . . , Hn (Ak ) ≤ nk } ↓ {N (A1 ) ≤ n1 , . . . , N (Ak ) ≤ nk } M/ i?2`27Q`2 lim P (Hn (A1 ) ≤ n1 , . . . , Hn (Ak ) ≤ nk ) = P (N (A1 ) ≤ n1 , . . . , N (Ak ) ≤ nk ) .

n↑∞

h?2 T`QQ7 Bb MQr HKQbi /QM2X Ai Dmbi `2KBMb iQ +QMbi`m+i i?2 b2[m2M+2 Q7 T`iB@ iBQMb {Kn }n≥1 X .2MQi2 #v B(a, r) i?2 +HQb2/ #HH Q7 +2Mi2` a M/ `/Bmb rX aBM+2 (E, d) Bb b2T`#H2- i?2`2 2tBbib  +QmMi#H2 b2i {a1 , a2 , . . . } i?i Bb /2Mb2 BM EX h?2 }`bi T`iBiBQM K1 +QMbBbib Q7 irQ b2ib K11 := B(a1 , 1) ,

K10 := E\K11 .

amTTQb2 i?i r2 ?p2 +QMbi`m+i2/ Kn−1 X h?2 M2ti T`iBiBQM Kn Bb +QMbi`m+i2/ b 7QHHQrbX G2iiBM;   Bn,i := B ai , 2−(n−i)

(i = 1, . . . , n) M/ Bn,0 := E\

n 

Bn,i ,

i=1

/2}M2  T`iBiBQM Cn = {Cn,i }ni=0 #v Cn,0 := Bn,0 ;

Cn,1 := Bn,1 ;

Cn,i := Bn,i \

i−1 

Bn,j

(i = 2, . . . , n) .

j=1

AM Q`/2` iQ Q#iBM i?2 T`iBiBQM Kn M2bi2/ BM Kn−1 - r2 BMi2`b2+i Cn M/ Kn−1 - i?i Bb Kn := {Cn,i ∩ Kn−1,j (j = 0, . . . , kn−1 , i = 0, . . . , n)} .  _2K`F RXjXkd AM i?2 +b2 E = Rm -  bBKTH2 b2[m2M+2 Q7 M2bi2/ T`iBiBQMb +QmH/ #2 i?2 7QHHQrBM;- bv rBi? m = 1 7Q` MQiiBQMH +QMp2MB2M+2,   Kn = (i2−n , (i + 1)2−n ] ; i ∈ Z .

RXjX h>1 .Aah_A"lhAPL P6  SPALh S_P*1aa

k8

.2}MBiBQM RXjXk3 h?2 +T+Biv 7mM+iBQMH Q7  TQBMi T`Q+2bb N bbQ+Bi2b rBi?  +QKT+i b2i C i?2 pHm2 T (C) := P (N (C) > 0) . P7 +Qm`b2- i?Bb 7mM+iBQMH Bb BMiBKi2Hv `2Hi2/ iQ i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM- bBM+2 T (C) = 1 − v(C)X AM T`iB+mH`- i?2 +T+Biv 7mM+iBQMH Q7  bBKTH2 TQBMi T`Q+2bb +?`+i2`Bx2b Bib /Bbi`B#miBQMX MQi?2` +HQb2Hv `2Hi2/ MQiBQM- i?Bb iBK2 7Q` biiBQM`v TQBMi T`Q+2bb2b- Bb i?2 7QHHQrBM;, .2}MBiBQM RXjXkN h?2 +QMi+i /Bbi`B#miBQM 7mM+iBQM Q7  biiBQM`v TQBMi T`Q+2bb N ≡ {Xn }n≥1 QM Rm Bb i?2 7mM+iBQM F : R+ → [0, 1] /2}M2/ #v F (r) := P (d(u, N ) ≤ r) ,

URXRyV

r?2`2 7Q` u ∈ E- d(u, N ) := inf n≥1 d(u, Xn )X h?2 biiBQM`Biv Q7 N ;m`Mi22b i?i i?2 `B;?i@?M/ bB/2 Q7 URXRyV Bb BM/2T2M/2Mi Q7 uX q2 ?p2 i?i F (r) = P (N (B(u, r) > 0) = T (B(u, r)) = T (B(0, r)) . h?2`27Q`2- BM i?2 biiBQM`v +b2- i?2 +QMi+i /Bbi`B#miBQM 7mM+iBQM /2i2`KBM2b i?2 +T+Biv 7mM+iBQMH 7Q` +QKT+i b2ib i?i `2 +HQb2/ /BbFbX >Qr2p2` Bi Bb MQi bm{+B2Mi iQ /2i2`KBM2 i?2 +T+Biv 7mM+iBQMH 7Q` HH +QKT+i b2ibX 1tKTH2 RXjXjy, *QMi+i .Bbi`B#miBQM 6mM+iBQM Q7  >QKQ;2M2Qmb SQBbbQM S`Q+2bbX A7 N Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv λF (r) = P (N (B(u, r)) > 0) = 1 − P (N (B(u, r)) = 0) = 1 − e−λ

m (B(u,r))

= 1 − e−λκd r , m

r?2`2 κm Bb i?2 m@pQHmK2 Q7 i?2 mMBi #HH B(0, 1) ⊂ Rm X G2i W #2  +QKT+i bm#b2i Q7 Rm Q7 }MBi2 G2#2b;m2 K2bm`2X h?2 7QHHQrBM; 2biBKiQ` Q7 F (r) 1 1{d(u,N )≤r} du , () F (r) := d

(W ) W Bb mM#Bb2/ b i?2 7QHHQrBM; 2H2K2Mi`v +QKTmiiBQM b?Qrb, 

   1 1 E E F (r) = d 1{d(u,N )≤r} du = d E 1{d(u,N )≤r} du

(W )

(W ) W  W  1 1 = d P ({d(u, N ) ≤ r) du = d F (r) du = F (r) .

(W ) W

(W ) W

ke

*>Sh1_ RX :1L1_GAhA1a

_2K`F RXjXjR AM TTHB+iBQMb- W Bb i?2 Ubv- +QKT+iV Q#b2`piBQM rBM/Qri?i Bb- QMHv TQBMib Q7 N HvBM; BM W `2 Q#b2`p2/X AM T`iB+mH`- F (r) +MMQi #2 +QKTmi2/ r?2M i?2`2 `2 TQBMib Q7 N QmibB/2 W i?i `2 i  /BbiM+2 7`QK W H2bb i?M rX M Q#pBQmb +Q``2+iBQM Q7 i?Bb 2/;2 2z2+i Bb iQ `2TH+2 BM i?2 `B;?i@?M/ bB/2 Q7 UV W #v Wr := {x ∈ W ; d(x, W ) ≤ r} . h?2 +Q``2bTQM/BM; 2biBKi2 F(r) :=

1 m

(Wr )

 1{d(u,N )≤r} du

()

Wr

rBHH i?2M #2 mM#Bb2/X >Qr2p2`- BMimBiBp2Hv- i?2 p`BM+2 Q7 i?Bb 2biBKiQ` rBHH #2 H`;2 7Q` H`;2 pHm2b Q7 r BM +QKT`BbQM rBi? i?2 bBx2 Q7 W UHQQF i i?2 2ti`2K2 +b2 Q7 pHm2b Q7 r H`;2` i?M i?2 `/Bmb Q7  ;Bp2M #HH +QMiBMBM; i?2 +QKT+i rBM/Qr W VX

RX9 *QMp2`;2M+2 BM .Bbi`B#miBQM M/ o`BiBQM h?Bb b2+iBQM ;Bp2b  b?Q`i BMi`Q/m+iBQM iQ i?2 i?2Q`B2b Q7 r2F +QMp2`;2M+2Ry M/ Q7 +QMp2`;2M+2 BM p`BiBQM Q7 TQBMi T`Q+2bb2bX *QMp2`;2M+2 BM .Bbi`B#miBQM q2F +QMp2`;2M+2 Q7 TQBMi T`Q+2bb2b Bb BM i2`Kb Q7 #QmM/2/ M/ +QMiBMmQmb 7mM+@ iBQMb f : Mp (E) → R+ X aBM+2 +QMiBMmBiv Bb /2}M2/ rBi? `2bT2+i iQ  iQTQHQ;v- Qm` }`bi ibF Bb i?2`27Q`2 iQ /2}M2 bm+?  iQTQHQ;v QM Mp (E)- M/ KQ`2 ;2M2`HHv QM M (E)X h?Bb iQTQHQ;v Bb i?2 p;m2 iQTQHQ;vX  b2[m2M+2 {μn }n≥1 Q7 K2bm`2b Q7 M (E) Bb bB/ iQ +QMp2`;2 p;m2Hv iQ μ ∈ M (E) B7 7Q` HH MQM@M2;iBp2  +QMiBMmQmb 7mM+iBQMb f : E → R rBi? +QK@ T+i bmTTQ`i- limn↑∞ E f dμn = E f dμX h?Bb Bb /2MQi2/ #v v

μn → μ . h?2Q`2K RX9XR h?2 7QHHQrBM; `2 2[mBpH2Mi, v

UBV μn → μX UBBV lim supn μn (K) ≤ μ(K) 7Q` HH `2HiBp2Hv +QKT+i b2ib K ∈ B(E)X UBBBV lim inf n μn (G) ≥ μ(G) 7Q` HH QT2M b2ib G ∈ B(E)X o;m2 +QMp2`;2M+2 Bb BM 7+i +QMp2`;2M+2 rBi? `2bT2+i iQ  iQTQHQ;v U+HH2/ i?2 p;m2 iQTQHQ;vV QM M (E) i?i +M #2 K2i`Bx2/ b  b2T`#H2 K2i`B+ bT+2X h?2 σ@H;2#` M(E) +M #2 b?QrM iQ #2 i?2 "Q`2H }2H/ QM M (E) 7Q` i?Bb iQTQHQ;v, M(E) ≡ B(M (E))X h?2 bK2 /2}MBiBQMb TTHv M/ i?2 bK2 `2bmHib ?QH/ rBi? Ry JQbiHv rBi?Qmi T`QQ7bc b22 (L2p2m- RNdd) UBM 6`2M+?V- (.H2v M/ o2`2@CQM2b- kyyj- kyy3) M/ (EHH2M#2`;- kyRd)X (_2bMB+F- RN3d) +QMiBMb  bm++BM+i ++QmMi Q7 i?2 i?2Q`vX

RX9X *PLo1_:1L*1 AL .Aah_A"lhAPL L. o_AhAPL

kd

M (E) `2TH+2/ #v Mp (E)X AM T`iB+mH` Mp (E) ≡ B(Mp (E))X Ai im`Mb Qmi i?i Mp (E) Bb  p;m2Hv +HQb2/ M/ i?2`27Q`2 K2bm`#H2 bm#b2i Q7 M (E) M/ i?i Mp (E) Bb i?2 i`+2 Q7 M(E) QM Mp (E)X o;m2 +QMp2`;2M+2 BM Mp (E) ?b  `i?2` bBKTH2 M/ Mim`H BMi2`T`2iiBQM BM i2`Kb Q7 i?2 TQBMib Q7 i?2 TQBMi K2bm`2b BMpQHp2/X h?2Q`2K RX9Xk h?2 b2[m2M+2 {μn }n≥1 Q7 K2bm`2b Q7 Mp (E) +QMp2`;2b p;m2Hv iQ μ ∈ Mp (E) Bz 7Q` Mv +QKT+i bm#b2i K Q7 E bm+? i?i μ(∂K) = 0 U∂K /2MQiBM; i?2 #QmM/`v Q7 KV- i?2`2 2tBbib M n(K) ∈ N bm+? i?i 7Q` n ≥ n(K)i?2`2 2tBbib  H#2HHBM; {xi (n); 1 ≤ i ≤ q} Q7 i?2 TQBMib Q7 μn BM K M/  H#2HHBM; {xi ; 1 ≤ i ≤ q} Q7 i?2 TQBMib Q7 μ BM K bm+? i?i μn (· ∩ K) =

q 

εxi (n) (·) M/ μ(· ∩ K) =

i=1

q 

εxi (·)

()

i=1

M/ (1 ≤ i ≤ q) .

lim xi (n) = xi

n↑∞

()

h?2 #Qp2 +?`+i2`BxiBQM Q7 p;m2 +QMp2`;2M+2 Bb mb27mH 7Q` T`QpBM; i?2 +QM@ iBMmBiv Q7  7mM+iBQM f : Mp (Rm ) → RX 1tKTH2 RX9Xj, hrQ 2tKTH2b Q7 +QMiBMmQmb KTTBM;bX 6Q` Mv #QmM/2/ K2bm`#H2 b2i C ∈ Rm - i?2 KTTBM; μ → μ(C) 7`QK Mp (E) iQ R Bb +QMiBMmQmbX v hQ T`Qp2 i?Bb- r2 ?p2 iQ b?Qr i?i B7 μn → μ- i?2M μn (C) → μ(C)X b C Bb #QmM/2/ M/ μ Bb  σ@}MBi2 TQBMi K2bm`2- QM2 +M Hrvb }M/  +QKT+i b2i K ⊃ C bm+? i?i μ(∂K) = 0X lbBM; QMHv UV Q7 h?2Q`2K RX9Xk- r2 b22 i?i 7Q` HH n ≥ n(K) Ub /2}M2/ BM i?2 bii2K2Mi Q7 i?2 i?2Q`2KV- μn (C) = μ(C)X  bBKBH` `;mK2Mi  Ui?Bb iBK2 mbBM; #Qi? UV M/ UVV rBHH b?Qr i?i i?2 KTTBM; μ → μ(g) = Rm g(x) μ(dx) Bb +QMiBMmQmb r?2M2p2` g : Rm → R Bb  +QMiBMmQmb 7mM+iBQM rBi? +QKT+i bmTTQ`iX .2}MBiBQM RX9X9 G2i {Nn }n≥1 #2  b2[m2M+2 Q7 `M/QK K2bm`2b UTQbbB#Hv /2}M2/ QM /Bz2`2Mi T`Q##BHBiv bT+2b {(Ωn , Fn , Pn )}n≥1 V M/ H2i {Pn }n≥1 #2 i?2B` `2bT2+iBp2 /Bbi`B#miBQMbX h?2 b2[m2M+2 {Pn }n≥1 Bb bB/ iQ +QMp2`;2 r2FHv iQ i?2 T`Q##BHBiv P QM (M (E), M(E)) B7 7Q` HH #QmM/2/ M/ +QMiBMmQmb 7mM+iBQMb f : M (E) → R  lim f (μ) Pn (dμ) = f (μ) P(dμ) , n↑∞

M (E)

M (E)

i?i Bb- 2[mBpH2MiHv f (μ) P(dμ) .

lim En [f (Nn )] =

n↑∞ w

M (E) w

h?Bb Bb /2MQi2/ #v Nn → P Q` Pn → PX

k3

*>Sh1_ RX :1L1_GAhA1a w

h?2Q`2K RX9X8 Nn → P B7 M/ QMHv B7 7Q` HH MQM@M2;iBp2 +QMiBMmQmb 7mM+iBQMb ϕ : E → R rBi? +QKT+i bmTTQ`i   LNn (ϕ) → LP (ϕ) := e− E ϕ(x) μ(dx) P (dμ) . M (E)

M TT`2MiHv r2F2` +QM+2Ti Q7 +QMp2`;2M+2 Bb r2F +QMp2`;2M+2 BM i?2 b2Mb2 Q7 }/B U}MBi2@/BK2MbBQMHV /Bbi`B#miBQMbX  +QMiBMmBiv b2i Q7  TQBMi T`Q+2bb N U`2bTX- Q7  T`Q##BHBiv P QM (Mp (E), Mp (E))V Bb  b2i A ∈ B(Rm ) bm+? i?i P (N (∂A) > 0) = 0 U`2bTX P ({μ ; μ(∂A) > 0}) = 0VX .2}MBiBQM RX9Xe h?2 b2[m2M+2 Q7 HQ+HHv }MBi2 TQBMi T`Q+2bb2b {Nn }n≥1 Bb bB/ iQ +QMp2`;2 BM i?2 b2Mb2 Q7 }/B /Bbi`B#miBQMb B7 7Q` HH A1 - A2 - Ę- Ak i?i `2 +QMiBMmBiv b2ib Q7 N - i?2 DQBMi /Bbi`B#miBQM Q7 (Nn (A1 ), . . . , Nn (Ak )) +QMp2`;2b r2FHv b n ↑ ∞ iQ i?2 /Bbi`B#miBQM Q7 (N (A1 ), . . . , N (Ak ))X 6Q` i?Bb Bi bm{+2b i?i limn↑∞ LNn (ϕ) = LN (ϕ) 7Q` HH ϕ Q7 i?2 7Q`K ϕ(x) = i=1 ti 1Ai (x) r?2`2 t1 , . . . , tk ∈ R+ X h?2`27Q`2,

k

h?2Q`2K RX9Xd PM Mp (E)- r2F +QMp2`;2M+2 Bb 2[mBpH2Mi iQ r2F +QMp2`;2M+2 BM i?2 b2Mb2 Q7 i?2 }/B /Bbi`B#miBQMbX 1tKTH2 RX9X3, SQBbbQM S`Q+2bb b GBKBi Q7 "BMQKBH SQBMi S`Q+2bb2bX AM i?2 1tKTH2 RXjXRy- H2i p = pn M/ +HH Nn i?2 +Q``2bTQM/BM; #BMQKBH TQBMi T`Q+2bb- Q7 GTH+2 i`Mb7Q`K  n  LNn (ϕ) = 1 − pn e−ϕ(x) Q(dx) . E

amTTQb2 MQr i?i npn → λ b n ↑ ∞X hFBM; HQ;`Bi?Kb    log LNn (ϕ) = n log 1 + pn e−ϕ(x) Q(dx) → (e−ϕ(x) − 1) λQ(dx) . E

E

h?2`27Q`2 i?2 HBKBi BM /Bbi`B#miBQM Q7 i?2 b2[m2M+2 {Nn }n≥1 Bb  SQBbbQM T`Q+2bb Q7 K2M K2bm`2 ν(·) = λQ(·)X

1tKTH2 RX9XN, h?BMMBM; M/ *QMi`+iBQMX G2i N #2  TQBMi T`Q+2bb QM Rm M/ H2i p ∈ (0, 1)X *QMbi`m+i  TQBMi T`Q+2bb Np #v i?BMMBM; rBi? T`Q##BHBiv p i?2 T`Q+2bb N +QMi`+i2/ #v p U7Q` r?B+? i?2 MmK#2` Q7 TQBMib BM C Bb N (p−1 C)VX lM/2` i?2 +QM/BiBQM i?i 7Q` Mv #QmM/2/ C ∈ B(Rm ) lim pN (p−1 C) = λ m (C) , p↓0

i?2 TQBMi T`Q+2bb Np +QMp2`;2b r2FHv iQ M ?TT Q7 BMi2MbBiv λ b p → 0X

RX9X *PLo1_:1L*1 AL .Aah_A"lhAPL L. o_AhAPL

kN

S`QQ7X "v URX3V  LNp (ϕ) = E exp

Rm

  log 1 − p(1 − e−ϕ(x) ) N (p−1 dx)



.

LQr rBi? ϕ(x) = ki=1 ti 1Ai (x)- r?2`2 i?2 Ai Ƕb `2 #QmM/2/ b2ib M/ i?2 ti Ƕb `2 MQM@M2;iBp2 MmK#2`b- r2 ?p2 i?i k    log 1 − p(1 − e−ϕ(x) ) = p(e−ϕ(x) − 1) + o(p) = p (e−ti − 1)1Ai (x) + o(p) i=1

M/ i?2`27Q`2- b p ↓ 0 Rm

k    log 1 − p(1 − e−ϕ(x) N (p−1 dx) = (e−ti − 1)pN (p−1 Ai )



i=1 k 

(e−ti − 1)λ m (Ai )

i=1

M/





E exp −

k 

 Np (Ai )

i=1

→ exp

 k 

 (e

−ti

− 1)λ (Ai ) m

.

i=1

h?2`27Q`2- Np +QMp2`;2b r2FHv iQ  SQBbbQM T`Q+2bb Q7 BMi2MbBiv λX



h?2 7QHHQrBM; `2bmHi Bb  +?`+i2`BxiBQM Q7 i?2 *Qt T`Q+2bb2b BM i2`Kb Q7 i?BMMBM;X h?2Q`2K RX9XRy URR V 6Q`  TQBMi T`Q+2bb N QM E- i?2 7QHHQrBM; irQ bii2K2Mib `2 2[mBpH2Mi, UBV N Bb  *Qt T`Q+2bbX  X UBBV 6Q` 2p2`v p ∈ (0, 1]- i?2`2 Bb  TQBMi T`Q+2bb N  bm+? i?i N = Nthin,p

h?2 T`QQ7 rBHH #2 #b2/ QM i?2 7QHHQrBM; H2KK, G2KK RX9XRR URk V G2i {Nn }n≥1 #2  b2[m2M+2 Q7 TQBMi T`Q+2bb2b QM E M/ H2i {pn }n≥1 #2  b2[m2M+2 Q7 `2H MmK#2`b BM (0, 1] bm+? i?i limn↑+∞ pn = 0 mMB7Q`KHvX G2i Nn #2 7Q` 2+? n ≥ 1  pn @i?BMMBM; Q7 Nn X h?2 irQ 7QHHQrBM; +QM/BiBQMb `2 2[mBpH2Mi, UBV h?2 b2[m2M+2 {Nn }n≥1 +QMp2`;2b BM /Bbi`B#miBQM iQ bQK2 TQBMi T`Q+2bb N  X UBBV h?2`2 2tBbib  `M/QK K2bm`2 ν QM E bm+? i?i i?2 b2[m2M+2 Q7 `M/QK K2bm`2b {pn Nn }n≥1 +QMp2`;2b BM /Bbi`B#miBQM iQ  `M/QK K2bm`2 M X A7 i?2b2 +QM/BiBQMb `2 biBb}2/- N  Bb  *Qt T`Q+2bb /B`2+i2/ #v M X RR Rk

(J2+F2- RNe3)X (EHH2M#2`;- RNd8)X

jy

*>Sh1_ RX :1L1_GAhA1a

S`QQ7X 6Q` i?2 2[mBpH2M+2- Bi bm{+2b iQ Q#b2`p2 i?i 7Q` 2+? MQM@M2;iBp2 7mM+@ iBQM ϕ : E → R   LNn (ϕ) = LNn − log 1 − pn − pn e−ϕ    LNn pn (1 − e−ϕ ) = Lpn Nn (1 − e−ϕ ) . SbbBM; iQ i?2 HBKBi ;Bp2b LN  (ϕ) = LM (1 − e−ϕ ) r?B+? Bb BM/22/ i?2 GTH+2 i`Mb7Q`K Q7  *Qt T`Q+2bb /B`2+i2/ #v M X



q2 +M MQr `2bmK2 i?2 T`QQ7 Q7 h?2Q`2K RX9XRy, S`QQ7X hQ T`Qp2 i?i UBV BKTHB2b UBBV iF2 7Q` N  i?2 *Qt T`Q+2bb rBi? /B`2+iBM; K2bm`2 p−1 M - r?2`2 M Bb i?2 /B`2+iBM; K2bm`2 Q7 N X 6Q` i?2 T`QQ7 i?i UBBV BKTHB2b UBV- Q#b2`p2 i?i 7Q` 2+? n ≥ 1- N ?b i?2 bK2 /Bbi`B#miBQM b i?2 1/n@ i?BMMBM; Q7 bQK2 TQBMi T`Q+2bb Nn M/ TTHv G2KK RX9XRRX  *QMp2`;2M+2 BM o`BiBQM a2+iBQM XR Q7 i?2 TT2M/Bt `2pB2rb i?2 #bB+ i?2Q`v Q7 +QMp2`;2M+2 BM p`BiBQM M22/2/ BM i?Bb b2+iBQMX .2}MBiBQM RX9XRk G2i P1 M/ P2 #2 irQ T`Q##BHBiv /Bbi`B#miBQMb (M (E), M(E))X h?2 /BbiM+2 BM p`BiBQM #2ir22M P1 M/ P2 Bb i?2 [mMiBiv

QM

dV (P1 , P2 ) := sup (P1 (Γ) − P2 (Γ)) . Γ∈M(E)

1[mBpH2MiHv dV (P1 , P2 ) = sup (P r(N1 ∈ Γ) − P r(N2 ∈ Γ)) Γ∈M(E)

7Q` Mv TB` (N1 , N2 ) rBi? ;Bp2M K`;BMH /Bbi`B#miBQMb P1 M/ P2 X "v h?2Q`2K XRXkydV (P1 , P2 ) ≤ P (N1 = N2 ) URXRRV M/- KQM; HH TB`b (N1 , N2 ) rBi? ;Bp2M K`;BMH /Bbi`B#miBQMb P1 M/ P2 - i?2`2 2tBbib i H2bi QM2 iiBMBM; 2[mHBiv BM URXRRVX AM Q`/2` iQ +QKTmi2 i?2 `B;?i@?M/ bB/2 Q7 URXRRV M/ i?2`27Q`2 Q#iBM  #QmM/ 7Q` i?2 /BbiM+2 BM p`BiBQM #2ir22M P1 M/ P2 -  +H2p2` +?QB+2 Q7 N1 M/ N2 Bb M22/2/- b i?2 7QHHQrBM; 2tKTH2 b?QrbX 1tKTH2 RX9XRj, o`BiBQM .BbiM+2 #2ir22M hrQ SQBbbQM S`Q+2bb2bX q2 b22F  #QmM/ 7Q` i?2 /BbiM+2 BM p`BiBQM #2ir22M irQ SQBbbQM /Bbi`B#miBQMb QM i?2 HBM2 rBi? i?2 `2bT2+iBp2 BMi2MbBiv 7mM+iBQMb λ1 (t) M/ λ2 (t)X 6Q` i?Bb- r2 +QMbi`m+i Ub22 1t2`+Bb2 RXdXRyV irQ SQBbbQM T`Q+2bb2b N1 M/ N2 QM i?2 HBM2 rBi?

RX9X *PLo1_:1L*1 AL .Aah_A"lhAPL L. o_AhAPL

jR

i?2 `2bT2+iBp2 BMi2MbBiv 7mM+iBQMb λ1 (t) M/ λ2 (t) 7`QK  biM/`/ ?TT N QM R2 b 7QHHQrb Ni (a, b] = N ({(t, z) ; (t, z) ∈ (a, b] × {z ; 0 ≤ z ≤ λi (t)}}) (i = 1, 2) . *H2`Hv- N1 = N2 B7 M/ QMHv B7 i?2`2 `2 TQBMib Q7 N #2ir22M  i?2 +m`p2b z = λ1 (t)  M/ z = λ2 (t)X h?2 T`Q##BHBiv 7Q` i?Bb iQ ?TT2M Bb 1−exp − R |λ1 (t) − λ2 (t)| dt X h?2`27Q`2- #v h?2Q`2K XRXky   dV (P1 , P2 ) ≤ 1 − exp − |λ1 (t) − λ2 (t)| dt . R

LQi2 i?i BM i?Bb 2tKTH2- +?QQbBM; N1 M/ N2 BM/2T2M/2Mi rQmH/ H2/ iQ i?2 i`BpBH #QmM/ 1X .2}MBiBQM RX9XR9  b2[m2M+2 Q7 `M/QK K2bm`2b {Nn }n≥1 rBi? `2bT2+iBp2 /Bb@ i`B#miBQMb {Pn }n≥1 Bb bB/ iQ +QMp2`;2 BM p`BiBQM iQ i?2 `M/QK K2bm`2 N rBi? i?2 T`Q##BHBiv /Bbi`B#miBQM P B7 lim dV (Pn , P) = 0 .

n↑∞

h?2 M2ti `2bmHi Bb 2bT2+BHHv /Ti2/ iQ i?2 +QmTHBM; Q7 irQ biQ+?biB+ T`Q+2bb2bQM2 Q7 r?B+? Bb 2`;Q/B+X Ai rBHH #2 mb2/ BM i?2 Hbi +?Ti2`X h?2Q`2K RX9XR8 URj V G2i (Ω, F, P ) #2 bQK2 T`Q##BHBiv bT+2 2M/Qr2/ rBi?  }Hi`iBQM {Ft }t≥0 M/ H2i X := {X(t)}t≥0 M/ Y := {Y (t)}t≥0 #2 irQ Ft @/Ti2/ T`Q+2bb2bX bbmK2 i?i 7Q` HH s ≥ 0P (X ≡ Y QM [s, ∞) | Fs ) ≥ Z(s) − ε(s) ,

URXRkV

r?2`2 {ε(t)}t≥0 Bb `2H@pHm2/ T`Q+2bb bm+? i?i ε(s) i2M/b iQ 0 XbX b s → ∞- M/ r?2`2 Z := {Z(t)}t≥0 Bb  `2H@pHm2/ 2`;Q/B+ T`Q+2bb bm+? i?i P (Z(0) > 0) > 0 .

URXRjV

h?2M {X(t)}t≥0 M/ {Y (t)}t≥0 +QmTH2 BM XbX }MBi2 iBK2X S`QQ7X UQ7 G2KK RX9XR8V ++Q`/BM; iQ URX9XR8V- i?2`2 2tBbib  β > 0 bm+? i?i P (Z(s) > β) ≥ βX h?2 2p2Mi As := {X ≡ Y QM [s, ∞)} BM+`2b2b b s → ∞ iQ i?2 2p2Mi A∞ := {X M/ Y +QmTH2} r?B+? Bb F∞ @K2bm`#H2 bBM+2 #Qi? X M/ Y `2 Ft @/Ti2/X 1[miBQM URXRkV i?2`27Q`2 BKTHB2b 1 P (A∞ | Fs ) ≥ β1[β,∞) (Z(s))1[−∞, 1 β] (ε(s)) . 2 2 AMi2;`iBM; #2ir22M s M/ s + tRj

(JbbQmHBû- RNN8)X

jk

*>Sh1_ RX :1L1_GAhA1a 1 t



s+t s

1 1 P (A∞ | Fu ) du ≥ β1[−∞, 1 β] (sup ε(u)) 2 2 t u≥s



s+t

1[β,∞) (Z(u)) du .

URXR9V

s

aBM+2 {P (A∞ | Fu )}u≥0 Bb  mMB7Q`KHv BMi2;`#H2 K`iBM;H2- Bi +QMp2`;2b HKQbi bm`2Hv iQ 1A∞ b u → ∞- M/ i?2`27Q`2 i?2 H27i@?M/ bB/2 Q7 URXR9V HbQ +QMp2`;2b XbX iQ 1A∞ X "v i?2 2`;Q/B+Biv Q7 Z- i?2 *2b¨`Q K2M QM i?2 `B;?i@?M/ bB/2 Q7 URXR9V +QMp2`;2b XbX iQ P (Z(0) ≥ β) ≥ βX h?2`27Q`2 1 1A∞ ≥ β 2 1[−∞, 1 β] (sup ε(u)) . 2 2 u≥s b s → ∞ i?2 BM/B+iQ` QM i?2 `B;?i@?M/ bB/2 Q7 i?2 #Qp2 2[miBQM i2M/b iQ 1 XbX bQ i?i i?2 BM/B+iQ` Q7 i?2 2p2Mi A∞ Bb XbX bi`B+iHv TQbBiBp2- M/ Bb i?2`27Q`2 XbX 2[mH iQ 1- i?i Bb- i?2 T`Q+2bb2b X M/ Y +QmTH2X 

RX8 *Hmbi2` SQBMi S`Q+2bb2b _Qm;?Hv bT2FBM;-  +Hmbi2` TQBMi T`Q+2bb +QMbBbib Q7  TQBMi T`Q+2bb Ui?2 ;2`KV 2+? TQBMi Q7 r?B+? Bb bm``QmM/2/ #v  TQBMi T`Q+2bb Ui?2 +Hmbi2` i i?Bb TQBMiVX JQ`2 ;2M2`HHv- i?2 +Hmbi2`b Kv #2 `M/QK K2bm`2bX 6Q`KHHv MQr- M/ KQ`2 ;2M2`HHv, G2i N0 #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM i?2 HX+X/X#X bT+2 E rBi? b2@ [m2M+2 Q7 TQBMib {X0,n }n∈N M/ HQ+HHv }MBi2 BMi2MbBiv K2bm`2 ν0 X G2i {(x, ω, C) ∈ E × Ω × B(E) → Zn (x, ω, C)}n≥1 #2 KTTBM;b bm+? i?i Zn (x, ω, ·) ∈ M (E) ((x, ω) ∈ E × Ω) M/ bm+? i?i 7Q` 2+? C ∈ B(E) i?2 KTTBM; (x, ω) ∈ E × Ω → Zn (x, ω, C) Bb B(E) ⊗ F @K2bm`#H2X q`Bi2 Zn (x)(ω, C) := Zn (x, ω, C) Ui?2`27Q`2 Zn (x) Bb  `M/QK K2bm`2 QM (E, B(E)VVX amTTQb2 i?i 7Q` 2+? x ∈ E- i?2 b2[m2M+2 Q7 `M/QK K2bm`2b {Zn (x)}n∈Z Bb BB/- BM/2T2M/2Mi Q7 N0 M/ bm+? i?i E [Z1 (x, C)] = KZ (x, C) 7Q` bQK2 K2bm`#H2 F2`M2H KZ 7`QK (E, B(E)) iQ (E, B(E)) bm+? i?i 7Q` Mv +QKT+i b2i C ∈ B(E) KZ (x, C − x) ν0 (dx) < ∞ .

URXR8V

E

.2}MBiBQM RX8XR h?2 `M/QK K2bm`2 N QM E /2}M2/ #v  Zn (X0,n , C − X0,n ) (C ∈ B(E)) N (C) :=

URXReV

n∈N

Bb +HH2/  +Hmbi2` `M/QK K2bm`2 rBi? ;2`K UTQBMi T`Q+2bbV N0 X h?2 `M/QK K2bm`2 Zn (X0,n , · − X0,n ) Bb i?2 +Hmbi2` i i?2 TQBMi X0,n X h?2 BMi2MbBiv K2bm`2 ν Q7 N Bb

RX8X *Glah1_ SPALh S_P*1aa1a

jj

 KZ (y, C − y) ν0 (dy) (C ∈ B(E)) .

ν(C) =

URXRdV

E

AM 7+i- #v *KT#2HHǶb 7Q`KmH

 E





 



Zn (X0,n , C − X0,n ) = E E

n∈N

n∈N





=E







E Zn (X0,n , C − X0,n ) | F

n∈N

 =E

Zn (X0,n , C − X0,n ) | F

N0

N0





 KZ (X0,n , C − X0,n )

n∈N

KZ (y, C − y)) ν0 (dy) .

= E

A7 i?Bb BMi2MbBiv K2bm`2 Bb HQ+HHv }MBi2- i?2 `M/QK K2bm`2 +QMbB/2`2/ Bb  `M/QK 2H2K2Mi Q7 M (E)X q?2M 7Q` HH x ∈ E M/ HH ω- Z1 (x, ω, ·) ∈ Mp (E)- URXReV /2}M2b  +Hmbi2` TQBMi T`Q+2bbX h?Bb TQBMi T`Q+2bb Bb bBKTH2 B7- 7Q` BMbiM+2- ν0 Bb  MQM@iQKB+ K2bm`2X 1tKTH2 RX8Xk, aT+2 >QKQ;2M2Qmb *Hmbi2`bX q?2M Z1 dz/Q2b MQi /2T2M/ QM xǴ- i?i Bb- r?2M Bi Bb  `M/QK K2bm`2 QM (E, B(E)) UBM r?B+? +b2 KZ (x, ·) = νZ (·)- i?2 BMi2MbBiv K2bm`2 Q7 Z1 V- r2 mb2 i?2 MQiiBQM N = N0 ∗ Z- r?2`2 Z biM/b 7Q` i?2 dz;2M2`B+Ǵ +Hmbi2`- i?i Bb- Mv `M/QK K2bm`2 rBi? i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 Zn ǶbX AKTHB+Bi BM i?Bb MQiiBQM Bb i?2 bbmKTiBQM i?i i?2 K`Fb Zn Q7 N0 `2 BB/ M/ BM/2T2M/2Mi Q7 N0 X AM i?Bb +b2ν = ν0 ∗ νZ ,

URXR3V

r?2`2 νZ Bb i?2 +QKKQM BMi2MbBiv K2bm`2 Q7 i?2 Zn Ƕb M/ ∗ /2MQi2b i?2 mbmH +QMpQHmiBQM T`Q/m+i Q7 irQ K2bm`2bX

_2K`F RX8Xj LQi2 i?i BM i?2 +b2 Q7 TQBMi T`Q+2bb +Hmbi2`b- i?2 Zn Ƕb Kv ?p2  TQBMi i 0 BM r?B+? +b2 bQK2 Q` HH TQBMib Q7 i?2 ;2`K TQBMi T`Q+2bb `2 T`i Q7 i?2 +Hmbi2` TQBMi T`Q+2bbX q?2M E = Rm -  bm{+B2Mi +QM/BiBQM 7Q` i?2 +Hmbi2` TQBMi T`Q+2bb iQ #2 bBKTH2 Bb i?i Bib BMi2MbBiv K2bm`2 #2 /Bzmb2X h?Bb Bb i?2 +b2 r?2M2p2` QM2 Q7 i?2 K2bm`2b Q7 i?2 +QMpQHmiBQM URXR3V Bb  KmHiBTH2 Q7 i?2 G2#2b;m2 K2bm`2 M/ i?2 Qi?2` Bb  }MBi2 K2bm`2X 6Q` BMbiM+2- B7 i?2 BMi2MbBiv K2bm`2 Q7 i?2 ;2`K TQBMi T`Q+2bb Bb Q7 i?2 7Q`K ν0 (dx) = λ0 m (dx)- i?2M 



ν(C) = Rm

λ0 m (C − x)νZ (dx) =

λ0 m (C)νZ (dx) = λ0 νZ (E) m (C) . Rm

j9

*>Sh1_ RX :1L1_GAhA1a

1tKTH2 RX8X9, h?2 *Qt *Hmbi2` SQBMi S`Q+2bbX A7 7Q` HH x ∈ E- Z1 (x, ·) Bb  SQBbbQM T`Q+2bb- i?2 +Hmbi2` TQBMi T`Q+2bb Bb +HH2/  *Qt +Hmbi2` TQBMi T`Q+2bbX

1tKTH2 RX8X8, h?2 L2vKMėa+Qii *Hmbi2` SQBMi S`Q+2bbXR9 AM i?Bb KQ/2H- i?2 ;2M2`B+ +Hmbi2` Bb Q7 i?2 7Q`K Z := {U1 , U2 , . . . , UT } r?2`2 T Bb M BMi2;2`@pHm2/ `M/QK p`B#H2 M/ {Un }n≥1 Bb M BB/ b2[m2M+2 Q7 `M/QK p2+iQ`b Q7 Rm - BM/2T2M/2Mi Q7 T X

1tKTH2 RX8Xe, h?BMMBM;X A7 i?2 ;2M2`B+ +Hmbi2` 2Bi?2` ?b QMHv QM2 TQBMi i 0 UrBi? T`Q##BHBiv pV Q` Bb 2KTiv UrBi? T`Q##BHBiv 1 − pV- i?2 +Hmbi2` TQBMi T`Q+2bb N = N0 ∗ Z Bb BM 7+i M BM/2T2M/2Mi p@i?BMMBM; Q7 N0 , 2+? TQBMi Q7 N0 ?b #22M 2`b2/ UrBi? T`Q##BHBiv 1 − pV Q` F2Ti UrBi? T`Q##BHBiv pV BM/2T2M/2MiHv Q7 i?2 Qi?2` TQBMibX 1tKTH2 RX8Xd, .BbTH+BM;X A7 i?2 ;2M2`B+ +Hmbi2` Z ?b 2t+iHv QM2 TQBMi i HQ+iBQM V - i?2 `2bmHiBM; +Hmbi2` TQBMi T`Q+2bb N = N0 ∗ Z Bb Q#iBM2/ #v BM/2@ T2M/2Mi `M/QK /BbTH+2K2Mib- i?2 ;2M2`B+ /BbTH+2K2Mi #2BM; V X h?2 TQBMib Q7 N = N0 ∗ Z `2 {X0,n + Vn }n∈N - r?2`2 i?2 Vn Ƕb 7Q`K M BB/ b2[m2M+2 Q7 i?2 bK2 /Bbi`B#miBQM b i?2 ;2M2`B+ /BbTH+2K2Mi V X

1tKTH2 RX8X3, GTH+2 6mM+iBQMH Q7  *Hmbi2` SQBMi S`Q+2bbX *QM@ bB/2` i?2 TQBMi T`Q+2bb N /2}M2/ #v URXReV- bbmK2/ bBKTH2 M/ rBi?  HQ+HHv }MBi2 BMi2MbBiv K2bm`2X amTTQb2 KQ`2Qp2` i?i Zn (x, ·) = Zn (·) Ui?2 ;2M2`H +b2 r?2`2 i?2 /Bbi`B#miBQMb Q7 i?2 +Hmbi2`b /2T2M/ QM i?2B` TQbBiBQMb +M Q7 +Qm`b2 #2 i`2i2/ bBKBH`HvVX .2MQi2 #v LN0 M/ LZ i?2 GTH+2 i`Mb7Q`Kb Q7 N0 M/ Mv Zn `2bT2+iBp2HvX h?2M     LN (ϕ) := E exp − ϕ(x + X0,n ) Zn (dx)  =E

E

n∈N



   exp − ϕ(x + X0,n ) Zn (dx) E

n∈Z

     N0 exp − ϕ(x + X0,n ) Zn (dx) | F =E E  

n∈Z



E

  

  N0 =E E exp − ϕ(x + X0,n ) Zn (dx) | F .

n∈N R9

(L2vKM M/ a+Qii- RN83)X

E

RX8X *Glah1_ SPALh S_P*1aa1a

j8

h?2`27Q`2- bBM+2  

 E exp − ϕ(x + X0,n ) Zn (dx) | F N0 = LZn (ϕ(· + X0,n )) = LZ (ϕ(· + X0,n )) , E

r2 Q#iBM

 LN (ϕ) = E 





LZ (ϕ(· + X0,n )

n∈N



= E exp



 log LZ (ϕ(· + X0,n )

.

n∈N

6BMHHvURXRNV

LN (ϕ) = LN0 (−ψ) , r?2`2 ψ(x) := log LZ (ϕ(· + x)) .

Bb SQBbbQM AM i?2 bT2+BH +b2 r?2`2 E = Rm M/ r?2`2 i?2 ;2`K TQBMi  T`Q+2bb  T`Q+2bb Q7 HQ+HHv }MBi2 BMi2MbBiv K2bm`2 μ - LN0 (ϕ) = exp Rm e−ϕ(x) − 1 μ (dx) M/ i?2`27Q`2        LN (ϕ) = exp E e− Rm ϕ(y+x) Z(dy) − 1 μ (dx) . URXkyV Rm

1tKTH2 RX8XN, pQB/M+2 S`Q##BHBiv 6mM+iBQM Q7  SQBbbQM *Hmbi2` S`Q+2bbX *QMbB/2` i?2 bT+2@?QKQ;2M2Qmb +Hmbi2` TQBMi T`Q+2bb Q7 1tKTH2 RX8XkrBi? i?2 //BiBQMH bT2+B}+iBQM i?i i?2 ;2`K N0 Bb  SQBbbQM T`Q+2bbX q2 +QKTmi2 Bib pQB/M+2 T`Q##BHBiv 7mM+iBQMX 6Q` i?Bb- r2 }`bi Q#b2`p2 i?i P (N (C) = 0) = lim E e−tN (C) . t↑∞

LQr

−tN (C)

E e





=E e

−t

 n

Zn (C−Xn )



 =E

  =E E  =E

 e

−tZn (C−Xn )





n



e

−tZn (C−Xn )

|F

n







−tZ1 (C−Xn )

E e

n





N0

=E 

= E exp







−tZn (C−Xn )

E e

|F

N0

n





log E e

−tZ1 (C−Xn )







 .

n

aBM+2 N0 Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν0 - i?2 Hbi i2`K Q7 i?2 #Qp2 b2[m2M+2 Q7 2[mHBiB2b Bb    −tZ1 (C−x)  exp E e − 1 ν0 (dx) . () Rm

je

*>Sh1_ RX :1L1_GAhA1a

"mi

lim E e−tZ1 (C−x) − 1 = vZ (C − x) − 1 , t↑∞

r?2`2 vZ Bb i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM Q7 Z1 X h?2`27Q`2 iFBM; i?2 HBKBi b t ↑ ∞ BM UV vB2H/b #v /QKBMi2/ +QMp2`;2M+2,   (vZ (C − x) − 1) ν0 (dx) . vN (C) = exp Rm

RXe h?2 aiB2HiD2bĜG2#2b;m2 *H+mHmb  H`;2 MmK#2` Q7 biQ+?biB+ T`Q+2bb2b QM i?2 `2H HBM2 ?p2 bKTH2 Ti?b Q7 #QmM/2/ p`BiBQMX h?2 ivTB+H +b2 Bb i?i Q7  TB2+2rBb2 +QMiBMmQmb T`Q+2bb r?Qb2 /Bb+QMiBMmBiB2b Q++m` i i?2 2p2Mi iBK2b Q7  TQBMi T`Q+2bb QM i?2 `2H HBM2X hQ K@ MBTmHi2 BMi2;`Hb rBi? `2bT2+i iQ i?Bb ivT2 Q7 T`Q+2bb2b QM2 ?b i /BbTQbBiBQM  bBKTH2 b2i Q7 `mH2b- bm+? b i?2 T`Q/m+i `mH2 M/ i?2 2tTQM2MiBH `mH2- r?B+? `2 [mBi2 +QMp2MB2Mi iQ 2tT`2bb i?2 2pQHmiBQM 2[miBQMb Q++m``BM; BM i?2 bim/v Q7 /v@ MKB+H bvbi2Kb- 2bT2+BHHv BM i?2 i?2Q`v Q7 biQ+?biB+ BMi2MbBiv U*?Ti2` 8V M/ BM i?2 i?2Q`v Q7 SHK T`Q##BHBiv U*?Ti2` dVX h?2 S`Q/m+i _mH2 6B`bi `2+HH i?2 /2}MBiBQM Q7  #QmM/2/ p`BiBQM 7mM+iBQM M/ i?i Q7  aiB2HiD2bĜ G2#2b;m2 BMi2;`H rBi? `2bT2+i iQ bm+? 7mM+iBQMbX .2}MBiBQM RXeXR G2i f : [0, ∞) → R #2  7mM+iBQM bm+? i?i 7Q` HH t ≥ 0Vf (t) := sup D

N 

|f (ti ) − f (ti−1 )| < ∞,

i=1

r?2`2 D `M;2b Qp2` HH i?2 bm#/BpBbBQMb Q7 [0, t], 0 = t0 < t1 < · · · < t N = t . h?2 7mM+iBQM f Bb i?2M bB/ iQ #2 Q7 #QmM/2/ p`BiBQM Qp2` }MBi2 BMi2`pHbX h?2 [mMiBiv Vf (t) Bb +HH2/ i?2 p`BiBQM Q7 f Qp2` i?2 BMi2`pH [0, t]X t 1tKTH2 RXeXk, hrQ 2tKTH2bX h?2 7Q`KmH f (t) = 0 g(s) ds- r?2`2 g : [0, ∞) → R Bb  HQ+HHv BMi2;`#H2 7mM+iBQM- /2}M2b  7mM+iBQM f Q7 #QmM/2/ p`B@ iBQM Qp2` }MBi2 BMi2`pHbX HbQ  MQM@/2+`2bBM; 7mM+iBQM a : [0, ∞) → R Bb Q7 #QmM/2/ p`BiBQM Qp2` }MBi2 BMi2`pHb M/ Vf (t) = a(t) − a(0)X  +HbbB+H `2bmHi Q7 MHvbBbR8 bvb i?i Mv `B;?i@+QMiBMmQmb 7mM+iBQM f : [0, ∞) → R Q7 #QmM/2/ p`BiBQM Qp2` }MBi2 BMi2`pHb +M #2 2tT`2bb2/ b i?2 /Bz2`2M+2 #2ir22M irQ MQM@/2+`2bBM; `B;?i@+QMiBMmQmb 7mM+iBQMb, R8

a22 7Q` BMbiM+2 (hvHQ`- RN38)- *?Ti2` NX

RXeX h>1 ahA1GhC1aĜG1"1a:l1 *G*lGla

jd

f (t) = f (0) + a(t) − b(t). 6Q` BMbiM+2-

URXkRV

b(t) = −f (t) + f (0) + Vf (t).

a(t) = Vf (t);

h?Bb T`iB+mH` /2+QKTQbBiBQM Bb `272``2/ iQ b i?2 +MQMB+H /2+QKTQbBiBQM Q7 f X t 6Q` BMbiM+2- B7 f (t) = 0 g(s) ds r?2`2 g : [0, ∞) → R Bb  HQ+HHv BMi2;`#H2 t t 7mM+iBQM- i?2M f (t) = 0 |g(s)| ds − 0 2g − (s) dsX AM i?2 #b2M+2 Q7 2tTHB+Bi K2MiBQM Q7 i?2 +QMi``v- i?2 #QmM/2/ p`BiBQM 7mM+iBQMb ?2M+27Q`i? +QMbB/2`2/ `2 bbmK2/ +Q`HQH- i?i Bb- `B;?i@+QMiBMmQmb rBi? H27i@?M/ HBKBib f (t−) = limh↓0 f (t − h)X G2i f : [0, ∞) → R #2  +Q`HQH #QmM/2/ p`BiBQM 7mM+iBQM rBi? +MQMB+H /2+QKTQbBiBQM URXkRVX hQ a M/ b- r?B+? `2 +Q`HQH M/ MQM@/2+`2bBM;- `2 bbQ+Bi2/ i?2 K2bm`2b μa M/ μb QM ((0, ∞), B((0, ∞)) `2bT2+iBp2Hv- /2}M2/ #v, μa ((0, t]) = a(t) = Vf (t);

μb ((0, t]) = b(t) .

LQi2 i?i μb ≤ 2μa X q2 b?HH mb2 i?2 MQiiBQM μa (dt) := |df (t)|X  K2bm`#H2 7mM+iBQM u : [0, ∞) → R bm+? i?i  |u(s)| μa (ds) < ∞ (0,∞)

 UM/ i?2`27Q`2 (0,∞) |u(s)| μb (ds) < ∞V Bb bB/ iQ #2 G2#2b;m2ĜaiB2HiD2b BMi2;`#H2 rBi? `2bT2+i iQ f - M/ Bib G2#2b;m2ĜaiB2HiD2b BMi2;`H Bb- #v /2}MBiBQM- i?2 [mMiBiv    u(s) df (s) := u(s) μa (ds) − u(s) μb (ds). (0,∞)

(0,∞)

(0,∞)

qBi? i?Bb /2}MBiBQM- r2 +M i?2`27Q`2 r`Bi2    u(s) df (s) := u(s) da(s) − (0,∞)

(0,∞)

u(s) db(s). (0,∞)

G2i f1 , f2 : R+ → R #2 irQ +Q`HQH 7mM+iBQMb Q7 HQ+HHv #QmM/2/ p`BiBQM rBi? +MQMB+H `2T`2b2MiiBQMb fi (t) = fi (0) + ai (t) − bi (t)- i = 1, 2X h?2 bvK#QHB+ 2tT`2bbBQM   u(x, y) df1 (x) df2 (y) (0,∞)

()

(0,∞)

Bb iQ #2 BMi2`T`2i2/ #v r`BiBM; df1 (x) df2 (y) = μa1 (dx)μa2 (dy)+μb1 (dx)μb2 (dy)−μa1 (dx)μb2 (dy)−μa2 (dx)μb1 (dy). h?2 [mMiBiv UV Bb r2HH /2}M2/ B7 i?2 7Qm` +Q``2bTQM/BM; BMi2;`Hb `2 r2HH /2}M2/ M/ B7 KQ`2Qp2` i?2 //BiBQMb M/ bm#i`+iBQMb BMpQHp2/ /Q MQi H2/ iQ BM/2i2`KB@ Mi2 7Q`KbX AM pB2r Q7 i?Bb /2+QKTQbBiBQM- i?2 +QM/BiBQMb Q7 TTHB+iBQM Q7 6m#BMBǶb 7Q`KmH 7QHHQr 7`QK i?2 biM/`/ +b2 Q7 T`Q/m+i K2bm`2bX h?2Q`2K RXeXj G2i f, g : R+ → R #2 irQ +Q`HQH 7mM+iBQMb Q7 HQ+HHv #QmM/2/ p`BiBQMX h?2M- 7Q` HH t ≥ 0  f (t)g(t) = f (0)g(0) + f (s) dg(s) + g(s−) df (s) . URXkkV (0,t]

(0,t]

j3

*>Sh1_ RX :1L1_GAhA1a

S`QQ7X Ai Bb 2MQm;? iQ T`Qp2 i?Bb 7Q` MQM@M2;iBp2 7mM+iBQMb f M/ gX "v 6m#BMB    df (x) dg(y) = df (x) dg(y) . (f (t) − f (0))(g(t) − g(0)) = (0,t]

(0,t]

(0,t]

(0,t]

G2i D := (0, t] × (0, t] ,

D1 := {(x, y) ∈ D ; x ≤ y} ,

M/ i?2`27Q`2  

 

 

df (x) dg(y) = (0,t]

D2 := {(x, y) ∈ D ; x > y} ,

df (x) dg(y) .

df (x) dg(y) +

(0,t]

D1

D2

"v 6m#BMB      df (x) dg(y) = df (x) dg(y) D1 (0,t] (0,y]   (f (y) − f (0)) dg(y) = f (y) dg(y) − f (0)(g(t) − g(0)) . = (0,t]

(0,t]

aBKBH`Hv  





 df (x) dg(y) = D2

dg(y) 

(0,t]



(0,t]

df (x)

(0,x)

(g(x−) − g(0)) df (x)

=

g(x−) df (x) − g(0)(f (t) − f (0)).

= (0,t]

*QK#BMBM; i?2 #Qp2 2[mHBiB2b H2/b iQ i?2 MMQmM+2/ `2bmHiX



*Q`QHH`v RXeX9 G2i a : R+ → R #2  `B;?i@+QMiBMmQmb MQM@/2+`2bBM; 7mM+iBQMX h?2M- 7Q` HH n ≥ 1  an (t) − an (0) ≤ an−1 (s−) da(s) ≤ an−1 (s) da(s) . n (0,t] (0,t] S`QQ7X q2 b?HH mb2 i?2 MQiiBQM h− iQ `2T`2b2Mi i?2 7mM+iBQM t → h(t−) r?2M h Bb  +Q`HQH 7mM+iBQMX qBi? i?Bb MQiiBQM- URXkkV `2/b d(f g) = f− dg + g df X TTHvBM; i?Bb `mH2 iQ f = an−1 M/ g = a- r2 Q#iBM n−1 d(an ) = an−1 ) − da + a d(a n−1 n−2 = a− da + aa− da + a2 d(an−2 ) = (an−1 + aan−2 + · · · + an−1 ) . − −

h?2 `2bmHi 7QHHQrb bBM+2 a− ≤ aX



RXeX h>1 ahA1GhC1aĜG1"1a:l1 *G*lGla

jN

h?2 1tTQM2MiBH _mH2 G2i a : R+ → R #2  `B;?i@+QMiBMmQmb 7mM+iBQM Q7 #QmM/2/ p`BiBQM bm+? i?i a(0) = 0- M/ H2i u : R+ → R #2 bm+? i?i 7Q` HH t ≥ 0 |u(s)| da(s) < ∞ . (0,t]

h?2Q`2K RXeX8 h?2 2[miBQM

 x(s−)u(s) da(s)

x(t) = x(0) + (0,t]

/KBib  mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM ;Bp2M #v  t x(t) = x(0) (1 + u(s)Δa(s)) × e o ac (ds) , 0 n0 X amTTQb2 BM //BiBQM i?i QM i?2 BMi2`pHb [tn , tn+1 ) i?i HB2 BM R+ 

t

f (t) = f (tn ) +

f  (s) ds ,

tn

7Q` bQK2 HQ+HHv BMi2;`#H2 7mM+iBQM f  Ui?2 /2`BpiBp2VX G2i MQr G : R → R #2  /Bz2`2MiB#H2 7mM+iBQM rBi? /2`BpiBp2 G X h?2M 7Q` HH [a, b) ⊂ R+ -

9k

*>Sh1_ RX :1L1_GAhA1a

G(f (b)) = G(f (a)) +





b

(G(f (tn )) − G(f (tn −)) +

f  (s)G (f (s)) ds . URXk9V

a

tn ∈(a,b)

6Q` BMbiM+2- rBi? G(x) = eαx  b    eαf (b) = eαf (a) + f  (s)αeαf (s) ds . eαf (tn ) − eαf (tn −) + a

tn ∈(a,b)

AM T`iB+mH`- B7 tn = Tn r?2`2 {Tn }n≥1 Bb i?2 b2[m2M+2 Q7 TQBMib Q7  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb N QM R+  b   f  (s)αeαf (s) ds , eα(f (Tn )−f (Tn −)) − 1 eαf (Tn −) + eαf (b) = eαf (a) + a

n≥1

i?i Bb- rBi? i?2 BMi2;`H MQiiBQM  b  α(f (s)−f (s−))  eαf (b) = eαf (a) + e − 1 eαf (s−) N (ds) + f  (s)αeαf (s) ds . a

(a,b]

1tKTH2 RXeXN, h?2 HBF2HB?QQ/ `iBQ 7Q`KmH- iF2 RX h?2 iBiH2 Q7 i?2 2tKTH2 `272`b iQ  7mM/K2MiH 7Q`KmH Q7 TQBMi T`Q+2bb i?2Q`v i?i r2 b?HH 2M+QmMi2` BM am#b2+iBQM 8X8 Uh?2Q`2K 8X8XRVX G2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM i?2 HBM2- M/ H2i {λ(t)}t≥0 M/ {μ(t)}t≥0 #2 irQ MQM@M2;iBp2 HQ+HHv BMi2;`#H2 biQ+?biB+ T`Q+2bb2bX h?2 biQ+?biB+ T`Q+2bb {L(t)}t≥0 i?i Bb i?2 mMB[m2 bQHmiBQM Q7 i?2 2[miBQM  L(t) = L(0) + L(s−)(μ(s) − 1)dM (s) () r?2`2 M (t) := N (0, t] −

(0,t]

t

λ(s) ds- Bb    t L(t) = L(0) μ(s)N ({s}) e− 0 (μ(s)−1)λ(s) ds , 0

()

0 u}- r?B+? Bb BM/22/ 2[mH iQ a(u)X  *Q`QHH`v RXeXRk G2i a : R+ → R+ #2  MQM@/2+`2bBM; +QMiBMmQmb 7mM+iBQM bm+? i?i a(0) = 0X h?2M- 7Q` HH K2bm`#H2 7mM+iBQMb g : R+ → R+ 





a(∞)

g((a(t)) da(t) = 0

g(t) dt.

URXkeV

0

S`QQ7X P#b2`p2 i?i bBM+2 a Bb +QMiBMmQmb- a(c(t)) = t r?2M t ∈ [0, a(+∞)) M/ TTHv 7Q`KmH URXk8V rBi? f = g ◦ aX 

RXd

1t2`+Bb2b

1t2`+Bb2 RXdXRX LQi  bBKTH2 TQBMi T`Q+2bb5 aBKTHB+Biv Q7  TQBMi T`Q+2bb N QM Rm Bb MQi ;m`Mi22/ #v i?2 `2[mB`2K2Mi P (N ({a}) ≤ 1) = 1 7Q` HH a ∈ Rm X :Bp2  UbBKTH25V +QmMi2`@2tKTH2X 1t2`+Bb2 RXdXkX h?2 TQBMi b2[m2M+2 Q7  HQ+HHv 7BMBi2 TQBMi T`Q+2bb X G2i μ #2  HQ+HHv }MBi2 TQBMi K2bm`2 QM (R, B(R))X a?Qr i?i UBV i?2`2 2tBbib  MQM@/2+`2bBM; b2[m2M+2 {tn }n∈Z BM R rBi? Ui?2 +QMp2MiBQMV t0 ≤ 0 < t1 M/ bm+? i?i 7Q` HH K2bm`#H2 b2ib C εtn (C) μ(C) = n∈Z

99

*>Sh1_ RX :1L1_GAhA1a

Ur?2`2 εa Bb i?2 .B`+ K2bm`2 i a B7 a ∈ R- M/ i?2 MmHH K2bm`2 B7 a = +∞ Q` −∞VUBBV i?2 KTTBM;b μ → tn = tn (μ) `2 K2bm`#H2 7`QK (M (E), M(E)) iQ (R, B)M/ UBBBV B7- BM //BiBQM- μ Bb bBKTH2- i?2M |tn | < ∞ ⇒ tn < tn+1 X UAM Qi?2` rQ`/b{tn }n∈Z Bb bi`B+iHv BM+`2bBM; QM RXV "X S`Qp2 G2KK RXRX8 r?2M E = Rm X 1t2`+Bb2 RXdXjX J2bm`#BHBiv Q7 TQBMi T`Q+2bb BMi2;`Hb G2i N #2  `M/QK K2bm`2 QM (E, B(E)) M/ H2i ϕ: (E, E) → (R, B) #2  MQM@ M2;iBp2 K2bm`#H2 7mM+iBQMX a?Qr i?i N (ϕ) := E ϕ(t) N (dt) Bb r2HH /2}M2/ M/  `M/QK p`B#H2X 1t2`+Bb2 RXdX9X AMi2MbBiv J2bm`2 Q7  J`F2/ SQBMi S`Q+2bb rBi? AM/2T2M/2Mi BB/ J`Fb S`Qp2 i?i i?2 BMi2MbBiv K2bm`2 Q7 i?2 K`F2/ TQBMi T`Q+2bb Q7 .2}MBiBQM RXRXRd Bb i?2 T`Q/m+i K2bm`2 ν × QZ - r?2`2 ν Bb i?2 BMi2MbBiv K2bm`2 Q7 i?2 #b2 TQBMi T`Q+2bbX 1t2`+Bb2 RXdX8X SQBMib QM i?2 ;`B/ AM 1tKTH2 RXRXRk- M/ BM i?2 +b2 i?i {Xn1 ,n2 }n1 ,n2 ∈Z Bb M BM/2T2M/2Mi b2[m2M+2+M r2 bv i?i +QM/BiBQM limn1 ,n2 ↑∞ pn1 ,n2 = 0 BKTHB2b i?i i?2 +Q``2bTQM/BM; TQBMi T`Q+2bb Bb HQ+HHv }MBi2\ 1t2`+Bb2 RXdXeX h?2 n@i? TQBMi Q7  SQBbbQM T`Q+2bb G2i N #2  SQBbbQM T`Q+2bb QM R rBi? BMi2MbBiv 7mM+iBQM λ : R → RX .2MQiBM; #v Tn i?2 n@i? TQBMi Q7 N bi`B+iHv iQ i?2 `B;?i Q7 i?2 Q`B;BM- T`Qp2 i?i Tn Bb M #bQHmi2Hv +QMiBMmQmb `M/QK p`B#H2 M/ ;Bp2 Bib T`Q##BHBiv /2MbBivX 1tT`2bb BM i2`Kb Q7 i?2 BMi2MbBiv 7mM+iBQM i?2 +mKmHiBp2 T`Q##BHBiv /Bbi`B#miBQM FT1 Q7 T1 M/ Bib T`Q##BHBiv /2MbBiv 7mM+iBQM fT1 X o2`B7v i?i λ : R → R Bb i?2 ?x`/ `i2 7mM+iBQM Q7 T1 X 1t2`+Bb2 RXdXdX 6+iQ`BH K2bm`2b Q7 SQBbbQM T`Q+2bb2b S`Qp2 i?i i?2 n@i? 7+iQ`BH KQK2Mi K2bm`2 Q7  SQBbbQM T`Q+2bb QM Rm rBi? K2M K2bm`2 ν Bb ν ⊗n X 1t2`+Bb2 RXdX3X h?2 n@i? 7+iQ`BH KQK2Mi K2bm`2 Q7  i?BMM2/ TQBMi T`Q+2bbX G2i N #2 M n@i? Q`/2` TQBMi T`Q+2bb rBi? b2[m2M+2 Q7 TQBMib {Xn }n∈N X :Bp2 i?2 ! Q7 i?2 i?BMM2/ TQBMi T`Q+2bb Nthin,p(·) n@i? Q`/2` 7+iQ`BH KQK2Mi K2bm`2 M n Ua22 .2}MBiBQM RXjXkyVX .2/m+2 7`QK i?2 `2bmHi i?i  i?BMM2/ /2i2`KBMMiH TQBMi T`Q+2bb Bb HbQ  /2i2`KBMMiH TQBMi T`Q+2bbX 1t2`+Bb2 RXdXNX .2i2`KBMMiH T`Q+2bb2b `2 bBKTH2 S`Qp2 i?i  /2i2`KBMMiH TQBMi T`Q+2bb Bb bBKTH2X

RXdX 1s1_*Aa1a

98

1t2`+Bb2 RXdXRyX SQBbbQM TQBMib mM/2`  +m`p2  #2 M ?TT QM R2 rBi? BMi2MbBiv 1X G2i λ : R → R #2  MQM@M2;iBp2 HQ+HHv G2i N BMi2;`#H2 7mM+iBQMX .2}M2  TQBMi T`Q+2bb N QM R #v   N (C) = R

R

 (dt × dz) . 1C (t) 1[0,λ(t)] (z) N

S`Qp2 i?i N Bb  SQBbbQM T`Q+2bb QM R rBi? BMi2MbBiv 7mM+iBQM λ : R → RX 1t2`+Bb2 RXdXRRX JQK2Mi K2bm`2b Q7  TQBMi T`Q+2bb X G2i N #2  b2+QM/@Q`/2` TQBMi T`Q+2bb QM i?2 HX+X/X#X EX a?Qr i?i i?2 7Q`KmH M2 (A × B) := E [N (A) N (B)] mMB[m2Hv /2}M2b  K2bm`2 M2 QM (E × E, B(E × E)) M/ i?i i?Bb K2bm`2 Bb HQ+HHv }MBi2X "X G2i N #2  SQBbbQM T`Q+2bb QM i?2 HX+X/X#X E rBi? BMi2MbBiv K2bm`2 νX S`Qp2 i?i M2 (A1 × A2 ) = ν(A1 )ν(A2 ) − ν(A1 ∩ A2 ) M/ M2! (A1 × A2 ) = ν(A1 )ν(A2 ) . *X A7 N Bb M ?TT QM Rm Q7 BMi2MbBiv λ- T`Qp2 i?i i?2 n@i? 7+iQ`BH KQK2Mi K2bm`2 Mn! /KBib  /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 n ;Bp2M #v ρ!n (x1 , . . . , xn ) = λn .

1t2`+Bb2 RXdXRkX _M/QK TQBMib mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] *QMbi`m+i  TQBMi T`Q+2bb N QM R BM i?2 7QHHQrBM; rvX 6B`bi /`r  }MBi2 BMi2;2`@ pHm2/ `M/QK p`B#H2 T - M/ i?2M M BB/ b2[m2M+2 {Un }n≥1 mMB7Q`KHv /Bbi`B#mi2/ QM 0VX 6BMHHv- H2i N =

T[0, 1]- BM/2T2M/2Mi Q7 T X .2}M2 αk := P (T = k) Uk ≥ 0 ε r?2`2 ε Bb i?2 .B`+ K2bm`2 i aM/ r?2`2 U a k k=1 k=1 εUk Bb i?2 MmHH K2bm`2 #v +QMp2MiBQMX q?i Bb i?2 GTH+2 i`Mb7Q`K Q7 N \ q?i #Qmi i?2 +b2 r?2`2 T Bb  SQBbbQM p`B#H2 Q7 K2M θ\ 1t2`+Bb2 RXdXRjX GTH+2 i`Mb7Q`K Q7  i?BMM2/ TQBMi T`Q+2bb S`Qp2 h?2Q`2K RXjXkRX 1t2`+Bb2 RXdXR9X h?2 GTH+2 7mM+iBQMH Q7 i?2 L2vKMėa+Qii +Hmbi2` TQBMi T`Q+2bb *QKTmi2 i?2 GTH+2 7mM+iBQMH Q7 i?2 L2vKMĜa+Qii +Hmbi2` TQBMi T`Q+2bb Q7 1t@ KTH2 RX8X8 BM i2`Kb Q7 i?2 ;2M2`iBM; 7mM+iBQM Q7 T M/ Q7 i?2 +QKKQM T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 Un ǶbX

9e

*>Sh1_ RX :1L1_GAhA1a

1t2`+Bb2 RXdXR8X h?2 HBMF #2ir22M i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM M/ i?2 GTH+2 7mM+iBQMH G2i N #2  TQBMi T`Q+2bb QM Rm rBi? GTH+2 7mM+iBQMH LN X a?Qr i?i Bib pQB/@ M+2 T`Q##BHBiv 7mM+iBQM v Bb ;Bp2M #v v(C) = lim LN (t1C ) . t↑+∞

1t2`+Bb2 RXdXReX  +?`+i2`BxiBQM Q7 i?2 SQBbbQM T`Q+2bb 6Q`  bBKTH2 TQBMi T`Q+2bb N QM Rm iQ #2  SQBbbQM T`Q+2bb Q7 σ@}MBi2 BMi2MbBiv K2bm`2 ν Bi Bb M2+2bb`v M/ bm{+B2Mi i?i 7Q` HH "Q`2H b2i C ∈ Rm Q7 }MBi2 ν@K2bm`2- N (C) #2  SQBbbQM p`B#H2 Q7 K2M ν(C)X Uh?2 bm`T`BbBM; 7+i Bb i?i QM2 /Q2b MQi M22/ i?2 BM/2T2M/2M+2 bbmKTiBQM Q7 .2}MBiBQM RXRXNXV 1t2`+Bb2 RXdXRdX  T`QT2`iv Q7 i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM UV a?Qr i?i B7 v1 M/ v2 `2 i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQMb Q7 bQK2 TQBMi T`Q+2bb2b QM Rm - bQ Bb v1 × v2 X U#V G2i v #2 i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM Q7 bQK2 TQBMi T`Q+2bb QM Rm M/ H2i gT #2 i?2 ;2M2`iBM; 7mM+iBQM Q7 bQK2 `M/QK p`B#H2 iFBM; Bib pHm2b BM NX a?Qr i?i w(C) := gT (v(C)) Bb i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM Q7 bQK2 TQBMi T`Q+2bb QM Rm X 1t2`+Bb2 RXdXR3X *QKT`BbQM Q7 pQB/M+2 7mM+iBQMb *QKT`2 i?2 pQB/M+2 7mM+iBQMb Q7 M ?TT Q7 BMi2MbBiv λ M/ Q7  ?QKQ;2M2Qmb *Qt T`Q+2bb Q7 BMi2MbBiv Λ bm+? i?i E [Λ] = λX 1t2`+Bb2 RXdXRNX M 2tTQM2MiBH 7Q`KmH G2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM (a, b]X a?Qr i?i 7Q` HH u ∈ RHH t ∈ (a, b]- r2 ?p2 i?i UTv ii2MiBQM iQ i?2 T`2Mi?2b2b M/ b[m`2 #`+F2ibV  iuN ((a,b]) iu e = 1 + (e − 1) eiuN ((a,s)) N (ds) . (a,b]

1t2`+Bb2 RXdXkyX h?2 rQ`FHQ/ T`Q+2bb G2i σ #2  TQbBiBp2 MmK#2` M/ H2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ X G2i {W (t)}t∈R+ #2  `B;?i@+QMiBMmQmb biQ+?biB+ T`Q+2bb rBi? H27i@?M/ HBKBib M/ /2+`2bBM; +QMbiMiHv i `i2 1 2t+2Ti BM irQ +B`+mKbiM+2b, UBV i  DmKT Q7 N r?2`2 Bi DmKTb 7`QK W (t−) iQ W (t) = W (t−) + σ- M/ UBBV r?2M Bi `2+?2b i?2 pHm2 0- BM r?B+? +b2 Bi `2KBMb MmHH mMiBH i?2 M2ti 2p2Mi iBK2 Q7 N X a?Qr i?i 7Q` HH u ∈ R- HH t ∈ R+ 



eiuW (t) = eiuW (0) + (eiuσ − 1)

eiuW (s−) N (ds) − iu (0,t]

t

1{W (s)>0} . 0

RXdX 1s1_*Aa1a

9d

U*QKK2Mi, i?2 iBiH2 Q7 i?2 2t2`+Bb2 `272`b iQ  b2`pB+2 bvbi2K BM r?B+? 2+? +mb@ iQK2` 2Mi2`BM; i?2 bvbi2K `2[mB`2b  b2`pB+2 iBK2 σ iQ  mMB[m2 b2`p2` /2HBp2`BM; b2`pB+2 i mMBi iBK2 b HQM; i?2`2 Bb rQ`F iQ #2 /QM2- M/ r?2`2 i?2 +mbiQK2`bǶ ``BpH iBK2b 7Q`K  TQBMi T`Q+2bb N X h?2 [mMiBiv W (t) Bb i?2 iQiH `2KBMBM; rQ`F i iBK2 t- M/ W (0) Bb i?2 BMBiBH HQ/X Ua22 1tKTH2 8XRXjyXVV

*?Ti2` k SQBbbQM S`Q+2bb2b QM i?2 GBM2 SQBbbQM T`Q+2bb2b QM i?2 HBM2 #2HQM; Q7 +Qm`b2 iQ i?2 H`;2` +i2;Q`v Q7 bTiBH SQBb@ bQM T`Q+2bb2b i?i rBHH #2 i`2i2/ BM i?2 M2ti +?Ti2`X >Qr2p2`- i?2 TQBMi Q7 pB2r /QTi2/ BM i?Bb +?Ti2` Bb `/B+HHv /Bz2`2MiX h?2 T`BM+BTH `2bmHi Bb i?2 bKQQi?BM; 7Q`KmH- r?B+? QT2Mb i?2 rv iQ i?2 biQ+?biB+ +H+mHmb Q7 TQBMi T`Q+2bb2b rBi?  biQ+?biB+ BMi2MbBiv Q7 *?Ti2` 8X Ai rBHH #2 mb2/ BM a2+iBQM kX9 7Q`  T`QQ7 Q7 qiM#2Ƕb i?2Q`2K Q7 +?`+i2`BxiBQM Q7 M ?TT BM i2`Kb Q7 K`iBM;H2bX a2+iBQM kX8 b?Qrb ?Qr  +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J`FQp +?BM +M #2 /2b+`B#2/ BM i2`Kb Q7  biQ+?biB+ /Bz2`2MiBH 2[miBQM Ub/2V /`Bp2M #v SQBbbQM T`Q+2bb2bX UJQ`2 2tKTH2b QM i?2 b/2 TQBMi Q7 pB2r BM TQBMi T`Q+2bb i?2Q`v rBHH #2 ;Bp2M i p`BQmb TH+2b BM i?2 b2[m2HXV MQi?2` `2K`F#H2 `2T`2b2MiiBQM Q7  +QMiBMmQmb@iBK2 ?Q@ KQ;2M2Qmb J`FQp +?BM BM i2`Kb Q7  iBK2@b+H2/ ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Bb ;Bp2M BM a2+iBQM kXeX

kXR

*QmMiBM; S`Q+2bb M/ AMi2`pH a2[m2M+2

_2+HH i?i  bBKTH2 M/ HQ+HHv }MBi2 biQ+?biB+ TQBMi T`Q+2bb QM i?2 TQbBiBp2 ?H7@ HBM2 Bb  b2[m2M+2 {Tn }n≥1 Q7 TQbBiBp2 UTQbbB#Hv BM}MBi2V `M/QK p`B#H2b bm+? i?i- HKQbi bm`2HvUBV 0 < Tn < Tn+1 r?2M2p2` Tn < ∞ UbBKTHB+BivV- M/ UBBV limn↑∞ Tn = +∞ UHQ+H }MBi2M2bbVX h?Bb b2[m2M+2 Bb i?2 2p2Mi iBK2 b2[m2M+2X aQK2iBK2b- QM2 `272`b iQ Tn b i?2 n@i? dzTQBMiǴ Q7 i?2 TQBMi T`Q+2bbX h?2 BMi2;2`@pHm2/ `M/QK p`B#H2  N ((a, b]) = 1(a,b] (Tn ) n≥1

+QmMib i?2 2p2Mi iBK2b Q++m``BM; BM i?2 iBK2 BMi2`pH (a, b] ⊂ R+ X 6Q` ivTQ;`T?@ B+H bBKTHB+Biv- r2 Q++bBQMHHv /2MQi2 i?Bb p`B#H2 #v N (a, b]- i?mb QKBiiBM; i?2 2ti2`MH T`2Mi?2b2bX 6Q` t ≥ 0- H2i N (t) := N (0, t]. AM T`iB+mH`- N (0) = 0 M/ N (a, b] = N (b)−N (a)X aBM+2 i?2 BMi2`pH (0, t] Bb +HQb2/ QM i?2 `B;?i- i?2 i`D2+iQ`B2b UQ` bKTH2 Ti?bV t → N (t, ω) `2 `B;?i@+QMiBMmQmb-

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_2

9N

8y

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

MQM@/2+`2bBM;- ?p2 HBKBib QM i?2 H27i M/ DmKT QM2 mMBi mTr`/b i 2+? 2p2Mi iBK2 Q7 i?2 TQBMi T`Q+2bbX h?2 biQ+?biB+ T`Q+2bb {N (t)}t≥0 Bb +HH2/ i?2 +QmMiBM; T`Q+2bb Q7 i?2 TQBMi T`Q+2bbX aBM+2 i?2 b2[m2M+2 Q7 2p2Mi iBK2b +M #2 `2+Qp2`2/ 7`QK N - i?2 Hii2` HbQ `2+2Bp2b i?2 TT2HHiBQM dzTQBMi T`Q+2bbǴX q2 `2T2i .2}MBiBQM RXRXN BM i?2 T`iB+mH` b2iiBM; Q7 TQBMi T`Q+2bb2b QM i?2 `2H HBM2X .2}MBiBQM kXRXR  SQBbbQM T`Q+2bb QM R+ rBi? HQ+HHv }MBi2 BMi2MbBiv K2bm`2 ν Bb  TQBMi T`Q+2bb N QM R+ bm+? i?i (α) 7Q` HH k ∈ N+ - HH KmimHHv /BbDQBMi BMi2`pHb Ij := (aj , bj ] (1 ≤ j ≤ k)i?2 `M/QK p`B#H2b N (Ij ) (1 ≤ j ≤ k) `2 BM/2T2M/2Mi- M/ (β) 7Q` Mv BMi2`pH (a, b] ⊂ R+ - N (a, b] Bb  SQBbbQM `M/QK p`B#H2 Q7 K2M ν((a, b])X 1[mBpH2MiHv- 7Q` HH KmimHHv /BbDQBMi BMi2`pHb Ij = (aj , bj ] ⊂ R+ M/ HH `2H MmK#2`b uj U1 ≤ j ≤ kV E e

i

k

j=1

uj N (Ij )



 = exp

k  

e

iuj



− 1 ν((aj , bj ])

 .

j=1

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N (t) 

Zn

(t ≥ 0)

n=1

Bb +HH2/  +QKTQmM/ U?QKQ;2M2QmbV SQBbbQM T`Q+2bbX 1tKTH2 kXRX9, h?2 _BbF JQ/2H- hF2 RX PM2 Q7 i?2 }`bi bB;MB}+Mi mb2b Q7 SQBbbQM T`Q+2bb2b BM TTHB+iBQMb Q++m``2/ BM i?2 BMbm`M+2 `BbF KQ/2H Q7 *`Kû` M/ GmM/#2`;X AM i?2 BMbm`M+2 #mbBM2bb- QM2 Q7 i?2 +2Mi`H T`Q#H2Kb Bb i?2 2pH@ miBQM Q7 i?2 T`Q##BHBiv Q7 `mBM Q7  ;Bp2M BMbm`M+2 +QKTMvX h?2 biM/`/ KQ/2H 72im`2b i?2 +QKTQmM/ SQBbbQM T`Q+2bb Q7 .2}MBiBQM kXRXj- r?2`2 i?2 K`Fb ?p2 +QKKQM +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM G- bm+? i?i G(0) = 0- rBi? K2M μ < ∞ M/ p`BM+2 σ 2 < ∞X h?2 `BbF T`Q+2bb Bb /2}M2/ QM [0, ∞) #v

kXRX *PlLhAL: S_P*1aa L. ALh1_oG a1Zl1L*1 X(t) = ct −

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n=1

r?2`2 c Bb  TQbBiBp2 `2H +QMbiMiX h?2 BMi2`T`2iiBQM Bb i?i c Bb i?2 ;`Qbb `BbF T`2KBmK Ui?i Bb- i?2 `i2 Q7 BM+QK2 T2` mMBi iBK2V- Tn Bb i?2 iBK2 Q7 Q++m``2M+2 Q7  +HBK M/ Zn Bb i?2 bBx2 Q7 i?2 +HBKX G2i F N := σ(N (s) ; s ≥ 0) #2 i?2 σ@}2H/ `2+Q`/BM; HH i?2 2p2Mib Q7 N X _2+HH i?i i?2 ?TTb N1 , N2 , . . . `2 +HH2/ BM/2T2M/2Mi B7 i?2 σ@}2H/b F N1 , F N2 , . . . `2 BM/2T2M/2Mi- i?i Bb- B7 7Q` HH A1 ∈ F N1 , A2 ∈ F N2 , . . .- i?2 7KBHv Q7 2p2Mib (A1 , A2 , . . .) Bb BM/2T2M/2MiX h?2Q`2K kXRX8 G2i Ni (i ≥ 1) #2  7KBHv Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λi (i ≥ 1)X h?2MUBV irQ /BbiBM+i ?TTb Q7 i?Bb 7KBHv ?p2 MQ TQBMib BM +QKKQM- M/



UBBV B7 ∞ i=1 λi = λ < ∞- i?2 bmK N := i=1 Ni Bb M ?TT rBi? BMi2MbBiv λX S`QQ7X UBV Ai bm{+2b iQ T`Qp2 i?Bb 7Q` irQ ?TTb- M/ i?2 `2bmHi Bb i?2M  /B`2+i +QMb2[m2M+2 Q7 h?2Q`2K RXkX9X 6Q` UBBV- rBi 7Q` h?2Q`2K jXRXkX  L2ti `2bmHi Bb +HH2/ i?2 +QKT2iBiBQM i?2Q`2K 7Q` ?TTbX

∞ h?2Q`2K kXRXe AM i?2 bBimiBQM Q7 h?2Q`2K i=1 λi = λ < ∞ ∞ kXRX8- r?2`2 /2MQi2 #v Z i?2 }`bi 2p2Mi iBK2 Q7 N := i=1 Ni M/ #v J i?2 BM/2t Q7 i?2 ?TT `2bTQMbB#H2 7Q` BiX AM T`iB+mH`- Z Bb i?2 }`bi 2p2Mi Q7 NJ X h?2M J M/ Z `2 BM/2T2M/2Mi- P (J = i) = λλi M/ Z Bb M 2tTQM2MiBH `M/QK p`B#H2 Q7 K2M λ−1 X X1

1

J=2

3

X2 =Z X3

S`QQ7X *HH X1 , X2 , . . . i?2 }`bi 2p2Mi iBK2b Q7 N1 , N2 , . . .X h?2b2 `2 BM/2T2M/2Mi 2tTQM2MiBH p`B#H2b rBi? `2bT2+iBp2 T`K2i2`b λ1 , λ2 , . . .X aBM+2 i?2b2 `M/QK p`B#H2b `2 #bQHmi2Hv +QMiBMmQmb M/ BM/2T2M/2Mi- i?2 BM/2t J Bb HKQbi bm`2Hv mMK#B;mQmbHv /2}M2/X q2 ?p2P (J = i, Z ≥ a) = P (Xi ≥ a, Xj ≥ Xi 7Q` HH j = i)  ∞ = λi e−λi s P (Xj ≥ s 7Q` HH j = i) ds a  ∞  ∞ λi = λi e−λi s Πj=i e−λj s ds = λi e−λs ds = e−λa . λ a a

8k

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

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(β) i?2 7KBHv Sτ Nα Uα ∈ AV Bb M BM/2T2M/2Mi 7KBHv Q7 ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λα (α ∈ A)X S`QT2`iB2b (α) M/ (β) +QMbiBimi2 r?i Bb +HH2/ i?2 bi`QM; J`FQp T`QT2`iv Q7 ?TTbX S`QQ7X h?2 T`QQ7 Bb ;Bp2M 7Q`  bBM;H2 TQBMi ?TT N X h?2 ;2M2`H +b2 Bb +QKTH2i2Hv bBKBH`- QM+2 Bi Bb Q#b2`p2/ i?i i?2 BM/2T2M/2M+2 bii2K2Mib +QM+2`M QMHv }MBi2 +QHH2+iBQMb Q7 TQBMi T`Q+2bb2b Q7 i?2 7KBHv Nα Uα ∈ AVX AM i?2 T`QQ7- i?2 mb2 Q7 h?2Q`2Kb kXRX3 M/ kXRXN #2HQr Bb BKTHB+BiX Uh?2b2 `2bmHib HHQr mb iQ `2bi`B+i Qm` ii2MiBQM iQ }MBi2@/BK2MbBQMH /Bbi`B#miBQMb BM Q`/2` iQ T`Qp2 2Bi?2` B/2MiBiv Q7 i?2 /Bbi`B#miBQMb Q7 biQ+?biB+ T`Q+2bb2b Q` i?2B` BM/2T2M/2M+2XV h?2Q`2K kXRX3 G2i E #2  SQHBb? bT+2 M/ H2i E := B(E) #2 Bib "Q`2H σ@}2H/X G2i Q = {Q(t1 ,...,tk ) ; k ≥ 1, t1 , . . . , tk ∈ R} #2  7KBHv Q7 T`Q##BHBiv /Bbi`B#miBQMb QM (E k , E k ) biBb7vBM; i?2 7QHHQrBM; +QKTiB#BHBiv +QM/BiBQMb C1 X 6Q` HH (t1 , . . . , tk ) ∈ Rk - M/ Mv T2`KmiiBQM σ QM Rk Qσ(t1 ,...,tk ) = Qt1 ,...,tk ◦ σ −1 .

UkXRV

C2 X 6Q` HH (t1 , . . . , tk , tk+1 ) ∈ Rk+1 M/ HH A ∈ E k Q(t1 ,...,tk ) (A) = Q(t1 ,...,tk+1 ) (A × E).

UkXkV

kXRX *PlLhAL: S_P*1aa L. ALh1_oG a1Zl1L*1

8j

h?2M i?2`2 2tBbib  mMB[m2 T`Q##BHBiv K2bm`2 P QM i?2 +MQMB+H K2bm`#H2 bT+2 (E R , E R ) bm+? i?i i?2 +QQ`/BMi2 T`Q+2bb {πt }t∈R /KBib i?2 }MBi2@/BK2MbBQMH /Bbi`B#miBQM QX h?2Q`2K kXRXN hrQ biQ+?biB+ T`Q+2bb2b {X(t)}t∈R M/ {Y (t)}t∈R - rBi? pHm2b BM (E, E)- `2 BM/2T2M/2Mi B7 7Q` HH t1 , . . . , tk ∈ R M/ HH s1 , . . . , s ∈ R- i?2 p2+iQ`b (X(t1 ), . . . , X(tk )) M/ (Y (s1 ), . . . , Y (s )) `2 BM/2T2M/2MiX _2im`MBM; iQ i?2 T`QQ7 Q7 h?2Q`2K kXRXd- r2 }`bi b?Qr i?i Bi bm{+2b iQ T`Qp2 i?i 7Q` HH BMi2;2`b k, - HH u1 , . . . , uk - v1 , . . . , v - w ∈ R- HH KmimHHv /BbDQBMi BMi2`pHb (aj , bj ] U1 ≤ j ≤ kV- HH KmimHHv /BbDQBMi BMi2`pHb [cj , dj ] U1 ≤ j ≤ V M/ Mv `2H@pHm2/ `M/QK p`B#H2 Z i?i Bb G@K2bm`#H2- Bi ?QH/b i?i  k    τ E ei( j=1 uj (Sτ N )(aj ,bj ]+ j=1 vj N (cj ,dj ]+wZ ) 1{τ 1 aJPPh>AL: 6P_JlG

88

h?2 AMi2`@2p2Mi a2[m2M+2 h?2 b2[m2M+2 {Sn }n≥1 /2}M2/ #v T0 = 0 M/ Sn := Tn − Tn−1 U= ∞ B7 Tn−1 = ∞V Bb +HH2/ i?2 BMi2`@2p2Mi b2[m2M+2 Q`- BM i?2 TT`QT`Bi2 +QMi2tii?2 BMi2`@``BpH b2[m2M+2X h?2 bi`QM; J`FQp T`QT2`iv vB2H/b i?2 T`Q##BHBbiB+ /2b+`BTiBQM Q7 M ?TT BM i2`Kb Bib BMi2`@2p2Mi iBK2bX h?2Q`2K kXRXRR h?2 BMi2`@2p2Mi b2[m2M+2 {Sn }n≥1 Q7 M ?TT QM i?2 TQbBiBp2 ?H7@HBM2 rBi? i?2 BMi2MbBiv λ > 0 Bb BB/- rBi? 2tTQM2MiBH /Bbi`B#miBQM Q7 T`K2i2` λ- i?i Bb P (Sn ≤ t) = 1 − e−λt M/ BM T`iB+mH` E[Sn ] = λ−1 X S`QQ7X TTHv h?2Q`2K kXRXd rBi? G = (Ω, ∅) M/ τ = Tn - iQ Q#iBM i?i i?2 }`bi 2p2Mi iBK2 Q7 STn N - MK2Hv Sn+1 - Bb BM/2T2M/2Mi Q7 N Tn UM/ BM T`iB+mH` Q7 S1 , . . . , Sn V M/ Bb M 2tTQM2MiBH `M/QK p`B#H2 rBi? T`K2i2` λX h?Bb #2BM; i`m2 7Q` HH n ≥ 1- i?2 T`QQ7 Bb /QM2X 

kXk

h?2 aKQQi?BM; 6Q`KmH

_2+HH bQK2 MQiiBQM M/ i2`KBMQHQ;vX  ?BbiQ`v QM (Ω, F) Bb- #v /2}MBiBQM-  MQM@ /2+`2bBM; 7KBHv {Ft }t∈R Q7 bm#@σ@}2H/b Q7 FX G2i F∞ := ∨t∈R Ft #2 i?2 bKHH2bi σ@}2H/ +QMiBMBM; 7Q` HH t ∈ R HH b2ib Q7 Ft X  biQ+?biB+ T`Q+2bb {X(t)}t∈R Bb bB/ iQ #2 Ft @/Ti2/ B7 i?2 `M/QK p`B#H2 X(t) Bb Ft @K2bm`#H2 7Q` HH t ∈ RX 6Q`  ;Bp2M TQBMi T`Q+2bb N QM R- M/ t ∈ R- /2}M2 i?2 σ@}2H/ FtN := σ(N (a, b] ; (a, b] ⊆ (−∞, t]) `2+Q`/BM; i?2 2p2Mib Q7 N mT iQ iBK2 tX Mv ?BbiQ`v {Ft }t∈R bm+? i?i FtN ⊆ Ft Ut ∈ RV Bb +HH2/  ?BbiQ`v Q7 N X h?2 ?BbiQ`v {FtN }t∈R Bb +HH2/ i?2 BMi2`MH ?BbiQ`v Q7 N X AM i?2 #Qp2 /2}MBiBQMb- R Bb i?2 BM/2t b2i- #mi Q7 +Qm`b2 bBKBH` /2}MBiBQMb ?QH/ 7Q` T`Q+2bb2b M/ ?BbiQ`B2b BM/2t2/ #v Mv bm#b2i Q7 R UBM i?Bb b2+iBQM, R+ VX h?2Q`2K kXkXR G2i {Ft }t≥0 #2  ?BbiQ`v Q7 N -  SQBbbQM T`Q+2bb QM R+ rBi? BMi2MbBiv K2bm`2 νX amTTQb2 KQ`2Qp2` i?i 7Q` HH a > 0 M/ HH c, d (a ≤ c ≤ d)N ((c, d]) Bb BM/2T2M/2Mi Q7 Fa X G2i {Z(t)}t≥0 #2  H27i@+QMiBMmQmb +QKTH2t biQ+?biB+ T`Q+2bb /Ti2/ iQ {Ft }t≥0 X amTTQb2 i?i i H2bi QM2 Q7 i?2 irQ +QM/BiBQMb #2HQr Bb biBb}2/, UBV {Z(t)}t≥0 Bb `2H MQM@M2;iBp2X UBBV {Z(t)}t≥0 Bb +QKTH2t@pHm2/ M/ E h?2M-

 E (0,∞)

 (0,∞)

 Z(s)N (ds) = E 0

 |Z(t)| ν(dt) < ∞X



Z(s) ν(ds) .

UkXeV

8e

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

S`QQ7X X q2 }`bi i`2i i?2 +b2 r?2`2 {Z(t)}t≥0 Bb `2H- MQM@M2;iBp2 M/ #QmM/2/M/ T`Qp2 i?i 

 T

E Z(s)N (ds) = E Z(s) ν(ds) , () 0

(0,T ]

r?2`2 T < ∞X G2i 7Q` HH n ≥ 1- HH ω ∈ Ω M/ HH t ≥ 0Zn (t, ω) :=

n −1 2

  Z kT 2−n , ω 1(kT 2−n ,(k+1)T 2−n ] (t).

UkXdV

k=0

"v i?2 H27i@+QMiBMmBiv ?vTQi?2bBb- 7Q` HH ω ∈ Ω M/ HH t ∈ [0, T ]lim Zn (t, ω) = Z(t, ω).

n↑∞

q2 }`bi +?2+F i?i UV Bb i`m2 r?2M {Z(t)}t≥0 Bb `2TH+2/ #v Bib TT`QtBKiBQM {Zn (t)}t≥0 X AM/22/- i?2 H27i@?M/ bB/2 Q7 i?Bb 2[mHBiv Bb i?2M 2n −1  

   −n −n −n E N ((kT 2 , (k + 1)T 2 ]) Zn (t)N (dt) = E Z kT 2 (0,T ]

=

k=0 n −1 2

  E Z kT 2−n N ((kT 2−n , (k + 1)T 2−n ]) .

k=0

"mi Z (kT 2−n ) Bb BM/2T2M/2Mi Q7 N ((kT 2−n , (k + 1)T 2−n ]) bBM+2 Bi Bb FkT 2−n @ K2bm`#H2- bQ i?i i?2 Hbi i2`K BM i?2 #Qp2 +?BM Q7 2[mHBiB2b Bb 2[mH iQ n −1 2

  E Z kT 2−n E N ((kT 2−n , (k + 1)T 2−n ])

k=0

=

n −1 2

  E Z kT 2−n ν((kT 2−n , (k + 1)kT 2−n ])

k=0 2n −1 

=E

k=0  T

=E



Z kT 2

−n



 ν((kT 2

−n

−n

, (k + 1)kT 2

])

Zn (s) ν(ds) .

0

h?2`27Q`2-



 Zn (s)N (ds) = E

E (0,T ]

T 0

Zn (s) ν(ds) .

UkX3V

T .2MQiBM; #v K i?2 mTT2`  #QmM/ Q7 Z(t,  ω)- r2 ?p2 i?i 0 Zn (t) ν(dt) ≤ Kν((0, T ]) < ∞ M/ E (0,T ] Zn (t)N (dt) ≤ KE [N (T )] = Kν((0, T ]) < ∞X h?2`27Q`2- #v /QKBMi2/ +QMp2`;2M+2- H2iiBM; n ↑ ∞ BM #Qi? bB/2b Q7 UkX3V- r2 Q#iBM UVX

kXjX SPAaaPL J_hAL:G1a L. ahP*>ahA* ALh1:_Ga "X q2 MQr i`2i i?2 MQM@M2;iBp2 M/ ⎧ ⎪ ⎨1 gK (x) = 0 ⎪ ⎩ −x + K + 1

8d

mM#QmM/2/ +b2X .2}MBM; B7 x ≤ K, B7 x ≥ K + 1, B7 K ≤ x ≤ K + 1,

i?2 T`Q+2bb {gK (Z(t))}t≥0 biBb}2b i?2 +QM/BiBQMb 7Q` UV M/ i?2`27Q`2 

 gK (Z(s))N (ds) = E

E

T

gK (Z(s)) ν(ds) ,

0

(0,T ]

r?B+? ;Bp2b UV b K → ∞- #v KQMQiQM2 +QMp2`;2M+2X Ai i?2M bm{+2b iQ H2i T ↑ ∞ M/ iQ BMpQF2 i?2 KQMQiQM2 +QMp2`;2M+2 i?2Q`2K ;BM iQ Q#iBM UkXeV BM i?2 MQM@ M2;iBp2 mM#QmM/2/ +b2X LQi2 i?i i?2 [mMiBiB2b BM UkXeV +M MQr p2`v r2HH #2 BM}MBi2X *X h?2 `2H@pHm2/ +b2 7QHHQrb 2bBHv- #v }`bi +QMbB/2`BM; b2T`i2Hv i?2 TQbB@ iBp2 M/ M2;iBp2 T`ib Q7 i?2 BMi2;`M/X h?2 +QKTH2t +b2 Bb  /B`2+i +QMb2[m2M+2 Q7 i?2 `2H +b2 r?2M QM2 +QMbB/2`b b2T`i2Hv i?2 `2H M/ BK;BM`v T`ibX 

kXj

SQBbbQM J`iBM;H2b M/ aiQ+?biB+ AMi2;`Hb

G2i N #2  SQBbbQM T`Q+2bb QM R+ rBi? i?2 BMi2MbBiv K2bm`2 νX h?2M M (t) := N (0, t] − ν((0, t]) (t ≥ 0) /2}M2b M FtN @K`iBM;H2 {M (t)}t≥0 XR AM/22/ M (t) Bb BMi2;`#H2 7Q` HH t ≥ 0 M/ #v i?2 BM/2T2M/2M+2 T`QT2`iv Q7 SQBbbQM T`Q+2bb2b- 7Q` HH (a, b] ⊂ R+ E N (a, b] | FaN = E [N (a, b]] = ν((a, b]) . G`;2 .2pBiBQMb aBM+2 i?2 2tT2+iiBQM Q7 i?2 SQBbbQM K`iBM;H2 Bb MmHH- QM2 Kv #2 BMi2`2bi2/ BM i?2 H`;2 /2pBiBQMb 7`QK Bib K2M 0X h?2 M2ti i?2Q`2K ;Bp2b M 2biBKi2 i?i rBHH #2 mb2/ Hi2` QMX h?2Q`2K kXjXR G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv 1X h?2M- 7Q` Mv T > 0 M/ ε > 0 bm+? i?i Tε ≤ 1

 P

sup |N (t) − t| ≥ ε

≤ 2e−T h( T ) , ε

t∈[0,T ]

r?2`2 h(x) := (1 + x) log(1 + x) − xX R

a2+iBQM Xj ;Bp2b i?2 /2}MBiBQM M/ #bB+ T`QT2`iB2b Q7 K`iBM;H2bX

83

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

S`QQ7X q2 b?HH mb2 EQHKQ;Q`QpǶb bm#K`iBM;H2 BM2[mHBiv Uh?2Q`2K XjX9V r?B+? i2HHb mb i?i B7 {S(t)}t∈[0,T ] Bb  bm#K`iBM;H2,   1 sup |S(t)| ≥ a ≤ E [|S(T )|] . a t∈[0,T ] 6Q` θ > 0 P



sup |N (t) − t| ≥ ε t∈[0,T ]



≤P

sup (N (t) − t) ≥ ε 

=P



t∈[0,T ]

sup e

 sup (t − N (t)) ≥ ε

+P

t∈[0,T ]

 θ(N (t)−t)

≥e

θε



 +P

t∈[0,T ]



sup e

θ(t−N (t))

≥e

θε

=A+B.

t∈[0,T ]

aBM+2 {N (t) − t}t∈[0,T ] M/ {t − N (t)}t∈[0,T ] `2 K`iBM;H2b M/ bBM+2 θ > 0{eθ(N (t)−t) }t∈[0,T ] M/ {eθ(t−N (t) }t∈[0,T ] `2 bm#K`iBM;H2b Ub +QMp2t 7mM+iBQMb Q7 K`iBM;H2bV iQ r?B+? r2 TTHv EQHKQ;Q`QpǶb BM2[mHBiv iQ Q#iBM A ≤ e−θε E eθ(N (T )−T ) M/ B ≤ e−θε E eθ(T −N (T )) . hFBM; BMiQ ++QmMi i?2 7+i i?i N (T ) Bb  SQBbbQM p`B#H2 rBi? K2M T θ θ A ≤ e−θ(ε+T ) E eθN (T ) = e−θ(ε+T ) eT (e −1) = e−θ(ε+T )+T (e −1) .   h?Bb [mMiBiv Bb KBMBKBx2/ KQM; HH TQbBiBp2 pHm2b Q7 θ #v θ = log 1 + Tε X ε ε h?2`27Q`2 A ≤ e−T h( T ) M/- bBKBH`Hv- B ≤ e−T h(− T ) X h?2 +QM+HmbBQM 7QHHQrb #v Q#b2`pBM; i?i h(−x) ≥ h(x) r?2M x ∈ [0, 1]X



aiQ+?biB+ BMi2;`Hb rBi? `2bT2+i iQ i?2 SQBbbQM K`iBM;H2 aiQ+?biB+ BMi2;`Hb rBi? `2bT2+i iQ  SQBbbQM K`iBM;H2 vB2H/ K`iBM;H2bX JQ`2 T`2+Bb2Hv- r2 ?p2 i?2 7QHHQrBM; `2bmHi- r?Qb2 T`QQ7 Bb H27i b 1t2`+Bb2 kXdXNX h?2Q`2K kXjXk G2i {Ft }t≥0 #2  ?BbiQ`v Q7 N - M ?TT QM R+ Q7 BMi2MbBiv K2bm`2 νX amTTQb2 KQ`2Qp2` i?i 7Q` HH a > 0 M/ HH c, d (a ≤ c ≤ d)- N (c, d] Bb BM/2T2M/2Mi Q7 Fa X G2i {Z(t)}t≥0 #2  +QKTH2t biQ+?biB+ T`Q+2bb /Ti2/ iQ {Ft }t≥0 M/ rBi? H27i@+QMiBMmQmb i`D2+iQ`B2bX X A7

 E

t

|Z(s)| ν(ds) < ∞

(t ≥ 0) ,

()

0

i?2M



 (0,t]



t

Z(s) N (ds) −

Z(s) dM (s) :=

Y (t) :=

(0,t]

Z(s) ν(ds)

()

0

Bb  r2HH@/2}M2/ [mMiBiv UMQi  ∞ − ∞ 7Q`KV M/ i?2 biQ+?biB+ T`Q+2bb {Y (t)}t≥0 Bb M Ft @K`iBM;H2X

kX9X qhL"1Ƕa *>_*h1_AwhAPL

8N

"X G2i i?BM;b #2 b #Qp2- 2t+2Ti 7Q` +QM/BiBQM UV- MQr `2TH+2/ #v i?2 HQ+H BMi2;`#BHBiv Q7 {Z(t)}t≥0 - i?i Bb- HKQbi@bm`2Hv  ∞ |Z(t)| ν(dt) < ∞ 7Q` HH t ≥ 0 . 0

h?2M Y (t) ;Bp2M #v UV Bb  r2HH@/2}M2/ [mMiBiv UMQi  ∞ − ∞ 7Q`KV M/ i?2 biQ+?biB+ T`Q+2bb {Y (t)}t≥0 Bb  HQ+H Ft @K`iBM;H2X

kX9

qiM#2Ƕb *?`+i2`BxiBQM

h?2Q`2K kX9XR Uk V G2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ - M/ H2i {Ft }t≥0 #2  ?BbiQ`v Q7 N X G2i ν #2  /Bzmb2 HQ+HHv }MBi2 K2bm`2 QM R+ X amTTQb2 i?i 

 T

E Z(t)N (dt) = E Z(t) ν(dt) UkXNV 0

(0,T ]

?QH/b i`m2 7Q` HH T > 0- M/ 7Q` HH MQM@M2;iBp2 `2H@pHm2/ H27i@+QMiBMmQmb biQ+?biB+ T`Q+2bb2b {Z(t)}t≥0 /Ti2/ iQ {Ft }t≥0 X h?2M- N Bb  SQBbbQM T`Q@ +2bb rBi? i?2 BMi2MbBiv K2bm`2 ν- M/ 7Q` Mv BMi2`pH (a, b] ∈ R+ - N (a, b] Bb BM/2T2M/2Mi Q7 Fa X S`QQ7X h?2 T`QQ7 ;Bp2M #2HQr Bb ivTB+H Q7 i?2 biQ+?biB+ +H+mHmb Q7 ?TTbXj 6B`biMQi2 i?i 7Q`KmH UkXNV ?QH/b i`m2 7Q` HH T > 0- M/ 7Q` HH +QKTH2t@pHm2/ biQ+?b@ iB+ T`Q+2bb2b {Z(t)}t≥0 rBi? H27i@+QMiBMmQmb i`D2+iQ`B2b- /Ti2/ iQ {Ft }t≥0 M/ T bm+? i?i E 0 |Z(t)| ν(dt) < ∞ UbK2 `;mK2Mib b BM T`i * Q7 i?2 T`QQ7 Q7 h?2Q`2K kXkXRVX q2 b?Qr i?i Bi Bb 2MQm;? iQ T`Qp2 i?i iu E 1A eiuN (a,b] = P (A) e(e −1)ν((a,b]) , () r?2`2 (a, b] ⊆ R+ M/ u ∈ R `2 `#Bi``v- M/ A Bb M `#Bi``v 2p2Mi BM Fa X AM 7+i- rBi? A = Ω BM UVE[eiuN (a,b] ] = e(e

iu −1)ν((a,b])

,

r?B+? b?Qrb i?i N (a, b] Bb  SQBbbQM `M/QK p`B#H2 Q7 K2M ν((a, b])X 1[mHBiv UV i?2M `2/b E[1A eiuN (a,b] ] = P (A)E[eiuN (a,b] ] , 7`QK r?B+? Bi 7QHHQrb i?i N (a, b] M/ A `2 BM/2T2M/2MiX aBM+2 A Bb M `#Bi``v 2p2Mi BM Fa - r2 ?p2 T`Qp2/ i?i N (a, b] Bb BM/2T2M/2Mi Q7 Fa M/- bBM+2 Fa ⊇ FaN i?i N ?b BM/2T2M/2Mi BM+`2K2MibX 6Q` i?2 T`QQ7 Q7 UV- +QMbB/2` i?2 T`Q+2bb k

(qiM#2- RNe9)X h?Bb T`QQ7 U("`ûKm/- RNd8 #)V BKBii2b i?2 T`QQ7 Q7 GûpvǶb +?`+i2`BxiBQM Q7 i?2 "`QrMBM KQiBQM ;Bp2M BM (EmMBi M/ qiM#2- RNed)- h?KX kXj- TX kRdX h?2 BMi2`2bi Q7 i?Bb T`QQ7 HB2b BM i?2 2bv 2ti2MbBQMb Q7 i?2 i?2Q`2K i?i Bi HHQrb- BM+Hm/BM; i?2 T`2b2Mi QM2X a22 7Q` BMbiM+2 1t2`+Bb2 kXdXRk M/ h?2Q`2K 8XdXRX j

ey

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1 X(t) := 1A eiuN (a,t]

(t ≥ a) ,

M/ Q#b2`p2 i?i Bi Bb TB2+2rBb2 +QMbiMi M/ i?i HH Bib iBK2b Q7 /Bb+QMiBMmBiv `2 HQ+i2/ i i?2 2p2Mi iBK2b Q7 N X h?2`27Q`2 X(b) = X(a) + {X(Tn ) − X(Tn− )}1(a,b] (Tn ). n≥1

"mi X(0) = 1A M/- 7Q` Mv Tn ∈ (a, b] Uri+? i?2 T`2Mi?2b2b M/ i?2 b[m`2 #`+F2ibVX(Tn ) = eiuN (a,Tn ] = eiu(N (a,Tn )+1) = eiuN (a,Tn ) eiu = X(Tn− ) eiu . h?2`27Q`2X(b) = 1A +



X(Tn− )(eiu − 1)1(a,b] (Tn )

n≥1

Q`- BM i?2 BMi2;`H MQiiBQM X(s−)(eiu − 1) N (ds).

X(b) = 1A +

UkXRyV

(a,b]

aBM+2 Z(t) = X(t−)(eiu − 1) /2}M2b  #QmM/2/ H27i@+QMiBMmQmb +QKTH2t@pHm2/ biQ+?biB+ T`Q+2bb /Ti2/ iQ {Ft }t≥0 - 2[mHBiv UkXNV ?QH/b i`m2 #v ?vTQi?2bBbX h?2`27Q`2 

 b

iu iu E X(s−)(e − 1)N (ds) = E X(s−)(e − 1) ν(ds) . a

(a,b]

LQr- 7Q` 2+? ω ∈ Ω- X(t−, ω) = X(t, ω) 2t+2Ti QM  +QmMi#H2 b2i- M/ i?2`27Q`2 QM2 Kv `2TH+2 X(t−) #v X(t) BM i?2 `B;?i@?M/ bB/2 Q7 i?2 #Qp2 2[mHBiv U`2+HH i?i ν Bb bbmK2/ /Bzmb2VX hFBM; 2tT2+iiBQMb BM UkXRyV i?2`27Q`2 vB2H/b 

b

E[X(s)](eiu − 1) ν(ds) .

E[X(b)] = P (A) + a

h?Bb #2BM; i`m2 7Q` HH b ≥ a- UV 7QHHQrb Uh?2Q`2K RXeX8VX



 KQbi mb27mH ;2M2`HBxiBQM Q7 qiM#2Ƕb i?2Q`2K rBHH #2 ;Bp2M BM a2+iBQM 8Xd- h?2Q`2K 8XdXRX h?2 7QHHQrBM; +Q`QHH`v Q7 h?2Q`2K kX9XR Bb i?2 Q`B;BMH p2`bBQM Q7 qiM#2Ƕb `2bmHi- r?B+? rb ;Bp2M BM i2`Kb Q7 K`iBM;H2bX *Q`QHH`v kX9Xk G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ bm+? i?i i?2 T`Q+2bb M (t) := N ((0, t]) − ν((0, t]) (t ≥ 0) Bb M Ft @K`iBM;H2- r?2`2 ν Bb  HQ+HHv }MBi2 /Bzmb2 K2bm`2 QM R+ X h?2M N Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? i?2 BMi2MbBiv K2bm`2 ν- M/ 7Q` Mv BMi2`pH (a, b] ∈ R+ - N (a, b] Bb BM/2T2M/2Mi Q7 Fa X

kX9X qhL"1Ƕa *>_*h1_AwhAPL

eR

S`QQ7X "v h?2Q`2K kX9XR- Bi bm{+2b iQ b?Qr i?i 7Q` HH T > 0 M/ HH MQM@M2;iBp2 #QmM/2/ `2H@pHm2/ H27i@+QMiBMmQmb biQ+?biB+ T`Q+2bb2b {Z(t)}t≥0 /Ti2/ iQ {Ft }t≥0



 T

Z(t) N (dt) = E

E

Z(t) ν(dt)t .

()

0

(0,T ]

1[mHBiv UV Bb i`m2 7Q` Z(t, ω) := 1A (ω) 1(a,b] (t) 7Q` Mv BMi2`pH (a, b] ⊂ R+ M/ Mv A ∈ Fa - bBM+2 BM i?Bb +b2- UV `2/b E [1A N (a, b]] = E [1A ν((a, b])] , i?i Bb- bBM+2 A Bb `#Bi``v BM Fa E [N (a, b] | Fa ] = ν((a, b]), r?B+? Bb i?2 K`iBM;H2 ?vTQi?2bBbX h?2 T`QQ7 i?2M 7QHHQrb i?2 bK2 HBM2b b i?i Q7 h?2Q`2K kX9XR pB i?2 TT`QtBKiBQM UkXdVX  *Q`QHH`v kX9Xj G2i N #2  TQBMi T`Q+2bb QM R+ rBi? HQ+HHv }MBi2 /Bzmb2 BMi2MbBiv K2bm`2 ν bm+? i?i 7Q` HH #QmM/2/ /BbDQBMi b2ib A M/ B BM B(R+ )- N (A) M/ N (B) `2 BM/2T2M/2Mi p`B#H2bX h?2M N Bb  SQBbbQM T`Q+2bbX S`QQ7X h?2 bbmKTiBQMb BKTHv i?i 7Q` HH s, t U0 ≤ s ≤ tV i?2 `M/QK p`B#H2 N ((s, t]) Bb BM/2T2M/2Mi Q7 FsN - M/ i?2`27Q`2 E N ((s, t]) | FsN = E [N ((s, t])] = ν((s, t]) = E ν((s, t]) | FsN , M/ i?2`27Q`2 Bb M

M (t) := N ((0, t]) − ν((0, t])

(t ≥ 0)

FtN @K`iBM;H2X



ai`QM; J`FQp S`QT2`iv Q7 i?2 ?TT pB qiM#2Ƕb i?2Q`2K h?2 bi`QM; J`FQp T`QT2`iv Q7 ?TTb +M #2 Q#iBM2/ b  +QMb2[m2M+2 Q7 qiM@ #2Ƕb i?2Q`2KX 6Q` BMbiM+2- H2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ M/ H2i τ #2  }MBi2 FtN @biQTTBM; iBK2X .2}M2 i?2 TQBMi T`Q+2bb N τ QM R+ #v N τ (0, t] := N (τ, τ + t) (t ≥ 0) . h?2M N τ Bb M ?TT BM/2T2M/2Mi Q7 FτN X S`QQ7X G2i (a, b] ⊂ R+ M/ A ∈ FτN+a X h?2 T`Q+2bb Z(t) = 1A 1(τ +a,τ +b] (t) Bb H27i@+QMiBMmQmb M/ /Ti2/ iQ {FtN }t≥0 M/ i?2`27Q`2

ek

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1 

 Z(t) N (dt) = E

E R+

Z(t) λ dt ,

R+

r?B+? `2/b BM i?Bb T`iB+mH` +b2 E [1A N τ (a, b]] = E [1A λ(b − a)] . aBM+2 i?Bb ?QH/b 7Q` HH (a, b] ⊂ R+ M/ HH A ∈ FτN+a - {N τ (0, t] − λt}t≥0 Bb M FτN+t @K`iBM;H2X h?2 +QM+HmbBQM i?2M 7QHHQrb 7`QK *Q`QHH`v kX9XkX 

kX8 ?K+b M/ aiQ+?biB+ .Bz2`2MiBH 1[miBQMb h?Bb bm#b2+iBQM BMi`Q/m+2b i?2 MQiBQM Q7 biQ+?biB+ /Bz2`2MiBH 2[miBQMb /`Bp2M #v SQBbbQM T`Q+2bb2b #v K2Mb Q7 i?2 +HbbB+H 2tKTH2 Q7 +QMiBMmQmb@iBK2 ?Q@ KQ;2M2Qmb J`FQp +?BMb U?K+VX9 Ai +QM+2`Mb i?2 T`Q#H2K Q7 `2HBxiBQM Q7 M BM}MBi2bBKH ;2M2`iQ`- i?i Bb- i?2 T`Q#H2K Q7 bbQ+BiBM; iQ  ;2M2`iQ` A  +QMiBMmQmb@iBK2 ?K+ /KBiiBM; Bi b M BM}MBi2bBKH ;2M2`iQ`X JQ`2 T`2+Bb2HvH2i E #2  +QmMi#H2 bT+2 M/ H2i A = {qij }i,j∈E #2  Ki`Bt Q7 `2H MmK#2`b BM/2t2/ #v E- bm+? i?i 7Q` HH i, j ∈ E bm+? i?i i = jqi ∈ [0, ∞],

qij ∈ [0, ∞) .

A Bb i?2M +HH2/  ;2M2`iQ` UMQi v2i M BM}MBi2bBKH ;2M2`iQ`VX bbmK2 KQ`2Qp2` i?i i?Bb ;2M2`iQ` Bb bi#H2 M/ +QMb2`piBp2- i?i Bb qi < ∞, qi = qik , k∈E k=i

r?2`2 qi = −qii X LQi2 HbQ i?i r?2M bT2FBM; #Qmi  ;2M2`iQ` A- QM2 /Q2b MQi `272` iQ  ?QKQ;2M2Qmb +QMiBMmQmb@iBK2 J`FQp +?BMX >Qr2p2`- r2 +M +QMbi`m+i  +QMiBMmQmb@iBK2 ?K+ /KBiiBM; i?2 ;Bp2M ;2M2`iQ` b M BM}MBi2bBKH ;2M2`iQ`X hQ /Q i?Bb- bi`i rBi?  7KBHv Ni,j Ui, j ∈ E, i = j) Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b qij Ui, j ∈ E, i = j) M/ BM/2T2M/2Mi Q7 i?2 BMBiBH bii2 X(0) ∈ EX h?2 T`Q+2bb {X(t)}t≥0 Bb +QMbi`m+i2/ b  DmKT T`Q+2bb, X(t) = Xn 7Q` t ∈ [τn , τn+1 ) , r?2`2 i?2 b2[m2M+2 {τn , Xn }n≥0 Bb /2}M2/ `2+m`bBp2Hv b 7QHHQrbX 6B`bi- τ0 = 0X0 = X(0)X "27Q`2 T`Q+22/BM; iQ i?2 ;2M2`H bi2T- `2+HH i?i Sa N Ua ∈ R+ V Bb #v /2}MBiBQM i?2 TQBMi T`Q+2bb N QM i?2 HBM2 b?B7i2/ #v a- i?i Bb- 7Q` Mv "Q`2H b2i C Q7 i?2 HBM2(Sa N )(C) := N (C + a) . A7 τn < ∞ M/ Xn = X(τn ) = i ∈ E- i?2M τn+1 − τn Bb i?2 }`bi 2p2Mi Q7 i?2 7KBHv Q7 ?TTb Sτn Nij Uj ∈ E , j = i)X A7 τn+1 − τn < ∞- Xn+1 Bb i?2 BM/2t k = i bm+? 9

RkX

JQ`2 QM i?2 i?2K2 Q7 biQ+?biB+ /Bz2`2MiBH 2[miBQMb /`Bp2M #v TQBMi T`Q+2bb2b BM *?Ti2`

kX8X >J*a L. ahP*>ahA* .A661_1LhAG 1ZlhAPLa

ej

i?i Sτn Nik Bb i?2 }`bi KQM; i?2 ?TTb Sτn Nij Uj ∈ E , j = iV i?i T`Q/m+2b M 2p2MiX Ai Kv Q++m` i?i τn+1 − τn = ∞X h?Bb Bb i?2 +b2 B7 M/ QMHv B7 qi = 0 UMQ 2p2Mib5V- M/ i?2M i?2 +QMbi`m+iBQM 2M/b #v H2iiBM; Xn+m = i M/ τn+m = ∞ 7Q` HH m ≥ 1- M/ X(t) = Xn 7Q` HH t ≥ τn X h?2 T`Q+2bb +QMbi`m+i2/ BM i?Bb rv Bb /2}M2/ QM [0, τ∞ )- r?2`2 τ∞ = limn↑∞ τn X A7 τ∞ = ∞ S@Xb- i?2M i?2 T`Q+2bb Bb 7mHHv /2}M2/ QM R+ M/ Bb i?2`27Q`2  `2;mH` DmKT T`Q+2bb QM EX _2K`F kX8XR h?2 bK2 ivT2 Q7 +QMbi`m+iBQM rBHH #2 mb2/ 7Q`  KQ`2 ;2M2`H ivT2 Q7 biQ+?biB+ bvbi2Kb- i?2 SQBbbQM bvbi2Kb Q7 a2+iBQM RRXkX h?2Q`2K kX8Xk X 6Q` HH n ≥ 0, ;Bp2M Xn - i?2 `M/QK p`B#H2b Xn+1 M/ τn+1 − τn `2 BM/2T2M/2Mi Q7 X0 , . . . , Xn−1 M/ τ1 , . . . , τn - M/ 7Q` HH i, j ∈ Ei = j- HH a ≥ 0P (Xn+1 = j, τn+1 − τn > a | Xn = i) = e−qi a pij , r?2`2 pij =

qij qi

UkXRRV

B7 qi > 0- pij = 0 B7 qi = 0X UAM T`iB+mH`- {Xn }n≥0 Bb M ?K+XV

"X A7 P (τ∞ = ∞) = 1- i?2 T`Q+2bb {X(t)}t≥0 +QMbi`m+i2/ b #Qp2 Bb  `2;mH` DmKT ?K+ rBi? BM}MBi2bBKH ;2M2`iQ` AX   N S`QQ7X X G2i Gt := σ(X0 ) ∨ ∨i,j∈E Ft ij M/ Q#b2`p2 i?i i?2 τn Ƕb 7Q`K  b2[m2M+2 Q7 Gt @biQTTBM; iBK2bX h?2 MMQmM+2/ `2bmHi i?2M 7QHHQrb 7`QK i?2 bi`QM; J`FQp T`QT2`iv 7Q` ?TTb Uh?2Q`2K kXRXdV M/ i?2 +QKT2iBiBQM i?2Q`2K Uh?2Q`2K kXRXeVX "X "v +QMbi`m+iBQM- 7Q`  ;Bp2M iBK2 t- i?2 T`Q+2bb 7i2` iBK2 t /2T2M/b QMHv mTQM X(t) M/ i?2 ?TTb St Nij Ui, j ∈ E , i = j)X h?2 ?QKQ;2M2Qmb J`FQp T`QT@ 2`iv 7QHHQrb BKK2/Bi2Hv 7`QK i?Bb Q#b2`piBQMX Ai `2KBMb iQ p2`B7v i?i A Bb BM/22/ i?2 BM}MBi2bBKH ;2M2`iQ` Q7 i?2 ?K+X q2 }`bi +?2+F i?i- 7Q` i = j1 lim Pi (X(t) = j) = qij . t↓0 t 6Q` i?Bb- Q#b2`p2 i?i r?2M X(t) = X(0)- M2+2bb`BHv τ1 < t- M/ r`Bi2 UBMi`Q/m+BM; i?2 MQiiBQM Pi (·) := P (· | X(0) = i)VPi (X(t) = j) = Pi (τ2 ≤ t, X(t) = j) + Pi (τ2 > t, X(t) = j) = Pi (τ2 ≤ t, X(t) = j) + Pi (τ2 > t, X1 = j, τ1 < t) = Pi (τ2 ≤ t, X(t) = j) + Pi (X1 = j, τ1 < t) −Pi (τ2 ≤ t, X1 = j, τ1 < t) . "v h?2Q`2K kXRXe- Pi (X1 = j, τ1 < t) = (1 − e−qi t ) qiji - M/ i?2`27Q`2 q

1 lim Pi (X1 = j, τ1 < t) = qij . t↓0 t

e9

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

Ai i?2`27Q`2 `2KBMb iQ b?Qr i?i Pi (τ2 ≤ t, X(t) = j) M/ Pi (τ2 ≤ t, X1 = j, τ1 ≤ t) `2 o(t) UQ#pBQmb B7 qi = 0- M/ i?2`27Q`2 r2 bmTTQb2 qi > 0VX "Qi? i2`Kb `2 #QmM/2/ #v Pi (τ2 ≤ t)- M/  Pi (τ1 ≤ t, X1 = k, τ2 − τ1 ≤ t) Pi (τ2 ≤ t) ≤ Pi (τ1 ≤ t, τ2 − τ1 ≤ t) = k∈E k=j

 qik  qik (1 − e−qk t ). (1 − e−qi t ) (1 − e−qk t ) = (1 − e−qi t ) = q q i i k∈E k∈E k=j

"mi (1 − e−qi t ) Bb O(t) UB/2MiB+HHv MmHH B7 qi = 0V M/ limt↓0 #v /QKBMi2/ +QMp2`;2M+2X h?2`27Q`2 Pi (τ2 ≤ t) Bb o(t)X q2 MQr +?2+F i?i limt↓0

1−pii (t) t

k=i

k∈E k=i

qik (1 − e−qk t ) qi

= 0-

= qi X 6`QK

1 − pii (t) = 1 − Pi (X(t) = i) = 1 − Pi (X(t) = i, τ1 > t) − Pi (X(t) = i, τ1 ≤ t) = 1 − Pi (τ1 > t) − Pi (X(t) = i, τ1 ≤ t, τ2 ≤ t) = 1 − e−qi t − Pi (X(t) = i, τ1 ≤ t, τ2 ≤ t), M/ i?2 MMQmM+2/ `2bmHi 7QHHQrb 7`QK i?2 7+i UT`Qp2/ #Qp2V i?i Pi (τ2 ≤ t) Bb o(t)X  _2K`F kX8Xj G2i Zi (t) := 1{X(t)=i} X h?2 +QMbi`m+iBQM Q7 i?2 bii2 T`Q+2bb {X(t)}t≥0 QM [0, τ∞ ) Bb bmKK`Bx2/ #v i?2 7QHHQrBM; 2[miBQMb, 7Q` HH i ∈ E    Zi (t) = Zi (0) + Zj (s−) Nji (ds) − Zi (s−) Nij (ds) . UkXRkV j∈E;j=i

(0,t]

j∈E;j=i

(0,t]

a2+iBQM RRXk rBHH 72im`2 M 2ti2MbBQM Q7 `2T`2b2MiiBQM UkXRkV T`iB+mH`Hv /Ti2/ iQ  [m2m2BM; +QMi2tiX 1[miBQMb UkXRkV +QMbiBimi2  bvbi2K Q7 biQ+?biB+ /Bz2`2MiBH 2[miBQMb /`Bp2M #v i?2 SQBbbQM T`Q+2bb2b Nij Ui, j ∈ E , i = jV 7Q` i?2 T`Q+2bb2b {Zi (t)}t≥0 Ui ∈ EVX Ai Bb bQK2iBK2b +QMp2MB2Mi iQ bii2 +QM+HmbBQM " Q7 h?2Q`2K kX8Xk b 7QHHQrb, h?2Q`2K kX8X9 G2i {X(t)}t≥0 #2 M E@pHm2/ `2;mH` DmKT T`Q+2bb biBb7vBM; UkXRkV r?2`2 Zi (t) = 1{X(t)=i} M/ r?2`2 Ni,j (i, j ∈ E , i = j) Bb  7KBHv Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b qij (i, j ∈ E , i = j)- M/ BM/2T2M/2Mi Q7 i?2 BMBiBH bii2 X(0)X h?2M {X(t)}t≥0 Bb  `2;mH` DmKT ?K+ rBi? BM}MBi2bBKH ;2M2`iQ` AX _2K`F kX8X8 LQi2 i?i UkXRkV Bb 2[mBpH2Mi iQ i?2 `2[mB`2K2Mi i?i   f (X(t)) − f (X(0)) = {f (j) − f (i)} 1{X(s−)=i} dNij (s) i,j∈E i=j

UkXRjV

(0,t]

7Q` HH MQM@M2;iBp2 7mM+iBQMb f : E → RX q2 b?HH MQr 2tTHQBi i?Bb +MQMB+H `2T`2b2MiiBQMX

kX8X >J*a L. ahP*>ahA* .A661_1LhAG 1ZlhAPLa

e8

;;`2;iBQM Q7 aii2b *QMbB/2`  `2;mH` DmKT ?K+ {X(t)}t≥0 rBi? bii2 bT+2 E M/ BM}MBi2bBKH ;2M@ ˜ 2`iQ` AX G2i E˜ = {α, β, . . .} #2  T`iBiBQM Q7 E- M/ /2}M2 i?2 T`Q+2bb {X(t)} t≥0 iFBM; Bib pHm2b BM E˜ #v ˜ X(t) = α Bz X(t) ∈ α .

UkXR9V

˜ h?2 ?K+ {X(t)} t≥0 Bb i?2 ;;`2;i2/ +?BM Q7 {X(t)}t≥0 UrBi? `2bT2+i iQ i?2 ˜ T`iBiBQM EVX ˜ i ∈ α- β ∈ E˜ rBi? α = βh?2Q`2K kX8Xe amTTQb2 i?i 7Q` HH α ∈ E qij = q˜αβ .

UkXR8V

j∈β

Uh?Bb 2[mHBiv MQi QMHv /2}M2b i?2 [mMiBiv QM i?2 `B;?i@?M/ bB/2 #mi HbQ bvb ˜ BKTHB+BiHv i?i i?2 H27i@?M/ bB/2 Bb BM/2T2M/2Mi Q7 i ∈ αVX h?2M {X(t)} t≥0 Bb  ˜ rBi? Qz@ ˜ `2;mH` DmKT ?K+ rBi? bii2 bT+2 E M/ BM}MBi2bBKH ;2M2`iQ` A/B;QMH i2`Kb ;Bp2M #v UkXR8VX S`QQ7X h?Bb bii2K2Mi +QM+2`Mb i?2 /Bbi`B#miBQM Q7 {X(t)}t≥0 - M/ i?2`27Q`2 r2 Kv bmTTQb2 i?i i?Bb T`Q+2bb Bb Q7 i?2 7Q`K UkXRjVX h?2M- 7Q` f : E˜ → R M/ s ≤ t ˜ ˜ ˜ ˜ f (X(t)) = f (X(0)) + {f (X(u)) − f (X(u−))}1 {X(u−)=i} dNij (u) (0,t]

i,j∈E i=j

˜ = f (X(0)) +



{f (β) − f (α)}

˜ α,β∈E α=β

  

 1{X(u−)=i} (0,t]

i∈α

(0,t]

j∈β

 dNij (u)

j∈β

˜ α = β- i?2 TQBMi T`Q+2bb N ˜αβ #v .2}M2 7Q` HH α, β ∈ E,       ˜ 1{X(s−)=i} dNij (s) + Nαβ (0, t] = (0,t] i∈α



ˆ 1{X(s−) ˜ =α} dNα,β (s) ,

ˆαβ } α,β∈E˜ 7Q`K M BM/2T2M/2Mi 7KBHv Q7 r?2`2 i?2 dz/mKKvǴ TQBMi T`Q+2bb2b {N α=β

?TTb rBi? BMi2MbBiB2b {˜ qαβ } α,β∈E˜ - `2bT2+iBp2Hv- M/ `2 BM/2T2M/2Mi Q7 X(0) M/ {Nij } i,j∈E X h?2M

α=β

i=j

˜ ˜ f (X(t)) = f (X(0)) +

 ˜ α,β∈E α=β

 (f (β) − f (α)) (0,t]

˜αβ (u). 1{X(u−)=α} dN ˜

UkXReV

˜α,β Uα, β ∈ E˜ , α = βV Bb  7KBHv AM pB2r Q7 _2K`F kX8X8- Bi bm{+2b iQ T`Qp2 i?i N Q7 BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b q˜αβ Uα, β ∈ E˜ - α =  βVX 6Q` i?Bb-

.

ee

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

r2 TTHv qiM#2Ƕb i?2Q`2K Uh?2Q`2K kX9XRVX G2i {Z(t)}t≥0 #2  H27i@+QMiBMmQmb Ft @/Ti2/ biQ+?biB+ T`Q+2bb- r?2`2   ˜ ˜uv N N . Ft = σ(X(0)) ∨ Ft αβ ∨ ∨u,v∈E;(u,v) ˜ =(α,β) F∞ q2 Q#iBM



E

   ˜αβ (t) = Z(t)dN E

(0,T ]

i∈α j∈β



Z(t)1{X(t−)=i} dNij (t) (0,T ]

+E (0,T ]

ˆαβ , Z(t)1{X(t−) d N ˜ =α}

M/ i?Bb [mMiBiv Bb 2[mH- #v i?2 bKQQi?BM; 7Q`KmH Uh?2Q`2K kXkXRV- iQ  T

 T

 E Z(t)1{X(t−)=i} qij dt + E Z(t)1{X(t) ˜αβ dt ˜ =α} q 0

i∈α j∈β





T

Z(t)

=E 

0

=E



0

 qij





+ q˜αβ 1{X(t−) 1{X(t−)=α} ˜ ˜ =α} dt

j∈β

T

Z(t)˜ qαβ dt .

0

˜αβ Bb M ?TT rBi? BMi2MbBiv q˜αβ BM/2T2M/2Mi Q7 h?2`27Q`2- #v h?2Q`2K kX9XR- N  ˜ Nuv X  σ(X(0)) ∨ ∨u,v∈E;(u,v) ˜ =(α,β) F∞

kXe ?K+b M/ hBK2@a+H2/ ?TTb h?2 KBM `2bmHi Q7 i?Bb b2+iBQM8 2tTHQBib  `2T`2b2MiiBQM Q7  +QMiBMmQmb@iBK2 ?Q@ KQ;2M2Qmb J`FQp +?BM /Bz2`2Mi 7`QK i?2 QM2 +QMbB/2`2/ BM i?2 T`2pBQmb b2+iBQMBM i2`Kb Q7  iBK2@b+H2/ ?TTX aT2+B}+HHv- r2 `2 ;QBM; iQ +QMbi`m+i  ?QKQ;2M2Qmb +QMiBMmQmb@iBK2 J`FQp +?BM {X(t)}t≥0 rBi? pHm2b BM E = Zd 7Q` r?B+? HH TQbbB#H2 i`MbBiBQMb `2 Q7 i?2 ivT2 x → x + ei r?2`2 ei ∈ E U1 ≤ i ≤ kVX Aib BM}MBi2bBKH ;2M2`iQ` Q Bb ;Bp2M M/ bbmK2/ bi#H2- +QMb2`piBp2 M/ `2;mH`- M/ Bib MQM@/B;QMH 2H2K2Mib i?i `2 TQbBiBp2 `2 /2MQi2/ #v qx,x+ei X h?2 +QMbi`m+iBQM Bb b 7QHHQrb X(t) := X(0) +

k  i=1





t

e i Ni

qX(s),X(s)+ei ds

,

UkXRdV

0

r?2`2 Ni U1 ≤ i ≤ kV `2 BM/2T2M/2Mi biM/`/ ?TTb BM/2T2M/2Mi Q7 i?2 BMBiBH bii2 X(0) ∈ EX Ai 7QHHQrb 7`QK i?2 J`FQp T`QT2`iv Q7 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b M/ i?2 +QKT2iBiBQM i?2Q`2K i?i i?2 biQ+?biB+ T`Q+2bb bQ /2}M2/ Bb BM/22/ M ?K+ rBi? i?2 ;Bp2M BM}MBi2bBKH ;2M2`iQ` U1t2`+Bb2 kXdXReVX 8

(Em`ix- RNdR- RN3y)X

kXeX >J*a L. hAJ1@a*G1. >SSa

ed

1tKTH2 kXeXR,  *HbbB+H JQ/2H Q7 1TB/2Kv- iF2 RX h?2 #Qp2 +QM@ bi`m+iBQM rBHH #2 TTHB2/ iQ  bBKTH2 KQ/2H Q7 2TB/2Kv BM  TQTmHiBQM Q7 N BM/BpB/mHb r?Q #2HQM; iQ QM2 Q7 irQ +Hbb2b, i?2 BM72+i2/ M/ i?2 bmb+2TiB#H2b UMQi BM72+i2/- 2Bi?2` #2+mb2 i?2v ?p2 M2p2` #22M BM72+i2/ Q` #2+mb2 i?2v ?p2 `2@ +Qp2`2/- rBi? ?Qr2p2`  TQbbB#H2 `2HTb2VX h?2 bii2 Q7 i?2 2TB/2Kv i  ;Bp2M iBK2 Bb ;Bp2M #v i?2 MmK#2` i U0 ≤ i ≤ N V Q7 BM72+i2/ BM/BpB/mHb UM/ i?2`27Q`2 i?2 MmK#2` Q7 bmb+2TiB#H2b i i?i iBK2 Bb N − iVX G2i XN (t) #2 i?2 MmK#2` Q7 BM72+i2/ BM/BpB/mHb i iBK2 tX Mv ;Bp2M BM72+i2/ BM/BpB/mH K22ib MQi?2` K2K@ #2` Q7 i?2 TQTmHiBQM i `i2 βX h?2 BM/BpB/mH K2i #v i?Bb BM72+i2/ BM/BpB/mH Bb +?Qb2M i `M/QK BM i?2 TQTmHiBQM- M/ i?2`27Q`2 i?2 `i2 i r?B+? bQK2 ;Bp2M BM72+i2/ BM/BpB/mH K22ib  bmb+2TiB#H2 Bb β NN−i X h?2 `i2 i r?B+? i?2 BM72+i2/ TQTmHiBQM BM+`2b2b #v QM2 mMBi Bb i?2`27Q`2 iβ NN−i X LQr- i?2 `i2 i r?B+? bQK2 ;Bp2M BM72+i2/ BM/BpB/mH Bb +m`2/ U#2+QK2b  bmb+2TiB#H2V Bb δX h?2`27Q`2 i?2 `i2 i r?B+? i?2 BM72+i2/ TQTmHiBQM /2+`2b2b #v QM2 mMBi Bb δiX HH i?Bb DmbiB}2b i?2 KQ/2HBxiBQM Q7 i?2 biQ+?biB+ T`Q+2bb {XN (t)}t≥0 b M ?K+ rBi? BM}MBi2bBKH ;2M2`iQ` ;Bp2M #v β qi,i+1 = i(N − i), qi,i−1 = iδ . N

h?Bb T`iB+mH` 2tKTH2 }ib i?2 7QHHQrBM; ;2M2`H KQ/2H BMi`Q/m+2/ i i?2 #2;BMMBM; Q7 i?Bb b2+iBQM- M/ XN (t) = XN (0) +

k 





t

e i Ni

N λi (XN (s)/N ) ds

,

UkXR3V

0

i=1

r?2`2 ei ∈ Zd U1 ≤ i ≤ kV M/ λi : Rd → R+ U1 ≤ i ≤ kVX h?2 7QHHQrBM; ?vTQi?2b2b `2 K/2, >R λ := maxi∈{1,...,k} supx∈Rd λi (x) < ∞X

>k h?2 7mM+iBQM x → F (x) := ki=1 ei λi (x) Bb GBTb+?Bix- i?i Bb- i?2`2 2tBbib  }MBi2 +QMbiMi M bm+? i?i |F (x) − F (y)| ≤ M |x − y|

(x, y ∈ Rd ) .

>j 6BMHHv r2 bmTTQb2 i?i y(0) := lim

N ↑∞

XN (0) 2tBbib S@XbX N

h?2Q`2K kXeXk bbmK2 i?2 ?vTQi?2b2b >R- >k M/ >j p2`B}2/X G2i y : R+ → Rd #2 i?2 bQHmiBQM Q7 i?2 BMi2;`H 2[miBQM  t y(t) = y(0) + F (y(s)) ds . UkXRNV 0

h?2M 7Q` Mv }t2/ ε > 0 M/ T > 0 M/ 7Q` bm{+B2MiHv H`;2 N -

e3

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1  P

 & &    &1 & εe−M T & & sup & XN (t) − y(t)& ≥ ε ≤ 2k exp −N T λh , N 2kT λe

t∈[0,T ]

UkXkyV

max

r?2`2 emax := maxi∈{1,...,k} |ei |X JQ`2Qp2`- P @XbX & & & &1 lim sup && XN (t) − y(t)&& = 0 . N ↑∞ t∈[0,T ] N

UkXkRV

_2K`F kXeXj lM/2` i?2 GBTb+?Bix ?vTQi?2bBb >k- i?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 bQHmiBQM Q7 i?2 BMi2;`H 2[miBQM UkXRNV Bb ;m`Mi22/ #v i?2 7mM/K2MiH `2bmHi Q7 i?2 i?2Q`v Q7 /Bz2`2MiBH 2[miBQMbX 1tKTH2 kXeX9,  *HbbB+H JQ/2H Q7 1TB/2Kv- iF2 kX AM i?2 KQ/2H Q7 1tKTH2 kXeXR i?2`2 `2 QMHv irQ DmKT K;MBim/2b, e1 = +1 M/ e2 = −1- M/ λ1 (x) = βx(1 − x)- λ2 (x) = δx- M/ i?2`27Q`2 F (x) = βx(1 − x) − δx . "Qi? ?vTQi?2b2b >R M/ >k `2 biBb}2/X "v M TT`QT`Bi2 b+HBM;- r2 Kv bbmK2 i?i δ = 1X bbmKBM; ?vTQi?2bBb >j- i?2 bQHmiBQM Q7 UkXRNV Bb ;Bp2M 7Q` β = 1 #v (β − 1)y(0)e(β−1)t y(t) = . (β − 1) − βy(0)(1 − e(β−1)t ) AM T`iB+mH`- y(t) →

β−1 β

B7 β > 1- r?2`2b y(t) → 0 B7 β < 1X 1 X (t)N N

S`QQ7X AMi`Q/m+BM; i?2 MQiiBQM YN (t) := YN (t) = YN (0) +

k 



UkXR3V #2+QK2b 

t

e i Ni

N λi (YN (s)) ds 0

i=1

Q`- rBi? Mi (t) := Ni (t) − t Ut ≥ 0VYN (t) = YN (0) +

k 



t

e i Mi

  t nλi (YN (s)) ds + F (YN (s)) ds .

0

i=1

0

am#i`+iBM; UkXRNV M/ iFBM; #bQHmi2 pHm2b,  t |F (YN (s)) − F (y(s))| ds + · · · |YN (t) − y(t)| ≤ |YN (0) − y(0)| + 0

&   t & k  & |ei | && Mi N λi (YN (s)) ds && . ··· + N & 0

i=1

"v ?vTQi?2b2b >j M/ >k- 7Q` bm{+B2MiHv H`;2 N 

t

|YN (t) − y(t)| ≤ ε + M

|YN (s) − y(s)| ds + 0

k  1 αN,i (t) , N i=1

UkXkkV

kXeX >J*a L. hAJ1@a*G1. >SSa

eN

&   t & & & & λi (YN (s)) ds && . αN,i (t) = |ei | &Mi N

r?2`2

0

"v i?2 #QmM/ Q7 h?2Q`2K kXjXR     k k   1 Nε P sup αN,i (t) ≥ αN,i (t) ≥ ε ≤ P sup k t∈[0,T ] i=1 N t∈[0,T ] i=1   k  Nε P sup emax |M (t)| ≥ ≤ k t∈[0,T ] i=1    ε . ≤ 2k exp −N T λh kT λemax *QK#BMBM; i?2 Hbi irQ BM2[mHBiB2b vB2H/b      t P sup |YN (t) − y(t)| − M |YN (s) − y(s)| ds ≥ 2ε t∈[0,T ]



0



≤ 2k exp −N T λh

ε kT λemax

 UkXkjV

.

G2i ZN (t) := |YN (t) − y(t)|X "v :`QMrHHǶb H2KK Uh?2Q`2K RXeXeV- B7 7Q` HH t ∈ [0, T ]  t ZN (s) ds ZN (t) ≤ 2ε + M 0

i?2M 7Q` HH t ∈ [0, T ]- ZN (t) ≤ 2εeM t ≤ 2εeM T - M/ BM T`iB+mH` supt∈[0,T ] ZN (t) ≤ 2εeM T X h?2`27Q`2        t MT P sup ZN (t) − M ZN (s) ds ≤ 2ε ≤ P sup ZN (t) ≤ 2εe t∈[0,T ]

Q`- 2[mBpH2MiHv  P

t∈[0,T ]

0



sup ZN (t) > 2εeM T

 ≤P

t∈[0,T ]





sup ZN (t) > 2εe

MT

P

 sup ZN (t) > ε

  ≤ 2k exp −N T λh





≤ 2k exp −N T λh

t∈[0,T ]

AM T`iB+mH`

 N ≥1

 P

ZN (s) ds

> 2ε

.

0

t∈[0,T ]

Q`- Hi2`MiBp2Hv





t

ZN (t) − M

sup t∈[0,T ]

h?2`27Q`2- #v UkXkjV P



ε 2kT λemax

εeM T 2kT λemax



 .

 sup ZN (t) > ε

0) ,

t∈[0,T ]

r?B+?- #v i?2 "Q`2HĜ*Mi2HHB H2KK- BKTHB2b i?2 MMQmM+2/ +QMp2`;2M+2X



dy

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

kXd 1t2`+Bb2b 1t2`+Bb2 kXdXRX SQBbbQM BM #Qt2b G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λX G2i I1 - Ę- IK #2 /BbDQBMi BMi2`pHb Q7 R M/ /2MQi2 #v I i?2B` mMBQMX G2i n #2 M BMi2;2`X q?i Bb i?2 +QM/BiBQMH /Bbi`B#miBQM Q7 i?2 p2+iQ` (N (I1 ), . . . , N (IK )) ;Bp2M i?i N (I) = n\ 1t2`+Bb2 kXdXkX h?2 `BbF T`Q+2bb *QKTmi2 i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 i?2 `M/QK p`B#H2 X(t) Q7 1tKTH2 kXRX9X 1t2`+Bb2 kXdXjX "+Fr`/ M/ 7Q`r`/ `2+m``2M+2 iBK2b G2i {Tn }n≥1 #2 i?2 b2[m2M+2 Q7 2p2Mi iBK2b Q7 M ?TT QM R rBi? i?2 BMi2MbBiv λ > 0M/ H2i T0 ≡ 0X h?2 7Q`r`/ `2+m``2M+2 T`Q+2bb {A(t)}t≥0 M/ i?2 #+Fr`/ `2+m`@ `2M+2 T`Q+2bb {B(t)}t≥0 `2 /2}M2/ b 7QHHQrbX "Qi? T`Q+2bb2b `2 `B;?i@+QMiBMmQmb rBi? H27i@?M/ HBKBibX 6Q` n ≥ 0- i?2v ?p2 HBM2` i`D2+iQ`B2b BM (Tn , Tn+1 ) rBi? `2bT2+iBp2 bHQT2b −1 M/ +1- M/ i  `2M2rH TQBMi Tn A(Tn ) = Tn+1 − Tn , A(Tn+1 −) = 0 , B(Tn ) = 0 , B(Tn+1 −) = Tn+1 − Tn . 6Q` 0 ≤ t < T0 - A(t) = T0 − t M/ B(t) = tX 6Q` }t2/ t ∈ R- r?i Bb i?2 /Bbi`B#miBQM Q7 i?2 p2+iQ` (B(t), F (t))\ q?i B7 t ↑ ∞\ *QKTmi2 limt↑∞ E[B(t) + F (t)]X 1t2`+Bb2 kXdX9X 6HBT@7HQT G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ > 0X .2}M2 i?2 Ui2H2;`T? Q` ~BT@~QTV T`Q+2bb {X (t)}t≥0 rBi? bii2 bT+2 E = {+1, −1} #v X (t) := X(0) (−1)N (t) Ui?Bb biQ+?biB+ T`Q+2bb Hi2`Mi2b #2ir22M −1 M/ +1 i 2+? 2p2Mi Q7 N V- r?2`2 X (0) Bb M E@pHm2/ `M/QK p`B#H2 BM/2T2M/2Mi Q7 i?2 +QmMiBM; T`Q+2bb N X h?2 T`Q##BHBiv /Bbi`B#miBQM Q7 X(0) Bb `#Bi``vX RX *QKTmi2 P (X (t + s) = j|X (s) = i) 7Q` HH t, s ≥ 0 M/ HH i, j ∈ EX kX :Bp2 7Q` HH i ∈ E i?2 HBKBi Q7 P (X (t) = i) b t i2M/b iQ ∞X jX *QKTmi2 P (X (t) = i) 7Q` HH i ∈ E M/ HH t ≥ 0 r?2M P (X (0) = i) = 7Q` HH i ∈ EX

1 2

1t2`+Bb2 kXdX8X 1`;Q/B+ BMi2`T`2iiBQM Q7 i?2 BMi2MbBiv Q7 M ?TT G2i N #2 M ?TT QM R rBi? TQbBiBp2 }MBi2 BMi2MbBiv λX a?Qr i?i lim

t→∞

N (t) 1 = - S@XbX t E[S1 ]

G2i MQr N #2  ?QKQ;2M2Qmb *Qt QM R rBi? U`M/QKV BMi2MbBiv ΛX *QK@   T`Q+2bb Tmi2 limt→∞ Nt(t) M/ limt→∞ E N t(t) X

kXdX 1s1_*Aa1a

dR

1t2`+Bb2 kXdXeX JmimHHv bBM;mH` ?TTb G2i N #2  TQBMi T`Q+2bb QM R /2}M2/ QM  K2bm`#H2 bT+2 (Ω, F)X G2i P1 M/ P2 #2 irQ T`Q##BHBiv K2bm`2b QM (Ω, F) i?i KF2 Q7 N ?TTb Q7 BMi2MbBiB2b λ1 > 0 M/ λ2 > 0 `2bT2+iBp2HvX a?Qr i?i B7 λ1 = λ2 - P1 M/ P2 `2 KmimHHv bBM;mH`¯ = 1X i?i Bb iQ bv- i?2`2 2tBbib  b2i A ∈ F bm+? i?i P1 (A) = 1 M/ P2 (A) 1t2`+Bb2 kXdXdX h?2 r`2?Qmb2 G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ > 0X G2i T #2  }t2/ }MBi2 TQbBiBp2 iBK2M/ H2i ST #2 i?2 +QHH2+iBQM Q7 FtN @biQTTBM; iBK2b bKHH2` i?M Q` 2[mH iQ T X q2 b22F iQ }M/- B7 Bi 2tBbib-  biQTTBM; iBK2 τ BM ST KBMBKBxBM; E [N (τ )(T − τ )]X U TQbbB#H2 BMi2`T`2iiBQM Bb i?2 7QHHQrBM;X N +QmMib i?2 Bi2Kb Q7  bT2+B}+ FBM/ 2Mi2`BM;  r`2?Qmb2X h?2 biQ`;2 722 Bb Q7 QM2 mMBi T2` Bi2K M/ T2` mMBi Q7 bQDQm`M iBK2 BM i?2 r`2?Qmb2X h?2`2 Bb MQ biQ`;2 722 7i2` iBK2 T bBM+2 HH i?2 Bi2Kb MQi /BbTi+?2/ `2 iF2M 2Hb2r?2`2 MvrvX i M BMi2`K2/B`v `M/QK iBK2 τ ∈ [0, T ]- i?2 N (τ ) Bi2Kb T`2b2Mi `2 iF2M rv 7`QK i?2 biQ`;2 ?Qmb2- bQ i?i MQ KQ`2 biQ`;2 +Qbi Bb BM+m``2/ #v i?2b2 Bi2KbX h?Bb bp2b N (τ )(T − τ ) mMBib Q7 biQ`;2 +QbiX h?2 BMi2`K2/B`v /BbTi+?BM; iBK2 Q7 +Qm`b2 +MMQi MiB+BTi2 i?2 7mim`2 iBK2 Q7 ``BpHb- i?Bb Bb r?v Bi Kmbi #2 M FtN @biQTTBM; iBK2XV 6BM/ i?2 QTiBKH biQTTBM; iBK2 τ ∗ X 1t2`+Bb2 kXdX3X aQK2 K`iBM;H2b G2i N #2  SQBbbQM T`Q+2bb QM R+ rBi? i?2 HQ+HHv BMi2;`#H2 BMi2MbBiv 7mM+iBQM λ : R+ → R+ X 6Q` Mv t ≥ 0- /2}M2 N (t) := N ((0, t])X G2i M (t) := N (t) − λtX S`Qp2 i?i {M (t)2 − λt}t≥0 Bb M FtN @K`iBM;H2X 1t2`+Bb2 kXdXNX aiQ+?biB+ BMi2;`Hb M/ K`iBM;H2b G2i {Ft }t≥0 #2  ?BbiQ`v Q7 N - M ?TT QM R+ Q7 BMi2MbBiv λX amTTQb2 KQ`2Qp2` i?i 7Q` HH a > 0 M/ HH c, d Ua ≤ c ≤ dV- N (c, d] Bb BM/2T2M/2Mi Q7 Fa X G2i {Z(t)}t≥0 #2  H27i@+QMiBMmQmb +QKTH2t biQ+?biB+ T`Q+2bb /Ti2/ iQ {Ft }t≥0 X X amTTQb2 i?i 7Q` HH t ≥ 0 

t

E

|Z(s)| ds < ∞ .

()

0

a?Qr i?i 

 (0,t]

 Z(s) N (ds) −

Z(s) dM (s) :=

Y (t) :=

(0,t]

t

Z(s) λ ds

()

0

Bb  r2HH@/2}M2/ [mMiBiv UMQi  ∞ − ∞ 7Q`KV M/ i?i i?2 biQ+?biB+ T`Q+2bb {Y (t)}t≥0 Bb M Ft @K`iBM;H2X "X G2i i?BM;b #2 b #Qp2- 2t+2Ti 7Q` +QM/BiBQM UV- MQr `2TH+2/ #v i?2 HQ+H BMi2;`#BHBiv Q7 {Z(t)}t≥0 - i?i Bb, 

∞ 0

|Z(t)| dt < ∞ 7Q` HH t ≥ 0 .

dk

*>Sh1_ kX SPAaaPL S_P*1aa1a PL h>1 GAL1

a?Qr i?i Y (t) ;Bp2M #v UV Bb  r2HH@/2}M2/ [mMiBiv UMQi  ∞ − ∞ 7Q`KV M/ i?i i?2 biQ+?biB+ T`Q+2bb {Y (t)}t≥0 Bb  HQ+H Ft @K`iBM;H2X 1t2`+Bb2 kXdXRyX AMi2;`Hb rX`XiX  K`iBM;H2 Q7 #QmM/2/ p`BiBQM G2i {M (t)}t∈R+ #2  +Q`HQH U+QMiBMmQmb QM i?2 `B;?i M/ rBi? HBKBib QM i?2 H27iV F  t @K`iBM;H2 Q7 #QmM/2/ p`BiBQM- M/ rBi? HQ+HHv BMi2;`#H2 p`BiBQM Ui?i Bb |dM (s)| < ∞ 7Q` HH t ≥ 0VX G2i {Z(t)}t∈R+ #2  H27i@+QMiBMmQmb Ft @/Ti2/ (0,t] `2H biQ+?biB+ T`Q+2bb2b bm+? i?i 

E |Z(s)||dM (s)| < ∞ (t ≥ 0) . (0,t]

S`Qp2 i?i i?2 biQ+?biB+ T`Q+2bb {m(t)}t∈R+ /2}M2/ #v m(t) := Bb M Ft @K`iBM;H2X

 (0,t]

Z(s) dM (s)

1t2`+Bb2 kXdXRRX Hi2`MiBM; ?TTb G2i N1 M/ N2 #2 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b QM R+ rBi? i?2 bK2 TQbBiBp2 BMi2MbBiv λX G2i 7Q` HH t ≥ 0- Ft := FtN1 ∨ FtN2 - M/ H2i {τn }n≥0 #2  MQM@/2+`2bBM; b2[m2M+2 Q7 Ft @biQTTBM; iBK2b- r?2`2 τ0 ≡ 0 M/ limn↑∞ τn = +∞X 2 QM R+ #v 1 M/ N .2}M2 i?2 TQBMi T`Q+2bb2b N 1 ((τ2n , t]) = N1 ((τ2n , t]) 7Q` HH t ∈ (τ2n , τ2n+1 ] (n ≥ 0) , N 1 ((τ2n+1 , t]) = N2 ((τ2n+1 , t]) 7Q` HH t ∈ (τ2n+1 , τ2n+2 ] (n ≥ 0) , N M/ 2 ((τ2n , t]) = N2 ((τ2n , t]) 7Q` HH t ∈ (τ2n , τ2n+1 ] (n ≥ 0) , N 2 ((τ2n+1 , t]) = N1 ((τ2n+1 , t]) 7Q` HH t ∈ (τ2n+1 , τ2n+2 ] (n ≥ 0) . N 2 `2 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b QM R+ rBi? 1 M/ N S`Qp2 i?i N BMi2MbBiv λX 1t2`+Bb2 kXdXRkX qiM#2ǰb i?2Q`2K 7Q` *Qt T`Q+2bb2b G2i N #2  bBKTH2 TQBMi T`Q+2bb QM R+ M/ H2i {λ(t)}t≥0 #2  HQ+HHv BMi2;`#H2 MQM@M2;iBp2 biQ+?biB+ T`Q+2bb bm+? i?i 

 T

E Z(t)N (dt) = E Z(t)λ(t) dt 0

(0,T ]

?QH/b i`m2 7Q` HH T > 0- M/ 7Q` HH MQM@M2;iBp2 `2H@pHm2/ H27i@+QMiBMmQmb biQ+?biB+ T`Q+2bb2b {Z(t)}t≥0 /Ti2/ iQ {Ft }t≥0 - r?2`2 Ft := FtN ∨ F λ

(t ≥ 0).

S`Qp2 i?i N Bb  /Qm#Hv biQ+?biB+ SQBbbQM T`Q+2bb UQ` *Qt T`Q+2bbV rBi? `2bT2+i iQ F λ rBi? i?2 +QM/BiBQMH BMi2MbBiv {λ(t)}t≥0 X

kXdX 1s1_*Aa1a

dj

1t2`+Bb2 kXdXRjX _2;mH`Biv bbmK2 i?2 Ki`Bt P /2}M2/ BM bii2K2Mi  Q7 h?2Q`2K kX8Xk Bb B``2/m+B#H2 M/ `2+m``2MiX a?Qr i?i τ∞ = ∞- HKQbi bm`2HvX 1t2`+Bb2 kXdXR9X EQHKQ;Q`Qpǰb 6Q`r`/ 1[miBQMb G2i {X(t)}t≥0 #2  `2;mH` DmKT ?K+ rBi? i`MbBiBQM b2KB@;`QmT {P(t)}t≥0 M/ BM}MBi2bBKH ;2M2`iQ` AX S`Qp2 i?i  t P(t) = I + P(s)Ads (t ≥ 0) . 0

1t2`+Bb2 kXdXR8X AM7BMBi2bBKH :2M2`iQ` M/ J`iBM;H2b ij i?2 TQBMi T`Q+2bb +QmMi@ _272` iQ h?2Q`2K kX8X9X AM i?2 `2;mH` +b2- /2MQi2 #v N BM; i?2 i`MbBiBQMb Q7 i?2 ?K+ {X(t)}t≥0 7`QK bii2 i iQ bii2 j = iX S`Qp2 i?i  t ij (t) := N ij (t) − M 1{X(s−)=i} qij ds 0

/2}M2b M

FtX @K`iBM;H2

{Mij (t)}t≥0 X a?Qr i?i 

Z(t) = Z(0) +

t

(t) , AZ(s) ds + M

UkXk9V

0

(t) = {M i (t)}i∈E M/ r?2`2 {Mi (t)}t≥0 Ui ∈ EV Bb M r?2`2 Z(t) = {Zi (t)}i∈E - M X Ft @K`iBM;H2X 1t2`+Bb2 kXdXReX Em`ixǰ `2T`2b2MiiBQM S`Qp2 i?2 bii2K2Mi 7QHHQrBM; 1[MX UkXRdVX

*?Ti2` j aTiBH SQBbbQM S`Q+2bb2b aTiBH SQBbbQM T`Q+2bb KQ/2Hb `2 TQTmH` T`iHv #2+mb2 Q7 i?2B` Ki?2KiB+H i`+i#BHBiv #b2/ QM  bBKTH2 M/ TQr2`7mH +H+mHmb r?Qb2 KBM BM;`2/B2Mib- #2@ bB/2b *KT#2HHǶb 7Q`KmH- `2 i?2 +Qp`BM+2 7Q`KmH M/ i?2 2tTQM2MiBH 7Q`KmH UHbQ FMQrM b *KT#2HHǶb b2+QM/ 7Q`KmHVX h?2b2 7Q`KmHb 2ti2M/ BKK2/Bi2Hv iQ K`F2/ SQBbbQM T`Q+2bb2b rBi? BM/2T2M/2Mi UQ` HQ+iBQM /2T2M/2MiV BB/ K`Fb b  +QMb2[m2M+2 Q7 i?2 7+i i?i bm+? K`F2/ SQBbbQM T`Q+2bb2b +M #2 /2b+`B#2/ b mMK`F2/ SQBbbQM T`Q+2bb2b BM  ?B;?2`@/BK2MbBQMH bT+2X h?Bb bHB;?i KQ/B@ }+iBQM Q7 TQBMi Q7 pB2r rBHH #2 TTHB2/ iQ i?2 bim/v Q7 i?2 TQBMi T`Q+2bb2b Q#@ iBM2/ #v K2Mb Q7 2H2K2Mi`v i`Mb7Q`KiBQMb Q7  #bB+ SQBbbQM T`Q+2bb- bm+? b i?BMMBM;- i`MbHiBQM Q` +Hmbi2`BM;X HH i?2 `2bmHib +QM+2`MBM; SQBbbQM T`Q+2bb2b }M/ bi`B;?i7Q`r`/ 2ti2MbBQMb +QM+2`MBM; *Qt T`Q+2bb2bX h?2 +H+mHmb Q7 bTiBH K`F2/ SQBbbQM Q` *Qt T`Q+2bb2b rBHH #2 TTHB2/ iQ  T`iB+mH` ivT2 Q7 biQ+?biB+ ;2QK2i`v U72im`BM; i?2 "QQH2M KQ/2HV M/ iQ i?2 2t+i bKTHBM; Q7 +Hmbi2` TQBMi T`Q+2bb2bX

jXR

aKTHBM;  SQBbbQM S`Q+2bb

h?2 M2ti `2bmHi +M #2 +QMbB/2`2/  T`QQ7 Q7 2tBbi2M+2 Q7 i?2 SQBbbQM T`Q+2bb Q7 .2}MBiBQM RXRXNX >Qr2p2`- Bib KBM BMi2`2bi HB2b BM i?2 7+i i?i Bi +M #2 mb2/ 7Q` bKTHBM; Ui?2 /Bbi`B#miBQM Q7V bm+?  TQBMi T`Q+2bb Ub22 .2}MBiBQM RXjXkVX G2i E #2 M HX+X/X#X bT+2 rBi? "Q`2H σ@}2H/ B(E)X h?2Q`2K jXRXR G2i {Zn }n≥1 #2 M BB/ E@pHm2/ `M/QK b2[m2M+2 rBi? +QKKQM /Bbi`B#miBQM QX G2i T #2  SQBbbQM `M/QK p`B#H2 rBi? K2M θ M/ BM/2T2M/2Mi Q7 {Zn }n≥1 X h?2 TQBMi T`Q+2bb N QM E /2}M2/ #v N (C) :=

T 

1C (Zn ) (C ∈ B(E))

n=1

Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν(·) = θ × Q(·)X S`QQ7X Ai bm{+2b iQ b?Qr i?i 7Q` Mv }MBi2 7KBHv C1 , . . . , CK Q7 TB`rBb2 /BbDQBMi K2bm`#H2 b2ib Q7 E rBi? }MBi2 ν@K2bm`2 M/ 7Q` HH MQM@M2;iBp2 `2H MmK#2`b t 1 , . . . , tK -

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_3

d8

de

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a E[e−

K

j=1 tj N (Cj )

  −tj ] = ΠK − 1) . j=1 exp ν(Cj )(e

q2 ?p2 K 

tj N (Cj ) =

j=1

K  j=1

 tj

T 

 1Cj (Zn )

=

n=1

 K T   n=1

 tj 1Cj (Zn )

=

T 

Yn ,

n=1

j=1

r?2`2 Yn = K j=1 tj 1Cj (Zn )X "v  +HbbB+H `2bmHi QM i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 `M/QK bmKbT E[e− n=1 Yn ] = gT (E[e−Y1 ]) , r?2`2 gT Bb i?2 ;2M2`iBM; 7mM+iBQM Q7 i?2 `M/QK p`B#H2 T X aBM+2 T Bb  SQBbbQM `M/QK p`B#H2 rBi? K2M θgT (z) = exp {θ(z − 1)} . h?2 `M/QK p`B#H2 Y1 iF2b i?2 pHm2b

t1 , . . . , tK M/ 0 rBi? i?2 `2bT2+iBp2 T`Q#@ #BHBiB2b Q(C1 )- Ę- Q(CK ) M/ 1 − K j=1 Q(Cj )X h?2`27Q`2 E[e−Y1 ] =

K 

e−tj Q(Cj ) + 1 −

j=1

K 

Q(Cj ) = 1 +

j=1

K  

 e−tj − 1 Q(Cj ) ,

j=1

7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX



h?2Q`2K jXR Bb  bT2+BH +b2 Q7 r?i Bb iQ #2 /QM2- i?i Bb- iQ +QMbi`m+i  SQBbbQM T`Q+2bb QM E rBi? M BMi2MbBiv K2bm`2 ν i?i Bb σ@}MBi2 UMQi Dmbi }MBi2VX am+?  K2bm`2 +M #2 /2+QKTQb2/ b ν(·) =

∞ 

θj × Qj (·) ,

j=1

r?2`2 i?2 θj Ƕb `2 TQbBiBp2 `2H MmK#2`b M/ i?2 Qj Ƕb `2 T`Q##BHBiv /Bbi`B#miBQMb QM EX PM2 +M +QMbi`m+i BM/2T2M/2Mi SQBbbQM T`Q+2bb2b Nj QM E rBi? `2bT2+iBp2 BMi2MbBiv K2bm`2b θj Qj (·)X h?2 MMQmM+2/ `2bmHi i?2M 7QHHQrb 7`QK i?2 7QHHQrBM;,

∞ h?2Q`2K jXRXk G2i ν #2  σ@}MBi2 K2bm`2 QM E Q7 i?2 7Q`K ν = i=1 νi r?2`2 i?2 νi Ƕb (i ≥ 1) `2 σ@}MBi2 K2bm`2b QM EX G2i Ni (i ≥ 1) #2  7KBHv Q7 BM/2T2M/2Mi SQBbbQM T`Q+2bb2b QM E rBi? `2bT2+iBp2 BMi2MbBiv K2bm`2b νi (i ≥ 1)X h?2M i?2 TQBMi T`Q+2bb ∞  N := Nj j=1

Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 νX S`QQ7X 6Q` KmimHHv /BbDQBMi K2bm`#H2 b2ib C1 - Ę- CK Q7 }MBi2 ν@K2bm`2b- M/ MQM@M2;iBp2 `2Hb t1 - Ę- tK -

jXRX aJSGAL:  SPAaaPL S_P*1aa

dd

 K   K  ∞ E e− =1 t N (C ) = E e− =1 t ( j=1 Nj (C ))   K n = E e− limn↑∞ =1 t ( j=1 Nj (C ))  K  n = lim E e− =1 t ( j=1 Nj (C )) , n↑∞

#v /QKBMi2/ +QMp2`;2M+2X "mi    n K   K n E e− =1 t ( j=1 Nj (C )) = E e− j=1 ( =1 t Nj (C )) =

n 

n  K    K E e− =1 t Nj (C ) = e−t Nj (C )

j=1

=

j=1 =1

n  K 





exp (e−t − 1)νj (C ) =

j=1 =1



= exp

K  

e

=1

−t

−1

 n  

n 

 exp

j=1





e−t − 1 νj (C )

=1



νj (C ))

K  

.

j=1

G2iiBM; n ↑ ∞- r2 Q#iBM- #v /QKBMi2/ +QMp2`;2M+2  K   K    − =1 t N (C ) −t E e e − 1 ν(C ) . = exp =1

h?2`27Q`2 N (C1 )- Ę- N (CK ) `2 BM/2T2M/2Mi SQBbbQM `M/QK p`B#H2b rBi? `2@ bT2+iBp2 K2Mb ν(C1 )- Ę- ν(CK )X  h?2Q`2K jXRXj G2i N #2  SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2bm`2 νX UV A7 ν Bb HQ+HHv }MBi2- i?2M N Bb XbX HQ+HHv }MBi2X U#V A7 ν Bb HQ+HHv }MBi2 M/ MQM@iQKB+- i?2M N Bb XbX bBKTH2X S`QQ7X UV A7 C Bb  #QmM/2/ K2bm`#H2 b2i- Bi Bb Q7 }MBi2 ν@K2bm`2- M/ i?2`27Q`2 E[N (C)] = ν(C) < ∞- r?B+? BKTHB2b i?i N (C) < ∞- P @HKQbi bm`2HvX U#V Ai bm{+2b iQ b?Qr i?Bb 7Q`  }MBi2 BMi2MbBiv K2bm`2 ν(·) = θ(·) Q r?2`2 θ Bb  TQbBiBp2 `2H MmK#2` M/ Q Bb  MQM@iQKB+ T`Q##BHBiv K2bm`2 QM Rm M/ i?2M mb2 i?2 +QMbi`m+iBQM Q7 h?2Q`2K jXRXRX AM im`M- Bi bm{+2b iQ b?Qr i?i 7Q` HH n ≥ 1- P (Zi = Zj 7Q` bQK2 TB` (i, j) (1 ≤ i < j ≤ n) | N (Rm ) = n) = 0X h?Bb Bb i?2 +b2 #2+mb2 7Q` BB/ p2+iQ`b Z1 , . . . , Zn rBi?  MQM@iQKB+ T`Q##BHBiv /Bbi`B#miBQM- P (Zi = Zj 7Q` bQK2 TB` (i, j) (1 ≤ i < j ≤ n)) = 0X  1tKTH2 jXRX9, h?2 biM/`/ bKTHBM; K2i?Q/ 7Q` M ?TTX h?2 +QM@ bi`m+iBQM Q7 h?2Q`2K jXRXR ;Bp2b  K2i?Q/ 7Q` bKTHBM;  SQBbbQM T`Q+2bb Q7 BMi2M@ bBiv K2bm`2 ν BM  rBM/Qr W ⊂ Rm Q7 }MBi2 ν@K2bm`2X 6B`bi bKTH2  SQBbbQM p`B#H2 T Q7 K2M θ := ν(W )X A7 T = k- bKTH2 k BB/ p2+iQ`b Z1 , . . . , Zk rBi? i?2

d3

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

+QKKQM /Bbi`B#miBQM P (Z1 ∈ C) = ν(C)/ν(W ) UC ⊂ W VX h?2 /2bB`2/ bKTH2 Bb i?2M i?2 b2i Q7 TQBMib {Z1 , . . . , Zk }X 6Q` i?2 bKTH2b Zn - mb2 QM2 Q7 i?2 KMv K2i?@ Q/b pBH#H2 UBMp2`b2 7mM+iBQM- ++2TiiBQM@`2D2+iBQM- 2i+XVX h?2 bKTHBM; Q7 T Kv i #2 /QM2 #v i?2 +HbbB+H BMp2`b2 7mM+iBQM K2i?Q/ b 7QHHQrbX G2iiBM; pi := e−θ θi! bKTH2  `M/QK p`B#H2 /Bbi`B#mi2/ QM [0, 1] M/ b2i T = k B7 U 7HHb

U mMB7Q`KHv

k BM i?2 BMi2`pH Ii := [ k−1 p , p ]X i=0 i i=0 i  +`m/2 p2`bBQM Q7 i?2 bKTHBM; H;Q`Bi?K +QMbBbib BM 2tKBMBM; i?2 BMi2`pHb Ii b2[m2MiBHHv mMiBH QM2 Bb 7QmM/ i?i +QMiBMb U X h?Bb rQmH/ `2[mB`2 QM i?2 p2`;2 1 + E[T ] = 1 + θ i`BHbX A7 θ Bb p2`v H`;2 KQ`2 2+QMQKB+H T`Q+2/m`2 Bb pBH#H2XR Ai iF2b BMiQ ++QmMi i?2 7+i i?i i?2 T`Q##BHBiv Kbb Q7  SQBbbQM p`B#H2 Bb KtBKH i  pHm2 i0 M2` i?2 p2`;2 pHm2 M/ /2+`2b2b b QM2 ;2ib 7`i?2` rv 7`QK i?Bb pHm2X h?2 2tTHQ`iBQM bi`ib rBi? i?2 pHm2 i0 - M/ i?2M T`Q+22/b iQ i0 − 1- i0 + 1- i0 − 2- i0 + 2- 2i+X h?2 p2`;2 MmK#2` Q7 i`BHb Bb i?2M `Qm;?Hv 2[mH iQ

√ |T − θ| √ . 1 + E [|T − θ|] = 1 + θE θ Bb TT`QtBKi2Hv /Bbi`B#mi2/ b  biM/`/ "v i?2 +2Mi`H HBKBi i?2Q`2K- Z := T√−θ θ :mbbBM p`B#H2X h?2`27Q`2 i?2 p2`;2 MmK#2` Q7 i`BHb 7Q` H`;2 θ Bb TT`QtB@ Ki2Hv √ √ 1 + θE [|N (0, 1)|]  1 + 0.82 θ .

MQi?2` rv Q7 biiBM; h?2Q`2K jXRXR Bb b 7QHHQrbX G2i ν #2  }MBi2 K2bm`2 QM E M/ H2i T #2  SQBbbQM p`B#H2 rBi? K2M ν(E)X 6Q` 2+? n ≥ 0- /2MQi2 #v Pn i?2 T`Q##BHBiv QM (Mp (E), Mp (E)) bm+? i?i i?2 +MQMB+H TQBMi T`Q+2bb ?b 2t+iHv X n TQBMib- TH+2/ BM/2T2M/2MiHv QM E ++Q`/BM; iQ i?2 T`Q##BHBiv /Bbi`B#miBQM ν(dx) ν(W ) q?i h?2Q`2K jXR b?Qrb Bb i?i i?2 T`Q##BHBiv P QM (Mp (E), Mp (E)) KFBM; Q7 i?2 +MQMB+H TQBMi T`Q+2bb  SQBbbQM T`Q+2bb rBi? U`2+HH, }MBi2V BMi2MbBiv K2bm`2 ν +M #2 `2T`2b2Mi2/ b  ν(E)n P = Pn . e−ν(E) UjXRV n! n=0 1tKTH2 jXRX8, _`2 S2`im`#iBQM a2MbBiBpBiv MHvbBbXk G2i N #2  bTiBH SQBbbQM T`Q+2bb QM 1 rBi? K2M K2bm`2 ν = αQ- r?2`2 α > 0 M/ Q(E) < ∞X .2MQi2 #v Pα #2 +Q``2bTQM/BM; T`Q##BHBivX G2i f (N ) #2 bQK2 #QmM/2/ [f (N )] K2bm`#H2 7mM+iBQMH Q7 N X h?2 ;QH Bb iQ }M/ M 2tT`2bbBQM 7Q` dEαdα X 6Q` i?Bbr2 mb2 i?2 `2T`2b2MiiBQM UjXRV- r?B+? ;Bp2b  Eα [f (N )] = P (Tα = n)En [f (N )] , UjXkV n=0

r?2`2 Tα Bb  SQBbbQM p`B#H2 rBi? K2M α M/ r?2`2 En /2MQi2b 2tT2+iiBQM rBi? `2bT2+i iQ i?2 T`Q##BHBiv Pn mM/2` r?B+? N ?b 2t+iHv n TQBMib BM/2T2M/2MiHv R k

(_Qbb- RNNk)- TTX 9NĜ8yX ("`ûKm/ M/ o€x[m2x@#/- RNNk)X

jXRX aJSGAL:  SPAaaPL S_P*1aa

dN

M/ B/2MiB+HHv /Bbi`B#mi2/ ++Q`/BM; iQ i?2 /Bbi`B#miBQM +QKTmiiBQM ;Bp2b

Q(dx) X Q(E)

dP (Tα = n) = −P (Tα = n) + P (Tα = n − 1) dα

 bi`B;?i7Q`r`/

(n ≥ 1) ,

M/ dP (Tα = 0) = −P (Tα = 0) dα

(n ≥ 1) ,

M/ i?2`27Q`2- mbBM; `2T`2b2MiiBQM UjXRVdEα [f (N )] = − P (Tα = 0)E0 (f (N )) dα + P (Tα = 0)E1 (f (N )) − P (Tα = 1)E1 (f (N )) + P (Tα = 1)E2 (f (N )) − P (Tα = 2)E2 (f (N )) + ··· _2``M;BM; i?2 i2`Kb- r2 Q#iBM dEα [f (N )] =P (Tα = 0) (E1 [f (N )] − E0 [f (N )]) dα + P (Tα = 1) (E2 [f (N )] E1 [f (N )]) + ··· LQr- 7Q` HH n ≥ 1En+1 [f (N )] =

1 Q(E)

 En [f (N + εx )] Q(dx) E

M/ i?2`27Q`2- }MHHv, dEα [f (N )] 1 = dα Q(E)

 (Eα [f (N + εx )] − Eα [f (N )]) Q(dx) . E

6`QK i?2 bBKmHiBQM TQBMi Q7 pB2r-j i?Bb /2`BpiBp2 +M #2 TT`QtBKi2/ #v i?2 2KTB`B+H K2M n 1 (f (Nk + εZk )) − (f (Nk )) n k=1 7Q` bm{+B2MiHv H`;2 n- r?2`2 {Nk }1≤k≤n Bb M BB/ b2[m2M+2 Q7 +QTB2b Q7 N mM/2` T`Q##BHBiv Pα - {Zk }1≤k≤n Bb M BB/ b2[m2M+2 Q7 E@pHm2/ `M/QK 2H2K2Mib rBi? i?2 /Bbi`B#miBQM Q(dx) M/ BM/2T2M/2Mi Q7 i?2 b2[m2M+2 {Nk }1≤k≤n X Q(E) j a22 (bKmbb2M M/ :HvMM- kyyd)- *?Ti2` oAA- 7Q` i?2 mb2 Q7 /2`BpiBp2b BM bBKmHiBQM M/ biQ+?biB+ QTiBKBxiBQMX

3y

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

jXk h?2 *Qp`BM+2 M/ 1tTQM2MiBH 6Q`KmHb G2i N #2  SQBbbQM T`Q+2bb QM i?2 HX+X/X#X bT+2 E- rBi? BMi2MbBiv K2bm`2 νX _2+HH *KT#2HHǶb i?2Q`2K Uh?2Q`2K RXkXRVX G2i ϕ : E → R #2  ν@BMi2;`#H2 K2bm`#H2 7mM+iBQMX h?2M N (ϕ) Bb  r2HH@/2}M2/ BMi2;`#H2 `M/QK p`B#H2 M/ 

 E ϕ(x) N (dx) = ϕ(x) ν(dx) . UjXjV E

E

h?2Q`2K jXkXR G2i N #2 b #Qp2X G2i ϕ, ψ : E → C #2 irQ ν@BMi2;`#H2 K2@ bm`#H2 7mM+iBQMb bm+? i?i KQ`2Qp2` |ϕ|2 M/ |ψ|2 `2 ν@BMi2;`#H2X h?2M N (ϕ) M/ N (ψ) `2 r2HHĜ/2}M2/ b[m`2@BMi2;`#H2 `M/QK p`B#H2b M/ r2 ?p2 i?2 +Qp`BM+2 7Q`KmH 7Q` SQBbbQM T`Q+2bb2b,     cov ϕ(x) N (dx), ψ(x) N (dx) = ϕ(x)ψ(x)∗ ν(dx) . UjX9V E

E

E

S`QQ7X Ai Bb 2MQm;? iQ +QMbB/2` i?2 +b2 Q7 `2H 7mM+iBQMbX 6B`bi bmTTQb2 i?i ϕ M/ ψ `2 bBKTH2 MQM@M2;iBp2 "Q`2H 7mM+iBQMb- bvϕ :=

K 

ah 1Ch M/

ψ :=

h=1

K 

b h 1 Ch ,

h=1

r?2`2 C1 , . . . , CK `2 /BbDQBMi K2bm`#H2 bm#b2ib Q7 E UQM2 Kv Hrvb bbmK2 i?i

K i?2 irQ 7mM+iBQMb ?p2 i?2 Ci Ƕb BM +QKKQMVX AM T`iB+mH`- ϕ(x)ψ(x) = h=1 ah bh 1Ch (x)X lbBM; i?2 7+ib i?i N (Ci ) M/ N (Cj ) `2 BM/2T2M/2Mi B7 i = j M/ i?i  SQBbbQM `M/QK p`B#H2 rBi? K2M θ ?b p`BM+2 θE[N (ϕ)N (ψ)] =

K 

ah bl E[N (Ch )N (Cl )]

h,l=1

=

K 

ah bl E[N (Ch )N (Cl )] +

h,l=1 h=l

=

K 

K 

al bl E[N (Cl )2 ]

l=1

ah bl E[N (Ch )]E[N (Cl )] +

h,l=1 h=l

K 

al bl E[N (Cl )2 ] ,

l=1

M/ i?2`27Q`2 E[N (ϕ)N (ψ)] =

K 

ah bl ν(Ch )ν(Cl ) +

h,l=1 h=l

=

k  h,l=1

k 

al bl [ν(Cl ) + ν(Cl )2 ]

l=1

ah bl ν(Ch )ν(Cl ) +

k  l=1

al bl ν(Cl ) = ν(ϕ)ν(ψ) + ν(ϕψ) .

jXkX h>1 *Po_AL*1 L. 1sSPL1LhAG 6P_JlGa

3R

G2i MQr ϕ- ψ #2 MQM@M2;iBp2 M/ H2i {ϕn }n≥1 - {ψn }n≥1 #2 MQM@/2+`2bBM; b2@ [m2M+2b Q7 bBKTH2 MQM@M2;iBp2 7mM+iBQMb- rBi? `2bT2+iBp2 HBKBib ϕ M/ ψX G2iiBM; n ;Q iQ ∞ BM i?2 2[mHBiv E[N (ϕn )N (ψn )] = ν(ϕn ψn ) + ν(ϕn )ν(ψn ) vB2H/b i?2 MMQmM+2/ `2bmHib- #v KQMQiQM2 +QMp2`;2M+2X G2i ϕ+ := ϕ 1{ϕ≥0} M/ ϕ− := −ϕ 1{ϕSh1_ jX aShAG SPAaaPL S_P*1aa1a

E[e

−N (ϕ)

 ] = E e



K

h=1

ah N (Ch ))



 =E

K 

 e

−ah N (Ch )

h=1

=

K 

K    E e−ah N (Ch ) = exp (e−ah − 1)ν(Ch )

h=1



= exp

h=1 K 

(e

−ah

− 1)ν(Ch )



  = exp ν(e−ϕ − 1) .

h=1

h?2 7Q`KmH Bb i?2`27Q`2 i`m2 7Q` MQM@M2;iBp2 bBKTH2 7mM+iBQMbX hF2 MQr  MQM@ /2+`2bBM; b2[m2M+2 {ϕn }n≥1 Q7 bm+? 7mM+iBQMb +QMp2`;BM; iQ ϕX 6Q` HH n ≥ 1  E[e−N (ϕn ) ] = exp ν(e−ϕn − 1) . "v KQMQiQM2 +QMp2`;2M+2- i?2 HBKBi b n i2M/b iQ ∞ Q7 N (ϕn ) Bb N (ϕ)X *QM@ b2[m2MiHv- #v /QKBMi2/ +QMp2`;2M+2- i?2 HBKBi Q7 i?2 H27i@?M/ bB/2 BM i?2 Hbi /BbTHv Bb E[e−N (ϕ) ]X h?2 7mM+iBQM gn = −(e−ϕn − 1) Bb  MQM@M2;iBp2 7mM+@ iBQM BM+`2bBM; iQ g = −(e−ϕ − 1)- M/ i?2`27Q`2- #v KQMQiQM2 +QMp2`;2M+2ν(e−ϕn − 1) = −ν(gn ) +QMp2`;2b iQ ν(e−ϕ − 1) = −ν(g)- r?B+? BM im`M BKTHB2b i?i i?2 `B;?i@?M/ bB/2 BM i?2 Hbi /BbTHv i2M/b iQ exp {ν(e−ϕ − 1)} b n i2M/b iQ ∞X  1tKTH2 jXkX9, h?2 GTH+2 6mM+iBQMH Q7  SQBbbQM S`Q+2bbX 6`QK h?2Q`2K jXkXk- r2 Q#iBM 7Q` i?2 GTH+2 7mM+iBQMH Q7  SQBbbQM T`Q+2bb N QM E rBi? BMi2MbBiv K2bm`2 ν,    LN (ϕ) = exp ν e−ϕ − 1 .

h?2 *KT#2HH 7Q`KmH BKTHB2b i?i 7Q`  SQBbbQM T`Q+2bb Q7 BMi2MbBiv K2bm`2 ν- B7 ν(ϕ) < ∞- i?2M N (ϕ) < ∞X AM 7+ih?2Q`2K jXkX8 P (N (ϕ) < ∞) = 1 ↔ ν(ϕ ∧ 1) < ∞ . S`QQ7X h?2 T`QQ7 mb2b i?2 7QHHQrBM; 7+ib, 1 − e−a ≤ a ∧ 1 7Q` HH a ≥ 0 M/   P (N (ϕ) < ∞) = lim e−tN (ϕ) = lim exp −ν(1 − e−tϕ . t↓0

t↓0

h?2`27Q`2- BM i?2 +b2 ν(ϕ ∧ 1) < ∞  P (N (ϕ) < ∞) = lim exp −ν(1 − e−tϕ ≥ lim exp (−ν(tϕ ∧ 1)) = 1 t↓0

t↓0

U#v /QKBMi2/ +QMp2`;2M+2 bBM+2 limt↓0 ν(tϕ∧1) = 0 M/ ν(tϕ∧1) ≤ ν(ϕ∧1) < ∞ 7Q` bm{+B2MiHv bKHH tVX

jXkX h>1 *Po_AL*1 L. 1sSPL1LhAG 6P_JlGa

3j

A7 ν(ϕ ∧ 1) = ∞   E e−N (ϕ) = exp −ν(1 − e−ϕ ) ≥ exp (−ν(ϕ ∧ 1)) = 0 r?B+? BKTHB2b i?i P (N (ϕ) = ∞) = 1X



1tKTH2 jXkXe, h?2 JtBKmK 6Q`KmHX G2i N #2  bBKTH2 SQBbbQM T`Q+2bb QM E rBi? BMi2MbBiv K2bm`2 ν M/ H2i ϕ : E → RX h?2M      P sup ϕ(Xn ) ≤ a = exp − 1{ϕ(x)>a} ν(dx) . n∈N

E

 /B`2+i T`QQ7 #b2/ QM i?2 +QMbi`m+iBQM Q7 i?2 SQBbbQM T`Q+2bb BM a2+iBQM RXR Bb TQbbB#H2 U1t2`+Bb2 jX3XjVX q2 iF2 MQi?2` Ti? M/ }`bi T`Qp2 i?i    lim E e−θ n∈N 1{ϕ(Xn )>a} = P sup ϕ(Xn ) ≤ a . () θ↑∞

n∈N

AM/22/- i?2 bmK n∈N 1{ϕ(Xn )>a} Bb bi`B+iHv TQbBiBp2- 2t+2Ti r?2M supn∈N ϕ(Xn ) ≤ a- BM r?B+? +b2 Bi Bb MmHHX h?2`27Q`2 lim e−θ

 n∈N

1{ϕ(Xn )>a}

θ↑∞

= 1{supn∈N ϕ(Xn )≤a} .

hFBM; 2tT2+iiBQMb vB2H/b UV- #v /QKBMi2/ +QMp2`;2M+2X LQr- #v h?2Q`2K jXkXk   −θ1   e {ϕ(x)>a} − 1 ν(dx) E e−θ n∈N 1{ϕ(Xn )>a} = exp E    −θ = exp e − 1 1{ϕ(x)>a} ν(dx) E

   M/ i?2 HBKBi Q7 i?2 Hii2` [mMiBiv b θ ↑ ∞ Bb exp − E 1{ϕ(x)>a} ν(dx) X h?2Q`2K jXkXd G2i Ni (i ∈ J) #2  }MBi2 +QHH2+iBQM Q7 bBKTH2 TQBMi T`Q+2bb2b QM EX A7 7Q` Mv +QHH2+iBQM ϕi : E → R+ (i ∈ J) Q7 MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb  −  Ni (ϕi )   −ϕi (x)  i∈J E e = e exp − 1 νi (dx) , UjX8V E

i∈J

r?2`2 νi (i ∈ J) Bb  +QHH2+iBQM Q7 σ@}MBi2 K2bm`2b QM E- i?2M Ni Ui ∈ JV Bb  7KBHv Q7 BM/2T2M/2Mi SQBbbQM T`Q+2bb2b rBi? `2bT2+iBp2 BMi2MbBiv K2bm`2b νi (i ∈ J)X S`QQ7X G2iiBM; HH i?2 ϕi 2t+2Ti ϕ1 #2 B/2MiB+HHv MmHH- r2 ?p2   −N1 (ϕ1 )   −ϕ1 (x) E e − 1 ν1 (dx) , = exp e E

r?B+? T`Qp2b i?i N1 Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν1 X aBKBH`Hv- 7Q` Mv i ∈ J- Ni Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 νi X AM/2T2M/2M+2 i?2M 7QHHQrb 7`QK h?2Q`2K RXjXRdX 

39

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

jXj J`F2/ aTiBH SQBbbQM S`Q+2bb2b h?2 2ti2MbBQM Q7 i?2 +Qp`BM+2 M/ 2tTQM2MiBH 7Q`KmHb iQ K`F2/ SQBbbQM T`Q@ +2bb2b /2T2M/ QM i?2 7mM/K2MiH `2bmHi #2HQr Uh?2Q`2K jXjXRVX G2i UαV N #2  bBKTH2 M/ HQ+HHv }MBi2 T`Q+2bb QM E rBi? TQBMi b2[m2M+2 {Xn }n∈N M/ H2i UβV {Zn }n∈N #2  b2[m2M+2 Q7 `M/QK 2H2K2Mib iFBM; i?2B` pHm2b BM i?2 K2@ bm`#H2 bT+2 (K, K)X h?2 b2[m2M+2 {Xn , Zn }n∈N Bb  K`F2/ TQBMi T`Q+2bb- rBi? i?2 BMi2`T`2iiBQM i?i Zn Bb i?2 K`F bbQ+Bi2/ rBi? i?2 TQBMi Xn X h?2 TQBMi T`Q+2bb N Bb +HH2/ i?2 #b2 TQBMi T`Q+2bb Q7 i?2 K`F2/ TQBMi T`Q+2bb M/ {Zn }n∈N Bb i?2 bbQ+Bi2/ b2[m2M+2 Q7 K`FbX AM Qi?2` rQ`/b- N Bb  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM E rBi? K`Fb {Zn }n∈N BM KX A7 KQ`2Qp2` URV N Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 νUkV {Zn }n∈N Bb M BB/ b2[m2M+2- M/ UjV {Zn }n∈N M/ N `2 BM/2T2M/2Mii?2 +Q``2bTQM/BM; K`F2/ TQBMi T`Q+2bb Bb +HH2/  SQBbbQM T`Q+2bb2b QM E rBi? BM/2T2M/2Mi BB/ K`FbX h?Bb KQ/2H +M #2 bHB;?iHv ;2M2`HBx2/ #v HHQrBM; i?2 K`F /Bbi`B#miBQM iQ /2T2M/ QM i?2 HQ+iBQM Q7 i?2 K`F2/ TQBMiX JQ`2 T`2+Bb2Hvr2 `2TH+2 UkV M/ UjV #v UkǶV {Zn }n∈N Bb- +QM/BiBQMHHv QM N - M BM/2T2M/2Mi b2[m2M+2UjǶV ;Bp2M Xn - i?2 `M/QK 2H2K2Mi Zn Bb BM/2T2M/2Mi Q7 Xk Uk ∈ N, k = nVM/ U9ǶV 7Q` HH n ∈ N M/ HH L ∈ KP (Zn ∈ L | Xn = x) = Q(x, L) , r?2`2 Q(·, ·) Bb  biQ+?biB+ F2`M2H 7`QK (E, B(E)) iQ (K, K)- i?i Bb-  7mM+iBQM 7`QK E×K iQ [0, 1] bm+? i?i 7Q` HH L ∈ K i?2 KTTBM; x → Q(x, L) Bb K2bm`#H2M/ 7Q` HH x ∈ E- Q(x, ·) Bb  T`Q##BHBiv K2bm`2 QM (K, K)X h?2Q`2K jXjXR G2i {Xn , Zn }n∈N #2 b BM UαV M/ UβV #Qp2 M/ /2}M2 i?2 TQBMi  QM E × K #v T`Q+2bb N  (A) := N



1A (Xn , Zn )

(A ∈ B(E) ⊗ K) .

UjXeV

n∈N

 Bb  bBKTH2 SQBbbQM A7 +QM/BiBQMb URV- UkǶV- UjǶV- M/ U9ǶV #Qp2 `2 biBb}2/- i?2M N T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν ;Bp2M #v  ν(C × L) = Q(x, L) ν(dx) (C ∈ B(E) , L ∈ K) . C

jXjX J_E1. aShAG SPAaaPL S_P*1aa1a

38

S`QQ7X AM pB2r Q7 h?2Q`2K RXjXR9- Bi bm{+2b iQ b?Qr i?i i?2 GTH+2 i`Mb7Q`K  ?b i?2 TT`QT`Bi2 7Q`K- i?i BbQ7 N           − 1 ν(dt × dz) E e−N (ϕ) = exp E K e−ϕ(t,z) 7Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM ϕ  : E × K → RX "v /QKBMi2/ +QMp2`@ ;2M+2       n ,Zn )    n ,Zn ) E e−N (ϕ) = lim E e− n≤k ϕ(X = E e− n∈N ϕ(X . k↑∞

6Q` i?2 iBK2 #2BM;- }t i?2 TQbBiBp2 BMi2;2` kX h?2M- iFBM; BMiQ ++QmMi bbmKT@ iBQMb UkǶV M/ UjǶV           n ,Zn )  n ,Zn )  n ,Zn ) E e− n≤k ϕ(X =E E =E e−ϕ(X e−ϕ(X | Xj ; j ≤ k  =E

n≤k

 n≤k

e

−ϕ(X  n ,z)





Q(Xn , dz) = E exp

K



n≤k

 n≤k





log

e

−ϕ(X  n ,z)

Q(Xn , dz)

K

   = E e− n≤k ψ(Xn ) ,

  r?2`2 ψ(x) := − log K e−ϕ(x,z) Q(x, dz)-  MQM@M2;iBp2 7mM+iBQMX G2iiBM; k ↑ ∞r2 Q#iBM #v /QKBMi2/ +QMp2`;2M+2 i?i      E e−N (ϕ) = E e− n∈N ψ(Xn ) = E e−N (ψ)    −ψ(x)  = exp e − 1 ν(dx) E 

  = exp e−ϕ(x,z) Q(x, dz) − 1 ν(dx) E K

  −ϕ(x,z)  = exp e  − 1 Q(x, dz) ν(dx) E  K    −ϕ(x,z) = exp − 1 ν(dx × dz) , e  r?2`2 i?2 7+i i?i

 K

E

K

Q(x, dz) = 1 rb iF2M BMiQ ++QmMiX



1tKTH2 jXjXk, h?2 Jf:Af∞ JQ/2H- hF2 RX h?Bb KQ/2H Bb Q7 BMi2`2bi BM [m2m2BM; i?2Q`v M/ BM i?2 i`{+ MHvbBb Q7 +QKKmMB+iBQMb M2irQ`FbX q2 /QTi i?2 [m2m2BM; BMi2`T`2iiBQMX G2i N #2 ?TT QM R rBi? BMi2MbBiv λ M/ H2i {σn }n∈Z #2  b2[m2M+2 Q7 `M/QK p`B#H2b iFBM; i?2B` pHm2b BM R+ rBi? i?2 T`Q##BHBiv /Bbi`B#miBQM QX bbmK2 KQ`2Qp2` i?i {σn }n∈Z M/ N `2 BM/2T2M/2MiX h?2 n@i? 2p2Mi iBK2 Tn Q7 N Bb BMi2`T`2i2/ b i?2 ``BpH iBK2 Q7 i?2 n@i? +mbiQK2` M/ σn  QM R × R+ /2}M2/ #v Bb ?2` b2`pB+2 `2[m2bi UBM iBK2 mMBibVX h?2 TQBMi T`Q+2bb N   (C) := 1C (Tn , σn ) (C ∈ B(R) ⊗ B(R+ )) N n∈Z

Bb- ++Q`/BM; iQ h?2Q`2K jXjXR-  bBKTH2 SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2

3e

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a ν(dt × dz) = λdt × Q(dz) .

AM i?2 Jf:Af∞ KQ/2H-9  +mbiQK2` ``BpBM; i iBK2 Tn Bb BKK2/Bi2Hv b2`p2/M/ i?2`27Q`2 /2T`ib 7`QK i?2 dzbvbi2KǴ i iBK2 Tn + σn X h?2 MmK#2` X(t) Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 t Bb i?2`27Q`2 ;Bp2M #v i?2 7Q`KmH  X(t) = 1(−∞,t] (Tn )1(t,∞) (Tn + σn ) . n∈Z

Uh?2 n@i? +mbiQK2` Bb BM i?2 bvbi2K i iBK2 t B7 M/ QMHv B7 b?2 ``Bp2/ i iBK2 Tn ≤ t M/ /2T`i2/ i iBK2 Tn + σn > tXV bbmK2 i?i i?2 b2`pB+2 iBK2b ?p2 }MBi2 2tT2+iiBQM, E [σ1 ] < ∞X h?2M- 7Q` HH t ∈ R- X(t) Bb  SQBbbQM `M/QK p`B#H2 rBi? K2M λE [σ1 ]X S`QQ7X LQiB+2 i?i

˜ (C(t)) , X(t) = N

r?2`2 C(t) := {(s, σ); s ≤ t, s + σ > t} ⊂ R × R+ Ui?2 b?/2/ `2;BQM BM 6B;m`2 jXjXRVX

C(t)

t

6B;m`2 jXjXR AM T`iB+mH`- X(t) Bb  SQBbbQM `M/QK p`B#H2 rBi? K2M     1{s+σ>t} 1{s≤t} ν˜(ds × dσ) = 1{s+σ>t} 1{s≤t} λ ds × Q(dσ) ν˜(C(t)) = R R+ R R+   t   1{s+σ>t} Q(dσ) 1{s≤t} λ ds = λ Q((t − s, +∞)) ds = −∞ R R+  ∞  ∞ Q((s, +∞)) ds = λ P (σ1 > s)ds = λE[σ1 ] . =λ 0

0

 Ai +M #2 b?QrM i?i i?2 /2T`im`2 T`Q+2bb D Q7 /2T`im`2 iBK2b /2}M2/ #v  D(C) := 1C (Tn + σn ) (C ∈ B(R)) n∈Z 9

dz∞Ǵ `2T`2b2Mib i?2 MmK#2` Q7 b2`p2`bX h?Bb KQ/2H Bb bQK2iBK2b +HH2/  dz[m2m2BM;Ǵ bvbi2KHi?Qm;? BM `2HBiv i?2`2 Bb MQ [m2m2BM;- bBM+2 +mbiQK2`b `2 b2`p2/ BKK2/Bi2Hv mTQM ``BpH M/ rBi?Qmi BMi2``mTiBQMX Ai Bb BM 7+i  dzTm`2 /2HvǴ bvbi2KX

jXjX J_E1. aShAG SPAaaPL S_P*1aa1a

3d

Bb M ?TT rBi? BMi2MbBiv λ U1t2`+Bb2 jX3XNVX 6Q`KmHb bm+? b *KT#2HHǶb }`bi 7Q`KmH M/ i?2 SQBbbQM 2tTQM2MiBH 7Q`KmH +M #2 bi`B;?i7Q`r`/Hv 2ti2M/2/ iQ +Qp2` i?2 +b2 Q7 K`F2/ TQBMi T`Q+2bb2bX AM i?2 bBimiBQM T`2pBHBM; BM h?2Q`2K jXjXR +QMbB/2`- 7Q`  7mM+iBQM ϕ˜ : E × K → RbmKb Q7 i?2 ivT2  ˜ (ϕ) N ˜ := ϕ(X ˜ n , Zn ) . UjXdV n∈N

LQi2 i?i- /2MQiBM; #v Z1 (x) Mv `M/QK 2H2K2Mi Q7 K rBi? i?2 /Bbi`B#miBQM Q(x, dz) 



ν˜(ϕ) ˜ =

ϕ(x, ˜ z)Q(x, dz) ν(dx) = E

K

E [ϕ(x, ˜ Z1 (x))] ν(dx) , E

r?2M2p2` i?2 [mMiBiB2b BMpQHp2/ `2 r2HH /2}M2/X lbBM; i?Bb Q#b2`piBQM- i?2 7Q`@ KmHb Q#iBM2/ BM i?2 T`2pBQmb bm#b2+iBQM +M #2 TTHB2/ BM i2`Kb Q7 K`F2/ TQBMi T`Q+2bb2b- M/ BM T`iB+mH` i?2 +Q`QHH`B2b #2HQr /Q MQi `2[mB`2 T`QQ7b- bBM+2 i?2v #2+QK2 `27Q`KmHiBQMb Q7 h?2Q`2K jXkXR- h?2Q`2K jXkXk M/ 1t2`+Bb2 jX3X8X G2i 0 < p < ∞X _2+HH i?i  K2bm`#H2 7mM+iBQM ϕ˜ : E × K → R U`2bTX → CV Bb bB/ iQ #2 BM LpR (˜ ν ) U`2bTX LpC (˜ ν )V B7   |ϕ(x, ˜ z)|p ν(dx) Q(x, dz) < ∞ . E

K

*Q`QHH`v jXjXj amTTQb2 i?i ϕ˜ ∈ L1C (˜ ν )X h?2M i?2 bmK UjXdV Bb r2HH /2}M2/- M/ KQ`2Qp2`     E ϕ(X ˜ n , Zn ) = E [ϕ(x, ˜ Z1 (x))] ν(dx) . E

n∈N

ν ) ∩ L2C (˜ ν )X h?2M G2i ϕ, ˜ ψ˜ : R × E → C #2 irQ K2bm`#H2 7mM+iBQMb BM L1C (˜  cov

 n∈N

ϕ(X ˜ n , Zn ),



 ˜ n , Zn ) ψ(X



  ˜ Z1 (x))∗ ν(dx) . E ϕ(x, ˜ Z1 (x))ψ(x,

= E

n∈N

*Q`QHH`v jXjX9 G2i ϕ˜ #2  MQM@M2;iBp2 7mM+iBQM 7`QK E × K iQ RX h?2M  ˜ n ,Zn ) E e− n∈N ϕ(X = exp



˜ 1 (x)) E e−ϕ(x,Z − 1 ν(dx) E

 .

33

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

jX9 PT2`iBQMb QM SQBbbQM S`Q+2bb2b h?2 QT2`iBQMb +QMbB/2`2/ `2 +QHQm`BM;- i?BMMBM;- i`MbTQ`iBM;- i`MbHiBM; M/ }Hi2`BM;X *QHQm`BM; h?2Q`2K jX9XR *QMbB/2` i?2 bBimiBQM /2TB+i2/ BM h?2Q`2K jXjXRX G2i I #2 M `#Bi``v BM/2t b2i M/ H2i {Li }i∈I #2  7KBHv Q7 /BbDQBMi K2bm`#H2 b2ib Q7 KX .2}M2 7Q` 2+? i ∈ I i?2 bBKTH2 TQBMi T`Q+2bb Ni QM Rm #v  Ni (C) = 1C (Xn )1Li (Zn ) . n∈N

Uh?2 TQBMib Q7 Ni `2+2Bp2 i?2 dz+QHQm`Ǵ iXV h?2M i?2 7KBHv Ni (i ∈ I) Bb M BM/2@ T2M/2Mi 7KBHv Q7 SQBbbQM T`Q+2bb2b rBi? `2bT2+iBp2 BMi2MbBiv K2bm`2b νi (i ∈ I) r?2`2 νi (dx) = Q(x, Li ) ν(dx) (i ∈ I) . S`QQ7X ++Q`/BM; iQ i?2 /2}MBiBQM Q7 BM/2T2M/2M+2- Bi bm{+2b iQ +QMbB/2`  }MBi2 ˜ QM Rm × K b BM UjXeVX h?2M N ˜ BM/2t b2i IX .2}M2 i?2 bBKTH2 TQBMi T`Q+2bb N  Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν˜(C × L) = C Q(x, L)ν(dx)X qBi?

˜ (ϕ)X ϕ(x, ˜ z) := i∈I ϕi (x)1Li (z)- r2 ?p2 i∈I Ni (ϕi ) = N ˜ h?2`27Q`2        −  Ni (ϕi ) ˜ ˜ ˜ = E e−N (ϕ) − 1 ν˜(dx × dz) = exp e−ϕ(x,z) E e i∈I Rm K      −ϕ(x,z) ˜ = exp − 1 Q(x, dz)ν(dx) e m R K   −  ϕi (x)1 (z)  Li = exp e i∈I − 1 Q(x, dz)ν(dx) m R K    = exp e−ϕi (x) − 1 1Li (z)Q(x, dz)ν(dx) 

Rm

e

= exp =



K i∈I



−ϕi (x)

Rm i∈I



Rm

 e−ϕi (x) − 1 Q(x, Li )ν(dx)

h?2`27Q`2



E e

 i∈I

Ni (ϕi )



=

− 1 Q(x, Li )ν(dx)



exp

i∈I











e

exp

i∈I

M/ i?2 `2bmHi 7QHHQrb 7`QK h?2Q`2K jXkXdX

Rm

−(ϕi )



 . 

− 1 νi (dx) 

_2K`F jX9Xk h?2 #Qp2 i?2Q`2K +QM+2`Mb BM T`iB+mH` i?BMMBM; Ui?2 QT2`iBQM Q7 `M/QKHv 2`bBM; TQBMibVX 6Q` BMbiM+2 i?2 TQBMi T`Q+2bb N1 Bb Q#iBM2/ #v i?BMMBM; Q7 N - 2+? TQBMi x Q7 r?B+? #2BM; bp2/ rBi? T`Q##BHBiv Q(x, L1 )X

jX9X PS1_hAPLa PL SPAaaPL S_P*1aa1a

3N

h`MbTQ`iiBQM h?2Q`2K jX9Xj G2i N #2  TQBMi T`Q+2bb QM bQK2 K2bm`#H2 bT+2 (E, E) M/ H2i T #2  K2bm`#H2 KTTBM; 7`QK (E, E) iQ MQi?2` K2bm`#H2 bT+2 (E  , E  )X amTTQb2 i?i N Bb  SQBbbQM T`Q+2bb QM E rBi? i?2 HQ+HHv }MBi2 BMi2MbBiv K2bm`2 ν M/ i?i i?2 K2bm`2 ν  r?B+? Bb i?2 BK;2 Q7 ν #v T - i?i Bbν  (C  ) = ν(T −1 (C  )) (C  ∈ E  ) , Bb σ@}MBi2X h?2M- i?2 TQBMi T`Q+2bb N  QM E  /2}M2/ #v N  (C  ) = N (T −1 (C  ))

(C  ∈ E  )

Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν  X S`QQ7X

     E e− E ϕ(x) N (dx) = e− n ϕ(T (Xn ))    = e− E ϕ(T (x)) N (dx)    −ϕ(T (x))  e − 1 ν(dx) = exp E   −ϕ(y)  = exp e − 1 ν  (dx) . E

 h?Bb i?2Q`2K +QK#BM2/ rBi? h?2Q`2K jXjXR Ur?B+? T2`KBib mb iQ pB2r  K`F2/ SQBbbQM T`Q+2bb b  SQBbbQM T`Q+2bb BM  H`;2` bT+2V +Qp2`b KMv bBimiBQMbX 1tKTH2 jX9X9, h?2 SQBMi S`Q+2bb Q7 J`FbX *QMbB/2` i?2 bBimiBQM /2@ TB+i2/ BM h?2Q`2K jXjXR- r?B+? r2 Kv BMi2`T`2i b 7QHHQrbX  TQBMi T`Q+2bb N ∗ QM K Bb 7Q`K2/ #v bbQ+BiBM; iQ  TQBMi Xn ∈ Rm  TQBMi Zn ∈ KX 6Q`KHHv  1L (Zn ) , N ∗ (L) = n∈N

r?2`2 L ∈ B(Rm )X q2 i?2M bv i?i N ∗ Bb Q#iBM2/ #v i`MbTQ`iBM; N pB i?2 biQ+?biB+ F2`M2H Q(x, ·)X bbmK2 i?i K Bb M HX+X/X#X bT+2 M/ i?i 7Q` HH +QKT+i bm#b2ib L ∈ K Rm

Q(x, L) ν(dx) < ∞ .

lM/2` i?2 #Qp2 +QM/BiBQMb- N ∗ Bb  SQBbbQM T`Q+2bb QM K rBi? BMi2MbBiv K2bm`2 ν ∗ ;Bp2M #v  ∗ Q(x, L) ν(dx) . ν (L) = Rm

S`QQ7X G2i ϕ∗ : K → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX q2 ?p2

Ny

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a  ∗ ∗ ∗ E e−N (ϕ ) = E e− n∈N ϕ (Zn )     −ϕ∗ (z)  e − 1 ν(dx)Q(x, dz) = exp m R K    −ϕ∗ (z)  = exp e −1 ν(dx)Q(x, dz) . Rm

K



1tKTH2 jX9X8, h`MbHiBQMX G2i N #2  SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2bm`2 ν M/ H2i {Vn }n∈N #2 M BB/ b2[m2M+2 `M/QK p2+iQ`b Q7 Rm rBi? +QKKQM /Bbi`B#miBQM QX 6Q`K i?2 TQBMi T`Q+2bb N ∗ QM Rm #v i`MbHiBM; 2+? TQBMi Xn Q7 N #v Vn X 6Q`KHHv 1C (Xn + Vn ) (C ∈ B(Rm )) . N ∗ (C) := n∈N

q2 `2 BM i?2 bBimiBQM Q7 h?2Q`2K jX9Xj rBi? T (Xn , Zn ) = Xn + Vn X AM T`iB+mH` Q(x, A) = Q(A − x)X Ai 7QHHQrb i?i N ∗ Bb  SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2bm`2  ν ∗ (L) = Q(L − x) ν(dx) , Rm

i?2 +QMpQHmiBQM Q7 ν M/ QX

SQBbbQM a?Qi LQBb2 U6BHi2`BM;V G2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N M/ rBi? K`Fb {Zn }n∈N BM i?2 K2bm`#H2 bT+2 (K, K)X G2i h : Rm × K → C #2  K2bm`#H2 7mM+iBQMX h?2 +QKTH2t@pHm2/ bTiBH biQ+?biB+ T`Q+2bb {X(y)}y∈Rm ;Bp2M #v  X(y) := h(y − Xn , Zn ) , UjX3V n∈N

r?2`2 i?2 `B;?i@?M/ bB/2 Bb bbmK2/ r2HH /2}M2/ U7Q` BMbiM+2- r?2M h iF2b `2H MQM@M2;iBp2 pHm2bV- Bb +HH2/  bTiBH b?Qi MQBb2 rBi? `M/QK BKTmHb2 `2bTQMb2X A7 N Bb  bBKTH2 M/ HQ+HHv }MBi2 SQBbbQM T`Q+2bb QM Rm rBi? BM/2T2M/2Mi BB/ K`Fb {Zn }n∈N - {X(y)}y∈Rm Bb +HH2/  SQBbbQM bTiBH b?Qi MQBb2 rBi? `M/QK BKTmHb2 `2bTQMb2 M/ BM/2T2M/2Mi BB/ K`FbX h?2 7QHHQrBM; `2bmHi Bb  /B`2+i TTHB+iBQM Q7 h?2Q`2Kb jXkXR M/ jXjXRX h?2Q`2K jX9Xe *QMbB/2` i?2 #Qp2 SQBbbQM bTiBH b?Qi MQBb2 rBi? `M/QK BK@ TmHb2 `2bTQMb2 M/ BM/2T2M/2Mi BB/ K`FbX amTTQb2 i?i 7Q` HH y ∈ Rm  E [|h(y − x, Z1 )|] ν(dx) < ∞ Rm

M/

jX8X *>L:1 P6 S_P""AGAhu  Rm

NR

E |h(y − x, Z1 )|2 ν(dx) < ∞ .

h?2M i?2 +QKTH2t@pHm2/ bTiBH biQ+?biB+ T`Q+2bb {X(y)}y∈Rm ;Bp2M #v UjX3V Bb r2HH /2}M2/- M/ 7Q` HH y, ξ ∈ Rm  E [X(y)] = E [h(y − x, Z1 )] ν(dx) Rm

M/

 cov(X(y + ξ), X(y)) = Rm

E [h(y − x, Z1 )h∗ (y + ξ − x, Z1 )] ν(dt) .

AM i?2 +b2 r?2`2 i?2 #b2 TQBMi T`Q+2bb N Bb M ?TT rBi? BMi2MbBiv λ E [X(y)] = λ E [h(x, Z1 )] dx Rm



M/

E [h(x, Z1 )h∗ (ξ + x, Z1 )] dx .

cov(X(y + ξ), X(y)) = λ Rm

P#b2`p2 i?i i?2b2 [mMiBiB2b /Q MQi /2T2M/ QM y ∈ Rm X h?2 T`Q+2bb {X(y)}y∈Rm Bb 7Q` i?i `2bQM +HH2/  rB/2@b2Mb2 biiBQM`v T`Q+2bbX

jX8

*?M;2 Q7 S`Q##BHBiv

G2i (Ω, F, P ) #2  T`Q##BHBiv bT+2 QM r?B+? Bb ;Bp2M  SQBbbQM T`Q+2bb N QM E rBi? HQ+HHv }MBi2 BMi2MbBiv K2bm`2 νX q2 b?HH `2TH+2 i?2 T`Q##BHBiv P #v MQi?2` T`Q##BHBiv P BM bm+?  rv i?i rBi? `2bT2+i iQ i?Bb M2r T`Q##BHBiv- i?2 bK2 TQBMi T`Q+2bb N Bb  SQBbbQM T`Q+2bb- #mi rBi? i?2 BMi2MbBiv K2bm`2 ν ;Bp2M #v  ν(C) = μ(x) ν(dx), UjXNV E

7Q` bQK2 MQM@M2;iBp2 K2bm`#H2 7mM+iBQM μ : E → RX h?2 +b2 Q7 }MBi2 BMi2MbBiv K2bm`2b h?Bb T`Q;`K Bb }`bi +``B2/ Qmi mM/2` i?2 7QHHQrBM; ?vTQi?2b2b, H1 , ν Bb  }MBi2 K2bm`2- M/ H2 , μ Bb ν@BMi2;`#H2 UQ`- 2[mBpH2MiHv- ν Bb }MBi2VX h?2 +?M;2 Q7 T`Q##BHBiv P → P rBHH #2 M #bQHmi2Hv +QMiBMmQmb QM2- i?i Bb7Q` HH A ∈ F UjXRyV P (A) = E[L 1A ], r?2`2 L Bb  MQM@M2;iBp2 `M/QK p`B#H2 bm+? i?i E[L] = 1 ,

UjXRRV

+HH2/ i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 Q7 P rBi? `2bT2+i iQ P - HbQ /2MQi2/ #v

dP X dP

Nk

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

G2KK jX8XR lM/2` ?vTQi?2b2b H1 M/ H2 - i?2 `M/QK p`B#H2       μ(Xn ) exp − (μ(x) − 1)ν(dx) L :=

UjXRkV

E

n∈N

biBb}2b i?2 `2[mB`2K2Mi UjXRRVX S`QQ7X G2i g(x) = log(μ(x)) M/ /2+QKTQb2 i?Bb 7mM+iBQM BMiQ Bib TQbBiBp2 M/ M2;iBp2 T`i- g = g+ − g− X "v h?2Q`2K jXkXk r2 ?p2 i?i   −N (g− )   −g− (x) E e − 1 ν(dx) = exp e E

M/



N (g+ )

E e







= exp

e

g+ (x)



− 1 ν(dx)

 .

E

G2i B1 = {x ∈ E ; g(x) > 0}X "v h?2Q`2K jX9XR- i?2 `2bi`B+iBQMb Q7 N iQ B1 M/ B2 = B¯1 `2 BM/2T2M/2Mi- M/ i?2`27Q`2 i?2 p`B#H2b e−N (g− ) M/ eN (g+ ) `2 BM/2T2M/2MiX AM T`iB+mH`- 7`QK i?2 irQ Hbi /BbTHvb   E μ(Xn ) n∈N

= E eN (log(μ)) = E eN (g) = E eN (g+ )−N (g− ) = E e−N (g− ) eN (g+ ) = E e−N (g− ) E eN (g+ )        g+ (x)  −g− (x) − 1 ν(dx) exp − 1 ν(dx) e e = exp E  E    g(x)  g(x)   = exp e e − 1 1{g(x)>0} ν(dx) exp − 1 1{g(x)≤0} ν(dx) E E    g(x)  g(x)   = exp e e − 1 1{g(x)>0} ν(dx) + − 1 1{g(x)≤0} ν(dx) E   E   g(x)  = exp e − 1 ν(dx) = exp (μ(x) − 1) ν(dx) . E

Rm

"v bbmKTiBQMb H1 M/ H2 i?2 Hbi [mMiBiv Bb }MBi2 M/ i?2`27Q`2- #v /BpB/BM; i?2 }`bi M/ Hbi i2`Kb Q7 i?2 #Qp2 +?BM Q7 2[mHBiB2b #v Bi- r2 Q#iBM UjXRRVX  h?2Q`2K jX8Xk lM/2` i?2 bbmKTiBQMb H1 M/ H2 - B7 r2 /2}M2 T`Q##BHBiv P #v UjXRyV M/ UjXRkV- N Bb mM/2` T`Q##BHBiv P  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν ;Bp2M #v UjXNVX

S`QQ7X .2MQi2 2tT2+iiBQM rBi? `2bT2+i iQ P #v EX Ai bm{+2b iQ b?Qr i?i i?2 GTH+2 i`Mb7Q`K Q7 N mM/2` T`Q##BHBiv P Bb i?i Q7  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 ν- i?i Bb- 7Q` Mv #QmM/2/ MQM@M2;iBp2 K2bm`#H2 7mM+iBQM ϕ : Rm → R-

jX8X *>L:1 P6 S_P""AGAhu E e−N (ϕ) = exp

Nj 

 e−ϕ(x) − 1 ν(dx) .

 E

"mi E e−N (ϕ) = E L e−N (ϕ)    = E eN (log(μ))− E (μ(x)−1)ν(dx) e−N (ϕ)  = E eN (−ϕ+log(μ)) e− E (μ(x)−1)ν(dx)        −ϕ(x)+log(μ(x)) = exp − 1 ν(dx) exp − (μ(x) − 1)ν(dx) e E    E   −ϕ(x)  = exp e μ(x) − 1 ν(dx) exp − (μ(x) − 1)ν(dx) E  E   −ϕ(x)  −ϕ(x)   = exp e e − 1 μ(x)ν(dx) = exp − 1 ν(dx) . E

E

 1tKTH2 jX8Xj, .2`BpiBp2 Q7 i?2 1tT2+iiBQM Q7  SQBbbQM 6mM+@ iBQMHX8 h?2 b2iiBM; Bb i?i Q7 1tKTH2 jXRX8- rBi? i?2 bK2 Q#D2+iBp2X h?Bb iBK2 r2 mb2 MQi?2` K2i?Q/X h?2 T`Q##BHBiv Pα Bb Q#iBM2/ 7`QK i?2 T`Q##BHBiv P1 KFBM; Q7 N  SQBbbQM T`Q+2bb QM E rBi? K2M K2bm`2 Q pB i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 dPα = αN (E) exp (−(α − 1)Q(E)) . dP1 h?2`27Q`2

dPα Eα [f (N )] = E1 f (N ) dP1   = E1 f (N ) αN (E) exp (−(α − 1)Q(E)) . .Bz2`2MiBiBM; mM/2` i?2 2tT2+iiBQM bB;M vB2H/b  

dPα 1 dEα [f (N )] = E1 f (N ) N (E) − Q(E) dα α dP  1  1 = Eα f (N ) N (E) − Q(E) . α U ;2M2`HBxiBQM Q7 i?Bb 2tKTH2 Bb ;Bp2M BM 1tKTH2 8X8X9XV h?2 +b2 Q7 HQ+HHv }MBi2 BMi2MbBiv K2bm`2b aQ 7` r2 ?p2 +QMbB/2`2/ i?2 +b2 r?2`2 P M/ P ?p2 }MBi2 BMi2MbBiv K2bm`2bX q2 MQr 2tKBM2 i?2 +b2 r?2`2 i?2b2 BMi2MbBiv K2bm`2b `2 QMHv HQ+HHv }MBi2X G2i N #2 i?2 +QQ`/BMi2 T`Q+2bb QM (Mp (E), Mp (E)) M/ H2i P #2 i?2 T`Q#@ #BHBiv K2bm`2 i?i KF2b Q7 Bi  SQBbbQM T`Q+2bb N QM E rBi? HQ+HHv }MBi2 8

("`ûKm/ M/ o€x[m2x@#/- RNNk)X

N9

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

BMi2MbBiv K2bm`2 νX G2i μ : E → R #2 bQK2 MQM@M2;iBp2 K2bm`#H2 7mM+iBQM HQ+HHv BMi2;`#H2 rBi? `2bT2+i iQ ν ;m`Mi22BM; i?i i?2 K2bm`2 ν ;Bp2M #v  ν(C) := μ(x) ν(dx) (C ∈ B(E)) UjXRjV E

Bb HbQ HQ+HHv }MBi2X G2i {Bn }n≥0 #2 M BM+`2bBM; b2[m2M+2 Q7 +QKT+i b2ib r?Qb2 mMBQM Bb E M/ bm+? i?i ν(Bn ) M/ ν(Bn ) `2 }MBi2X h?2 +QM/BiBQMb QM ν M/ μ ;m`Mi22 i?2 2tBbi2M+2 Q7 bm+?  b2[m2M+2X .2MQi2 #v Fn i?2 σ@}2H/ ;2M2`i2/ #v i?2 `2bi`B+iBQM Q7 N iQ Bn X G2i An := Bn − Bn−1 Un ≥ 1VX G2i PAn #2 i?2 `2bi`B+iBQM Q7 P iQ i?2 σ@}2H/ Gn ;2M2`i2/ #v i?2 `2bi`B+iBQM Q7 N iQ An X "v h?2Q`2K jX8Xk- i?2 T`Q##BHBiv PAn QM (Ω, Gn ) /2}M2/ #v      dPAn = μ(Xi )1An (Xi ) exp (μ(x) − 1) ν(dx) () dPAn An i Bb bm+? i?i i?2 `2bi`B+iBQM Q7 N iQ An Bb  SQBbbQM T`Q+2bb r?Qb2 BMi2MbBiv K2bm`2 Bb i?2 `2bi`B+iBQM Q7 ν iQ An X .2MQiBM; #v Zn i?2 [mMiBiv UV- i?2 b2[m2M+2 {Zn }n≥1 Bb- BM pB2r Q7 i?2 BM/2T2M/2M+2 T`QT2`iB2b Q7 SQBbbQM T`Q+2bb2b- BB/X :HmBM; iQ;2i?2` i?2 `2bi`B+iBQMb Q7 N iQ i?2 An Ƕb- r2 Q#iBM  T`Q##BHBiv K2bm`2 P QM (Ω, F) bm+? i?i n n   d PA k dP Ln := | Fn = = Zk . dP dPAk k=1 k=1 h?2 ;2M2`H `2bmHib Q7 a2+iBQM X8 i2HH mb i?i i?2`2 `2 QMHv irQ TQbbB#BHBiB2b, P  P Q` P ⊥P - M/ i?i P  P ⇐⇒

∞ 

1

E[Zn2 ] > 0 .

n=1

LQr-





 1 (μ(x) − 1) ν(dx) (μ(Xi ) )1An (Xi ) exp Zn = An 2 i      1 1 2 2 (μ(Xi ) )1An (Xi ) exp (μ(x) − 1) ν(dx) = 1 2



1 2



An

  1 1 (μ(x) − 1) ν(dx) exp{ −(μ(x) 2 − 1) ν(dx)} exp 2 An  An  1 1 n exp =L (μ(x) − 1) − (μ(x) 2 − 1) ν(dx) , An 2 

i



r?2`2 n := L 



 i

 1 2



1



(μ(x) 2 − 1) ν(dx)

(μ(Xi ) )1An (Xi ) exp An

n = 1 UTTHv G2KK jX8XR rBi? μ(x) 12 BMbi2/ Q7 μ(x)VX h?2`27Q`2 Bb bm+? i?i E L

jX8X *>L:1 P6 S_P""AGAhu

N8



1 1 (μ(x) − 1) − (μ(x) 2 − 1) ν(dx) 2 An  1 = exp (μ(x) 2 − 1)2 ν(dx)

1



E[Zn2 ] = exp

An

M/ BM T`iB+mH` ∞ 

1



E[Zn2 ] = exp

n=1

1 1 − (μ(x) 2 − 1)2 ν(dx) 2 E

 .

h?2`27Q`2, h?2Q`2K jX8X9  ' P  P ⇐⇒ ( μ(x) − 1)2 ν(dx) < ∞ . E

h?2 JBt2/ SQBbbQM *b2 G2i N #2  SQBbbQM T`Q+2bb QM E Q7 }MBi2 BMi2MbBiv K2bm`2 ν M/ H2i Λ #2  MQM@M2;iBp2 `M/QK p`B#H2 BM/2T2M/2Mi Q7 N X G2i L := ΛN (E) exp{−(Λ − 1)N (E)} . h?2 `;mK2Mib Q7 i?2 T`QQ7 Q7 G2KK jX8XR M/ h?2Q`2K jX8Xk `2 BKK2/Bi2Hv /Ti#H2 iQ b?Qr i?i EP [L] = 1 M/ i?i mM/2` i?2 T`Q##BHBiv K2bm`2 P  /2}M2/ #v ddPP = L- N Bb  *Qt T`Q+2bb U?2`2  KBt2/ SQBbbQM T`Q+2bbV rBi? σ(Λ)@ +QM/BiBQMH BMi2MbBiv K2bm`2 Λν(dx)X h?2Q`2K jX8X8 lM/2` i?2 #Qp2 +QM/BiBQMb- 7Q` Mv MQM@M2;iBp2 7mM+iBQM g : R+ → R g(λ)λN (E) e−λN (E) F (dλ) N . UjXR9V EP g(Λ) | F =  N (E) −λN (E) λ e F (dλ) S`QQ7X h?2 T`QQ7 Bb #b2/ QM i?2 7QHHQrBM; 7mM/K2MiH H2KK, G2KK jX8Xe G2i P M/ Q #2 irQ T`Q##BHBiv K2bm`2b QM i?2 K2bm`#H2 bT+2 dP (Ω, F) bm+? i?i P  Q M/ H2i L := dQ X G2i Z #2  MQM@M2;iBp2 `M/QK p`B#H2X 6Q` Mv bm#@σ@}2H/ G Q7 FEQ [L | G] EP [Z | G] = EQ [ZL | G]

Q@XbX

UjXR8V

Q`- 2[mBpH2MiHvEP [Z | G] =

EQ [ZL | G] EQ [L | G]

P @XbX

UjXReV

Ne

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

S`QQ7X "v /2}MBiBQM Q7 +QM/BiBQMH 2tT2+iiBQM- 7Q` HH A ∈ G 

 EP [Z | G] dP .

Z dP = A

A

"v /2}MBiBQM Q7 L M/ Q7 +QM/BiBQMH T`Q##BHBiv ;BM





Z dP = A

EQ [ZL | G] dQ .

ZL dQ = A

A

HbQ 

 EP [Z | G] dP = A

A

EP [Z | G] L dQ EP [Z | G] EQ [L | G] dQ .

= A

h?2`27Q`2



 EQ [ZL | G] dQ = A

EP [Z | G] EQ [L | G] dQ , A

r?B+? Bb 2[mBpH2Mi iQ UjXR8V #2+mb2 A Bb `#Bi``v BM GX aBM+2 P  Q i?Bb 2[mHBiv HbQ ?QH/b P @XbX hQ Q#iBM UjXReV- Bi `2KBMb iQ b?Qr i?i P (EQ [L | G] = 0) = 0X AM/22/ P (EQ [L | G] = 0) =



=  =

1{EQ [L|G]=0} dP 1{EQ [L|G]=0} LdQ 1{EQ [L|G]=0} EQ [L | G] dQ = 0 . 

q2 Kv MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K jX8X8X "v G2KK jX8Xe EP g(Λ)L | F N EP g(Λ) | F N = EP [L | F N ] M/ i?2`27Q`2- bBM+2 mM/2` P - N M/ Λ `2 BM/2T2M/2Mi

EP g(Λ) | F

N



 =

7`QK r?B+? i?2 `2bmHi 7QHHQrbX

R+



g(λ)λN (E) exp{−(λ − 1)N (E)} F (dλ)

R+

λN (E) exp{−(λ − 1)N (E)} F (dλ)

, 

jXeX 1s*h aJSGAL: P6 *Glah1_ SPALh S_P*1aa1a

jXe

Nd

1t+i aKTHBM; Q7 *Hmbi2` SQBMi S`Q+2bb2b

h?2 ;2M2`H b2iiBM; Q7 i?Bb bm#b2+iBQM Bb i?i Q7  +Hmbi2` `M/QK K2bm`2 Ub22 am#b2+iBQM RX8V r?Qb2 MQiiBQM Bb /QTi2/X h?2 T`Q#H2K +QMbB/2`2/ Bb i?i Q7 bKTHBM; i?2 /Bbi`B#miBQM Q7 NW - i?2 `2bi`B+iBQM Q7 N iQ  +QKT+i bm#b2i W ⊂ E Ui?2 dzrBM/QrǴVX U_2+HH i?i bKTHBM; i?2 /Bbi`B#miBQM Q7  `M/QK K2bm`2 K2Mb +QMbi`m+iBM;  `M/QK K2bm`2 rBi? i?2 ;Bp2M /Bbi`B#miBQMXV aKTHBM; N BM i?2 rBM/Qr W `2[mB`2b BM T`BM+BTH2 iQ ;2M2`i2 HH i?2 TQBMib X0,n Q7 i?2 ;2`K TQBMi T`Q+2bb N0 - bBM+2 i?2 bbQ+Bi2/ +Hmbi2`b Zn (X0,n , · − X0,n ) +M TQi2MiBHHv +QMiBM TQBMib BM i?2 rBM/Qr W X h?Bb Bb MQi 72bB#H2 B7 i?2`2 Bb M BM}MBi2 MmK#2` Q7 ;2`KbX PM2 bQHmiBQM Bb iQ TT`QtBKi2 NW #v iFBM; BMiQ ++QmMi QMHv i?2 TQBMib Q7 i?2 ;2`K TQBMi T`Q+2bb i?i `2 BM  dzbm{+B2MiHvǴ H`;2 rBM/Qr W  ⊃ W X "mi i?Bb BMi`Q/m+2b 2/;2 2z2+ib- BM T`iB+mH`  HQbb Q7 Kbb BM i?2 rBM/Qr W X h?Bb BHK2Mi }M/b  `2K2/v BM i?2 +b2 Q7 i?2 *Qt +Hmbi2` TQBMi T`Q+2bb rBi? i?2 "`BtĜE2M/HH 2t+i bKTHBM; K2i?Q/Xe h?2 7mM/K2MiH B/2 Q7 i?Bb K2i?Q/ +M #2 TTHB2/ iQ KQ`2 ;2M2`H +b2b- b r2 MQr b?Qr- #v K2Mb Q7 bBKTH2 +H+mHiBQMb QM GTH+2 i`Mb7Q`KbXd q2 +QKTmi2 i?2 GTH+2 i`Mb7Q`K Q7 i?2 +Hmbi2` TQBMi T`Q+2bb M/ i?2 GTH+2 i`Mb7Q`K b BM i?2 +QMbi`m+iBQM T`QTQb2/ BM i?2 "`BtĜE2M/HH +QMbi`m+iBQM- M/ b?Qr i?i i?2v H2/ iQ i?2 bK2 `2bmHiX RX lbBM; i?2 7+i i?i {Zn }n∈Z Bb- +QM/BiBQMHHv QM F N0 - M BB/ b2[m2M+2-





LNW (ϕ) := E exp −

n∈N

 



= E E exp −   =E E



 

ϕ(x) Zn (X0,n , dx − X0,n ) W

n∈N

ϕ(x) Zn (X0,n , dx − X0,n )

|F

N0

W

n∈N







 

    N0 exp − ϕ(x) Zn (X0,n , dx − X0,n ) | F W

     N0 . E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | F =E 



n∈N

W

LQr- rBi? An := {Zn (x, W − X0,n ) > 0}-

e d

("`Bt M/ E2M/HH- kyyk)X ("`ûKm/- kyRe)X

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 E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | F N0 W  

 = E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | X0,n  W 

= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) 1An + 1 − 1An | X0,n  W

 = E exp − ϕ(x) Zn (X0,n , dx − X0,n ) 1{Zn (W −X0,n )>0} | X0,n W

+ P (Zn (X0,n , W − X0,n ) = 0 | X0,n ) := g1 (X0,n ) + g2 (X0,n ) . h?2`27Q`2

 LNW (ϕ) = E



 (g1 (X0,n ) + g2 (X0,n ))

n∈N 

= E e n∈N log(g1 (X0,n )+g2 (X0,n ))    = E e E log(g1 (x)+g2 (x)) N0 (dx)    = E e E log(g1 (x)+P (Z1 (x,W −x)>0)) N0 (dx) . r?2`2 Z1 (x, · − x) Bb  ivTB+H +Hmbi2` Zn (·, ·)X LQr g1 (x) + P (Z1 (x, W − x) > 0) g1 (x) = P (Z1 (x, W − x) > 0) + P (Z1 (x, W − x) > 0) . P (Z1 (x, W − x) > 0) P#b2`p2 i?i    g1 (x) = E e− W ϕ(x) Nx (dx) 1{Z1 (x,W −x)>0} /P (Z1 (x, W − x) > 0) P (Z1 (x, W − x) > 0) Bb i?2 GTH+2 i`Mb7Q`K Q7 i?2 `M/QK K2bm`2 Z1 (x, · − x) +QM/BiBQM2/ iQ ?p2 i H2bi QM2 TQBMi BM W X W kX h?2 "`BtĜE2M/HH 2t+i bKTHBM; H;Q`Bi?K +QMbi`m+ib  TQBMi T`Q+2bb N QM W b 7QHHQrbX 6B`bi- i?2 TQBMi T`Q+2bb N0 Bb i?BMM2/-  TQBMi X0,n #2BM; `2iBM2/ rBi? T`Q##BHBiv p(X0,n ) r?2`2 p(x) := P (Z1 (x, W − x) > 0)X JQ`2 T`2+Bb2Hv- i?2 0 Bb /2}M2/ #v i?BMM2/ TQBMi T`Q+2bb N  0 (C) := N 1C (X0,n )Yn , n∈N

r?2`2 {Yn }n∈Z Bb- +QM/BiBQMHHv QM FN0 - M BM/2T2M/2Mi b2[m2M+2 rBi? pHm2b BM {0, 1}- M/ 7Q` 2+? n ∈ N- P (Yn = 1 | F N0 ) = P (Yn = 1 | X0,n ) = p(X0,n )X h?2M 7Q` 2+? n ∈ N- H2i Zn #2  TQBMi T`Q+2bb i?i ?b i?2 bK2 /Bbi`B#miBQM b Zn +QM/BiBQM2/ #v Zn (X0,n , W − X0,n ) > 0X Uh?2 TQBMi T`Q+2bb Zn Bb Q#@ iBM2/ #v bKTHBM; BM/2T2M/2Mi TQBMi T`Q+2bb2b Q7 i?2 ivT2 Zn mMiBH i?2 +QM/BiBQM

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NN

 Bb i?2M +QMbi`m+i2/ Zn (X0,n , W − X0,n ) > 0 Bb biBb}2/XV h?2 +M/B/i2 bKTH2 N b  W (C) = Yn Zn (X0,n , C − X0,n ) . N n∈N

Uh?2`27Q`2- QMHv i?2 Zn +Q``2bTQM/BM; iQ  TQBMi X0,n i?i ?b #22M `2iBM2/ rBHH W Bb i?2 /2bB`2/ 2t+i bKTH2- Bi M22/ iQ #2 bKTH2/XV AM Q`/2` iQ +?2+F i?i N Kmbi #2 T`Qp2/ i?i Bi ?b i?2 bK2 /Bbi`B#miBQM b NW X h?Bb Bb /QM2 #2HQr #v b?QrBM; i?i i?2v ?p2 i?2 bK2 GTH+2 7mM+iBQMHX q`Bi2      L  (ϕ) := E exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) NW

W

n∈N

    exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) =E 

W

n∈Z

  =E E  =E

    exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) | F N0



W

n∈Z



  

 N0  E exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) | F .

W

n∈N

LQr r`Bi2  

 N0  E exp − ϕ(x) Yn Zn (X0,n , dx − X0,n ) | F  W 

= E exp − ϕ(x) Zn (X0,n , dx − X0,n ) Yn | X0,n + E [1 − Yn | X0,n ]  W 

 = E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | X0,n E [Yn | X0,n ] + E [1 − Yn | X0,n ]  W

 = E exp − ϕ(x) Zn (X0,n , dx − X0,n ) | X0,n p(X0,n )) + (1 − p(X0,n ))) W

= g(X0,n )p(X0,n ) + 1 − p(X0,n ) , r?2`2 g(x) Bb i?2 GTH+2 7mM+iBQMH Q7 Z1 (x, · − x) +QM/BiBQM2/ iQ ?p2 i H2bi QM2 TQBMi BM W X h?2 `2bi Q7 i?2 p2`B}+iBQM Bb +QKTH2i2/ #v    LNW (ϕ) = E (g(X0,n )p(X0,n ) + 1 − p(X0,n )

n∈N





log(g(x)p(x) + 1 − p(x)) N0 (dx)

= E exp

,

E

M/ i?2 Q#b2`piBQM g(x)p(x) = g1 (x)X 1tKTH2 jXeXR, 1t+i aKTHBM; Q7  *Qt *Hmbi2` SQBMi S`Q+2bbX AM i?2 +b2 r?2`2 N0 Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 μ - i?2 2t+i bKTHBM; 0 Q7 N0 - BM i?Bb +b2  SQBbbQM T`Q+2/m`2 +QMbBbib Q7 +QMbi`m+iBM;  i?BMM2/ p2`bBQM N T`Q+2bb rBi? BMi2MbBiv K2bm`2 P (Z1 (x, W − x) > 0) μ(dx)- M/ 7`QK 2+? TQBMi

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0,n Q7 N 0 `2HBx2  p2`bBQM Q7 Zn X h?2`2 `2 irQ +QM/BiBQMb 7Q` i?Bb iQ T`Q/m+2 M X 2t+i bKTHBM; Q7 NW BM  }MBi2 MmK#2` Q7 QT2`iBQMbX h?2 MmK#2` Q7 TQBMib Q7 0 Kmbi #2 }MBi2-  bm{+B2Mi +QM/BiBQM 7Q` i?Bb #2BM; i?i N  P (Z1 (x, W − x) > 0) μ(dx) < ∞ . () E

6Q`  *Qt +Hmbi2` TQBMi T`Q+2bb P (Z1 (x, W − x) > 0) = 1 − e−K(x,W −x) M/ i?2`27Q`2 Rd P (Z1 (x, W − x) > 0) μ(dx) = Rd (1 − e−K(x,W −x) ) μ (dx) < ∞X h?2 Hbi BM2[mHBiv 7QHHQrb 7`QK +QM/BiBQM URXR8V M/ i?2 BM2[mHBiv 1−e−x ≤ x Ux ∈ RVX

1ti2MbBQM iQ LQM@SQBbbQMBM :2`K S`Q+2bb2b amTTQb2 ;`Mi2/ i?2 TQbbB#BHBiv Q7 bKTHBM;  +Hmbi2` 2bBHvX h?2M- b i?2 #Qp2 +H+mHiBQMb +QM}`K- i?2`2 `2 irQ BM;`2/B2Mib i?i KF2 i?BM;b rQ`FX 6B`bi Q7 HHi?2 i?BMMBM; T`Q##BHBiv 7mM+iBQM Kmbi #2 pBH#H2 BM +HQb2/ 7Q`KX h?Bb Bb MQi i?2 +b2 7Q`  >rF2b TQBMi T`Q+2bbX AM Q`/2` iQ #vTbb i?Bb /B{+mHiv- M /TiiBQM Q7 i?2 i?BMMBM; QT2`iBQM rBHH #2 ;Bp2M BM a2+iBQM RkX9X  b2+QM/ TQbbB#H2 /B{+mHiv r?2M ii2KTiBM; iQ 2ti2M/ i?2 "`BtĜE2M/HH K2i?Q/ iQ MQM@SQBbbQMBM ;2`K TQBMi T`Q+2bb2b +QM+2`Mb i?2 Q#i2MiBQM Q7  bKTH2 Q7 i?2 i?BMM2/ ;2`K TQBMi T`Q+2bbX UAM i?2 +b2 Q7  SQBbbQM ;2`K T`Q+2bb- i?2 i?BMM2/ T`Q+2bb Bb HbQ  SQBbbQM T`Q+2bb M/ i?2 /B{+mHiv /Q2b MQi 2tBbiXV 1tKTH2 jXeXk, h?BMMBM; i?2 :`B/X >2`2 Bb i?2 #bB+ B/2X *QMbB/2`  TQBMi T`Q+2bb QM N `2T`2b2Mi2/ #v  b2[m2M+2 {Xn }n≥0 Q7 BB/ {0, 1}@pHm2/ `M/QK p`B@ #H2b- rBi? i?2 +QKKQM /Bbi`B#miBQM ;Bp2M #v P (Xn = 1) = pn Un ≥ 0VX q2 `2 i?2`27Q`2 dzi?BMMBM; i?2 ;`B/Ǵ N U /2i2`KBMBbiB+ TQBMi T`Q+2bbV rBi? i?2 i?BMMBM;

T`Q##BHBiv 7mM+iBQM pn X amTTQb2 i?i n≥0 pn < ∞- r?B+? ;m`Mi22b i?i i?2 i?BMM2/ ;`B/ ?b HKQbi bm`2Hv  }MBi2 MmK#2` Q7 TQBMib M/ H2i T #2 Bib Hbi TQBMiX LQi2 i?i  P (T = n) = P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = pn (1 − pk ) () k≥n+1

M/ i?i- 7Q` 0 ≤ k ≤ n − 1P (Xk = 1, T = n) P (T = n) P (Xk = 1, Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) P (Xk = 1)P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = P (Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .) = P (Xk = 1) .

P (Xk = 1 | T = n) =

h?2`27Q`2- BM Q`/2` iQ bBKmHi2 i?2 i?BMM2/ ;`B/- QM2 Kv bi`i #v bKTHBM;  p`B#H2 T rBi? i?2 /Bbi`B#miBQM UV- M/ B7 T = n- b2i Xn = 1, Xn+1 = 0, Xn+2 = 0, . . .- M/ 7Q` 0 ≤ k ≤ n − 1- bKTH2 Xk rBi? i?2 /Bbi`B#miBQM P (Xk = 1) = pk X

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h?BMMBM; i?2 irQ@/BK2MbBQMH ;`B/ Z2 Bb +QM+2TimHHv i?2 bK2X >2`2 i?2 T`Q##BHBiv Q7 F22TBM; i?2 TQBMi (i, j) ∈ Z2 Bb pi,j r?2`2 Bi Bb bbmK2/ i?i

(i,j)∈Z2 pi,j < ∞ r?2`2#v ;m`Mi22BM; i?i i?2 MmK#2` Q7 TQBMib Q7 i?2 i?BMM2/ ;`B/ Bb }MBi2X Ai bm{+2b iQ TTHv #BD2+iBp2Hv Z2 QM N #v 2MmK2`iBM; i?2 TQBMib Q7 Z2 b {(in , jn )}n≥0 - /2}MBM; i?Bb #BD2+iBQM #v (in , jn ) → nX h?2 `2bi Bb i?2M Q#pBQmbX h?Bb K2i?Q/ Kv #2 mb27mH r?2M  bKTH2 Q7 i?2 ;2`K TQBMi T`Q+2bb Bb ;Bp2M U2tT2`BK2MiHHvVX h?2 #Qp2 i?BMMBM; T`Q+2/m`2 Bb 2bBHv /Ti#H2 iQ i?Bb +b2X h?2`2 Bb biBHH M Bbbm2 H27i bB/2 BM i?2 T`2b2MiiBQM Q7 i?2 i?BMMBM; T`Q+2/m`2 Q7 i?2 ;`B/ NX *M r2 `2HHv bKTH2 T \ AM 7+i QM2 M22/b iQ ?p2 i /BbTQbBiBQM  +HQb2/ 2tT`2bbBQM Q7 i?2 /Bbi`B#miBQM Q7 i?Bb p`B#H2- BM T`iB+mH` Q7 i?2 BM}MBi2 ! T`Q/m+i k≥n+1 (1 − pk )X A7 i?Bb Bb MQi TQbbB#H2- r2 Kv

#2 Hm+Fv 2MQm;? iQ }M/  /QKBMiBM; /Bbi`B#miBQM 7mM+iBQM q ≥ p bm+? i?i n n n qn < ∞ M/ bm+? i?i i?2 ! BM}MBi2 T`Q/m+i k≥n+1 (1−qk ) Bb +QKTmi#H2X PM2 rQmH/ i?2M bKTH2 i?2 i?BMM2/ ;`B/ rBi? i?BMMBM; T`Q##BHBiv 7mM+iBQM qn X  TQBMi Q7 i?Bb /QKBMiBM; ;`B/ HQ+i2/ i k rBHH i?2M #2 F2Ti rBi? T`Q##BHBiv pk /qk b  TQBMi Q7 i?2 /2bB`2/ bKTH2X _2K`F jXeXj MQi?2` 2tKTH2 r?2`2 i?BMMBM; Bb TQbbB#H2 +QM+2`Mb `2M2rH T`Q@ +2bb2bX Ai rBHH #2 i`2i2/ Hi2`- 7i2` i?2 bim/v Q7 `2M2rH T`Q+2bb2b M/ r?2M i?2 iQQHb Q7 biQ+?biB+ BMi2MbBiv i?2Q`v rBHH #2 pBH#H2 U1tKTH2 8XdX8VX a22 HbQ 1t@ KTH2 3X9XReX

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aiQ+?biB+ ;2QK2i`v +QM+2`Mb i?2 bim/v Q7 `M/QK b?T2bX h?2 2tKTH2b +QMbB/2`2/ #2HQr T2`iBM iQ  T`iB+mH` bQ`i Q7 biQ+?biB+ ;2QK2i`v- r?2`2 i?2 `M/QKM2bb Q7 i?2 b?T2b Bb /2T2M/2Mi QM i?2 TQbBiBQMb Q7 i?2 TQBMib Q7 M mM/2`HvBM; TQBMi T`Q+2bbX Hi?Qm;? i?2`2 2tBbib  bQmM/ Ki?2KiB+H i?2Q`v Q7 `M/QK b2ib-3 i?Bb i?2Q`v rBHH MQi #2 M2+2bb`v b HQM; b i?2 `M/QK b2ib +QMbB/2`2/ BM i?2 TTHB+iBQMb `2 dzMB+2Ǵ b2ib 7mHHv /2b+`B#2/ #v  `M/QK p2+iQ` Q7 }MBi2 /BK2MbBQM U+B`+H2- /BbFTQHv;QM- HBM2- b2;K2Mi- 2i+XVX q2 i?2M `2bQ`i iQ r?i +M #2 +HH2/ i?2 dzTQQ` KMǶb `M/QK b2i i?2Q`vǴ- BM r?B+?  `M/QK b2i Bb  b2i Q7 i?2 7Q`K S(Z) ⊆ Rm r?2`2 Z ∈ Rd Bb  `M/QK p2+iQ` M/ 7Q` 2+? z- S(z) Bb  K2bm`#H2 b2i M/ S(Z) Bb HbQ  K2bm`#H2 b2iX h?2b2 `2 KBMBKH `2[mB`2K2Mib mbmHHv biBb}2/ BM TTHB+iBQMbX q2 b?HH +QMbB/2` `2H@pHm2/ 7mM+iBQMb Q7 S- 7Q` BMbiM+2 g(S) = d (S) , g(S) = 1a∈S , 7Q` r?B+? i?2 2tT2+iiBQM Bb r2HH /2}M2/- b  g(S(z))P (Z ∈ dz) . E [g(S)] := E [g(S(Z))] = Rd

()

UjXRdV

q2 QMHv M22/ iQ 2Mbm`2 i?i i?2 7mM+iBQM z ∈ Rd → g(S(z)) ∈ R Bb K2bm`#H2 M/ Bib BMi2;`H rBi? `2bT2+i iQ i?2 /Bbi`B#miBQM Q7 Z r2HH /2}M2/ UBM i?2 2tKTH2b3

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i?Bb rBHH ;2M2`HHv i?2 +b2- #2+mb2 g rBHH #2  MQM@M2;iBp2 7mM+iBQM- b BM UVVX q2 b?HH mb2 7Q` UjXRdV i?2 ##`2pBi2/ MQiiBQM  g(s) Q(ds) , S

i?2`2#v T`2i2M/BM; i?i i?2`2 2tBbib  b2i S Q7 b?T2b rBi?  bmBi#H2 σ@}2H/ G QM Bi- M/ M /2[mi2 T`Q##BHBiv /Bbi`B#miBQM Q QM (S, G)X 1tKTH2 jXdXR, _M/QK .BbFX AM i?Bb 2tKTH2- S Bb i?2 +HQb2/ /BbF BM R2 +2Mi2`2/ QM i?2 Q`B;BM M/ rBi? `/Bmb Z-  MQM@M2;iBp2 `M/QK p`B#H2X q2 ?p2- rBi? g(S) = 2 (S) E [g(S)] = E 2 (S(Z)) = E πZ 2 , M/ rBi? g(S) = 1{a∈S} -

E [g(S)] = E 1{a∈S(Z)} = P (a ∈ S(Z)) = P (Z ≥ a) .

.2}MBiBQM jXdXk h?2 +T+Biv 7mM+iBQMH Q7 i?2 `M/QK b2i S Bb i?2 7mM+iBQM K → TS (K) UK +QKT+iV /2}M2/ #v TS (K) := P (S ∩ K = ∅) . 1tKTH2 jXdXj, *T+Biv 6mM+iBQMH Q7  SQBMi S`Q+2bbX  bBKTH2 TQBMi T`Q+2bb N +M HbQ #2 pB2r2/ b  `M/QK b2i S ≡ N X AM i?Bb +b2 i?2 +T+Biv 7mM+iBQMH Bb i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQM, TN (K) := P (N ∩ K = ∅) = P (N (K) = 0) = vN (K) .

1tKTH2 jXdX9, *T+Biv 7mM+iBQMH M/ pQB/M+2 S`Q##BHBiv 6mM+@ iBQMX  `M/QK b2i S ⊂ Rm i?i Bb HKQbi bm`2Hv i?2 +HQbm`2 Q7 Bib BMi2`BQ`N Kv #2 /2b+`B#2/ b i?2 bmTTQ`i Q7  `M/QK K2bm`2- MK2Hv i?2 K2bm`2 μS rBi? /2MbBiv 1S (x) rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 m μS (C) := m (C ∩ S) , C ∈ B(Rm ).  aBM+2 S ∩ K = ∅ Bb 2[mBpH2Mi iQ K 1S (x) dx = 0- i?i Bb μS (K) = 0P (S ∩ K = ∅) = 1 − P (μS (K) = 0) . h?2`27Q`2 1 − TS (K) Bb i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQMH Q7 i?2 `M/QK K2@ bm`2 μS 2pHmi2/ i K- i?i Bb- mbBM; i?2 T`2pBQmb MQiiBQM 7Q` i?2 pQB/M+2 T`Q##BHBiv 7mM+iBQMHTS (K) = vμS (K) .

N

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jXdX h>1 "PPG1L JP.1G

Ryj

_2K`F jXdX8 h?2 JƺM+?Ĝ_ûMvB i?2Q`2K ?b  ;2M2`HBxiBQM +QM+2`MBM; +HQb2/ `M/QK b2ib i?i bvb `Qm;?Hv i?i i?2 +T+Biv 7mM+iBQMH Q7  +HQb2/ `M/QK b2i mMB[m2Hv /2i2`KBM2b i?2 /Bbi`B#miBQM Q7 i?Bb `M/QK b2iXRy Uh?2 [mHB}+iBQM dz`Qm;?HvǴ `272`b iQ i?2 7+i i?i QM2 ?b iQ /2}M2 i?2 /Bbi`B#miBQM Q7  `M/QK +HQb2/ b2i- M/ BM T`iB+mH` /2}M2  K2bm`#H2 bT+2 (F, A) Q7 +HQb2/ b2ibXV q2 MQr BMi`Q/m+2 i?2 "QQH2M KQ/2HXRR b r2 b?HH b22- i?Bb Q#D2+i Bb +HQb2Hv `2Hi2/ iQ +Hmbi2` `M/QK K2bm`2bX G2i N #2  SQBbbQM T`Q+2bb QM Rm rBi?  MQM@ iQKB+ σ@}MBi2 BMi2MbBiv K2bm`2 νX .2MQi2 #v {Xn }n∈N Bib b2[m2M+2 Q7 TQBMibX G2i MQr {Sn }n∈N #2  b2[m2M+2 Q7 `M/QK K`Fb- BB/ M/ BM/2T2M/2Mi Q7 N X 1+? Sn Bb  +QKT+i `M/QK b2iX h?2 Xn Ƕb `2 +HH2/ i?2 ;2`Kb r?2`2b i?2 Sn Ƕb `2 +HH2/ i?2 ;`BMbX A7 A M/ B `2 bm#b2ib Q7 Rm M/ x ∈ Rm - x + A := {x + y ; y ∈ A}M/ A ⊕ B := {x + y ; x ∈ A, x ∈ B}X

6B;m`2 jXdXR PM2 Q7 i?2 [mMiBiB2b Q7 BMi2`2bi BM TTHB+iBQMb Bb i?2 T`Q##BHBiv Q7 BMi2`b2+iBQM Q7 i?2 `M/QK b2i Σ = ∪n∈N (Xn + Sn ) rBi?  ;Bp2M +QKT+i b2i K ⊂ Rm - i?i Bb TΣ (K) := P (Σ ∩ K = ∅) . AM Q`/2` iQ +QKTmi2 i?Bb [mMiBiv Ui?2 +T+Biv 7mM+iBQMH Q7 ΣV- r2 }`bi Q#b2`p2 i?i TΣ (K) = P (N (K) = 0) , r?2`2



N (K) =

1{(Xn +Sn )∩K=∅} .

n∈N

q2 b?Qr i?i N (K) Bb  SQBbbQM p`B#H2 rBi? K2M  θ(K) := P ((x + S1 ) ∩ K = ∅) ν(dx)

()

Rm

M/ i?2`27Q`2

  TΣ (K) = 1 − exp −

Rm

Ry RR

 P ((x + S1 ) ∩ K = ∅) ν(dx)

a22 7Q` BMbiM+2 (JQH+?MQp- kyy8)- h?K RXjXkyX (Ji?2`QM- RNed- RNd8)- (a2``- RN3k)X

.

UjXR3V

Ry9

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

S`QQ7X h?2 GTH+2 i`Mb7Q`K Q7 i?2 /Bbi`B#miBQM Q7 N (K) Bb ;Bp2M #v   E e−tN (K) = E e−t n∈N 1(Xn +Sn )∩K=∅ = E e−t n∈N f (Xn ,Sn ) , r?2`2 t ≥ 0 M/ f (x, s) := 1(x+s)∩K=∅ X hQ b22 i?Bb- mb2 i?2 7Q`KmH BM *Q`QHH`v jXjX9- r?B+? ;Bp2b     −tf (x,S1 )  E e−t n∈N f (Xn ,Sn ) = exp E e − 1 ν(dx) m R    −t1 (x+S1 )∩K=∅ = exp − 1 ν(dx) E e m  R   = exp (e−t − 1) P ((x + S1 ) ∩ K = ∅) ν(dx) . Rm

 A7 i?2 ;2`K T`Q+2bb Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ- i?2 K2M pHm2 θ(K) Q7 N (K) iF2b i?2 7Q`K θ(K) = λE [ m ((−S1 ) ⊕ K)] . AM/22/- 7`QK UV θ(K) = λ Rm



= λE

P ((x + S1 ) ∩ K = ∅) dx

 1(x+S1 )∩K=∅ dx = λE

Rm

Rm

1(−S1 )⊕K (x) dx .

h?2`27Q`2- BM i?2 ?QKQ;2M2Qmb +b2TΣ (K) = 1 − exp {λE [ m ((−S1 ) ⊕ K)]} .

UjXRNV

MQi?2` [mMiBiv Q7 BMi2`2bi r?2M i?2 ;2`K TQBMi T`Q+2bb Bb M ?TT rBi? BMi2MbBiv λ Bb i?2 pQHmK2 7`+iBQM p Q7 i?2 `M/QK b2i Σ- /2}M2/ #v p :=

E [ m (Σ ∩ B)] .

m (B)

Ai Bb BM/2T2M/2Mi Q7 B- M/ #v i`MbHiBQM BMp`BM+2 Q7 i?2 KQ/2H- Bi Bb 2[mH iQ  E B 1x∈Σ dx p= = P (0 ∈ Σ) = P (Σ ∩ {0} = ∅) .

m (B) h?2`27Q`2p = p({0}) = 1 − e− = 1 − e−



Rm

 Rm

P ((x+S1 )∩{0}=∅) λ dx

P ((x+S1 )∩{0}=∅) λ dx

= 1 − e−

 Rm

P (−x∈S1 ) λ dx

= 1 − e−λE[

h?2 +Qp`BM+2 7mM+iBQM C : Rm → R Q7 i?2 `M/QK b2i Σ Bb /2}M2/ #v

m (S

1 )]

.

jXdX h>1 "PPG1L JP.1G

Ry8

C(x) := P (0 ∈ Σ , x ∈ Σ) . Uh?Bb Bb i?2 +Qp`BM+2 7mM+iBQM- BM i?2 mbmH b2Mb2- Q7 i?2 rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb {1Σ (t)}t∈Rm XV AM i?2 ?QKQ;2M2Qmb +b2C(x) = 2p − 1 − (1 − p)2 exp {λE [ m (S1 ∪ (S1 − x))]} . S`QQ7X C(x) = P (0 ∈ Σ ∩ (Σ − x)) = P (0 ∈ Σ) + P (x ∈ Σ) − P (0 ∈ Σ ∪ (Σ − x)) = 2p − 1 + P (0 ∈ / Σ ∪ (Σ − x)) = 2p − 1 + P (Σ ∩ {o, x} = ∅) = 2p − 1 + TΣ ({0, x}) . 6`QK UjXRNVTΣ ({0, x}) = 1 − exp {λE [ m ((−S1 ) ⊕ {0, x})]} . LQr E [ m ((−S1 ) ⊕ {0, x})] = E [ m ((−S1 ) ∪ (−S1 + x))] = E [ m (−S1 )] + E [ m (−S1 + x)] − E [ m ((−S1 ) ∩ (−S1 + x))] = 2E [ m (−S1 )] − E [ m ((−S1 ) ∩ (−S1 + x))] . *QK#BMBM; i?2 #Qp2 2[mHBiB2b rBi? i?2 Q#b2`piBQM i?i 1 − p = exp {−λ m (S1 )} ;Bp2b i?2 MMQmM+2/ `2bmHiX  1tKTH2 jXdXe, "QmM/`v Q7  SQBbbQM *Hmbi2` Q7 .BbFbX G2i N #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R2 - Q7 BMi2MbBiv λX G2i {Xn }n∈N #2 Bib b2[m2M+2 Q7 TQBMibX .`r `QmM/ 2+? TQBMi Xn  +HQb2/ /BbF Q7 `/Bmb aX h?2 `2 BMbB/2 i?2 b[m`2 [0, T ] × [0, T ] i?i Bb MQi +Qp2`2/ #v  /BbF Bb /2HBKBi2/ #v  +m`p2X q2 b22F iQ +QKTmi2 Bib p2`;2 H2M;i?- 2t+Hm/BM; i?2 T`ib QM i?2 #QmM/`B2b Q7 [0, T ] × [0, T ]X

6B;m`2 jXdXkX  "QQH2M b2i M/ Bib #QmM/`v h?2 MmK#2` Q7 /BbFb +Qp2`BM;  ;Bp2M TQBMi y ∈ R2 Bb

Rye

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a Z(y) =



1{||Xn −y|| 0X 6Q` HH t ≥ 0- H2i 2ε2

Rj

+ S0,t := {(s, z) : 0 < s < t, t − s < z} , − S0,t := {(s, z) : −∞ < s < 0, −s < z < t − s}

M/ H2i

  + − Xε (t) := ε N ε (S0,t ) − N ε (S0,t ) .

URV a?Qr i?i Xε (t) Bb r2HH /2}M2/ 7Q` HH t ≥ 0X UkV *QKTmi2 i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 (Xε (t1 ), . . . , Xε (tn )) 7Q` HH 0 ≤ t1 ≤ t 2 . . . ≤ tn X Rj

(*BQ+x2F@:2Q`;2b M/ JM/2H#`Qi- RNN8)X

jX3X 1s1_*Aa1a

RRR

UjV a?Qr i?i 7Q` HH 0 ≤ t1 ≤ t2 . . . ≤ tn - (Xε (t1 ), . . . , Xε (tn )) +QMp2`;2b b ε ↓ 0 BM /Bbi`B#miBQM iQ (BH (t1 ), . . . , BH (tn ))- r?2`2 {BH (t)}t≥0 Bb  7`+iH "`QrMBM KQiBQM rBi? >m`bi T`K2i2` H = 1−θ M/ p`BM+2 E [BH (1)2 ] = θ−1 (1 − θ)−1 2 i?i Bb- {BH (t)}t≥0 Bb  +2Mi2`2/ :mbbBM T`Q+2bb bm+? i?i BH (0) = 0 M/ rBi? +Qp`BM+2 7mM+iBQM  1  2H E [BH (t)BH (s)] = |s| + |t|2H − |s − t|2H E BH (1)2 . 2

+ S0,t

− S0,t

t

0 6B;m`2 jX3XR

1t2`+Bb2 jX3X3X .Bbi`B#miBQM Q7 i?2 JtBKmK LmBbM+2 G2i N #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm Q7 TQbBiBp2 BMi2MbBiv λ M/ rBi? TQBMi b2[m2M+2 {Xn }n≥1 X G2i {Zn }n≥1 #2 M BB/ b2[m2M+2 Q7 `2H MQM@M2;iBp2 `M/QK p`B#H2b rBi? +QKKQM /Bbi`B#miBQM Q- M/ BM/2T2M/2Mi Q7 N X *QKTmi2 i?2 /Bbi`B#miBQM Q7 i?2 `M/QK p`B#H2 max Zn e−β||Xn || n≥1

(β > 0) .

Uh?2 iBiH2 Q7 i?2 2t2`+Bb2 `272`b iQ  +QKKmMB+iBQMb +QMi2ti r?2`2 Zn Bb i?2 MQBb2 BMi2MbBiv ;2M2`i2/ i TQBMi Xn - M/ e−β||Xn || Bb M ii2MmiBQM 7+iQ` 7Q`  `2+2Bp2` HQ+i2/ i 0XV 1t2`+Bb2 jX3XNX h?2 Jf:Af∞ JQ/2H- hF2 k AM 1tKTH2 jXjXkUBV T`Qp2 i?i i?2 /2T`im`2 T`Q+2bb Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2M@ bBiv λUBBV +QKTmi2 cov(X(t), X(t + τ )) 7Q` HH t, τ ∈ R- τ ≥ 0- M/ UBBBV BMi2`T`2i i?2 T`Q+2bb {X(t)}t∈R b  b?Qi MQBb2 BM Q`/2` iQ Q#iBM i?2 `2bmHib Q7 1tKTH2 jXjXk- M/ Q7 UBV M/ UBBV- 7`QK i?2 ;2M2`H `2bmHib Q7 h?2Q`2K jX9XeX 1t2`+Bb2 jX3XRyX GB7iBM; G2i N #2  SQBbbQM T`Q+2bb QM R rBi? i?2 UHQ+HHv BMi2;`#H2V BMi2MbBiv 7mM+iBQM λ : R → RX G2i {Tn }n∈Z #2 Bib b2[m2M+2 Q7 TQBMib M/ H2i {Un }n∈Z #2 M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X G2i N #2 M ?TT QM R × R+ - rBi? BMi2MbBiv 1 M/ BM/2T2M/2Mi Q7 N M/ Q7 {Un }n∈Z X .2}M2  TQBMi  QM R × R+ #v T`Q+2bb N

RRk

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a  (C) := N



1C ((Tn , Un λ(Tn ))) + N (C ∩ H)

n∈Z

r?2`2 H := {(t, z) ∈ R × R+ ; 0 ≤ z ≤ λ(t)} .  Bb M ?TT QM R × R+ rBi? BMi2MbBiv 1X a?Qr i?i N 1t2`+Bb2 jX3XRRX *QKTQmM/ SQBbbQM S`Q+2bb G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ M/ TQBMi b2[m2M+2 {Tn }n≥1 X G2i {Zn }n≥1 #2 M BB/ `2H@pHm2/ b2[m2M+2 BM/2T2M/2Mi Q7 N - rBi? +QKKQM +mKmHiBp2 /Bbi`B@ #miBQM 7mM+iBQM F X .2}M2 7Q` HH t ≥ 0 Y (t) = Zn 1(0,t] (Tn ) . n≥1

Uh?2 T`Q+2bb {Y (t)}t≥0 Bb +HH2/  +QKTQmM/ SQBbbQM T`Q+2bbXV a?Qr i?i E e−rY (t) = eλt(1−h(r)) ∞ r?2`2 h(r) := E e−rZ1 = 0 e−rx dF (x)X 1t2`+Bb2 jX3XRkX qi2` "QK#b uQm `2 BMBiBHHv HQ+i2/ i i?2 Q`B;BM (0, 0) Q7 i?2 THM2 i r?B+? Bb +2Mi2`2/  /BbF . Q7 `/Bmb RX uQm `mM BM  bi`B;?i HBM2 7`QK i?2 Q`B;BM iQ i?2 dzb?2Hi2` TQBMiǴ (0, R) i +QMbiMi bT22/ vX h?2 `2bQM r?v vQm `2 `mMMBM; Bb i?i ri2` #QK#b `2 /`QTT2/ QM i?2 /BbF .X h?2 iBK2b Q7 BKT+i 7Q`K M ?TT Q7 BMi2MbBiv λ- M/ 2+? BKT+i Bb HQ+i2/ BM/2T2M/2MiHv Q7 HH i?2 `2bi- mMB7Q`KHv QM i?2 /BbFX uQm rBHH ;2i r2i B7 i?2 BKT+i Q7 i?2 #QK# Bb rBi?BM /BbiM+2 a Q7 vQm` TQbBiBQM i i?2 iBK2 Q7 BKT+iX PM+2 vQm ``Bp2 i i?2 b?2Hi2` TQBMi (0, R)- i?2 #QK#BM; biQTbX q?i `2 vQm` +?M+2b Q7 MQi ;2iiBM; r2i\ q?i Bb- ;Bp2M i?i vQm /B/ ;2i r2ii?2 2tT2+i2/ iBK2 i?i vQm `2KBM2/ /`v\ 1t2`+Bb2 jX3XRjX aKQFBM; SQi i aBMi J`v@CM2ǰb *QHH2;2 aKQFBM; TQi `2+2MiHv ;Qi #MM2/ QM i?2 aBMi J`v@CM2Ƕb +QHH2;2 +KTmbX h?2 mi?Q`BiB2b MQiB+2/ i?i i?2 pBQHiQ`b Q7 i?2 #M KF2 mb2 Q7  `2bi`QQK BM  b2+Hm/2/ rBM; Q7 i?2 +KTmbX h?2v +QMb2[m2MiHv /2pBb2/  bi`i2;v iQ b2M/ dz+QTbǴ iQ +Tim`2 i?2 +mHT`BibX bbmK2 i?i i?2 b+?QQH#QvbǶ ``BpH iBK2b BM i?2 `2bi`QQK T`2KBb2b 7Q`K  SQBbbQM T`Q+2bb rBi? BM/2T2M/2Mi BB/ K`FbX G2i τn /2MQi2 i?2 n@i? ``BpH iBK2 Q7  b+?QQH#Qv BM i?2 TQi bM+im`v Ui?2 `2bi`QQKbV M/ H2i σn #2 i?2 iBK2 ?2 bT2M/b BM bKQFBM;X h?2 +QTbǶ ``BpH iBK2b HbQ 7Q`K  SQBbbQM T`Q+2bb rBi? BM/2T2M/2Mi BB/ K`FbX .2MQi2 i?2 k@i? ``BpH iBK2 Q7  +QT i i?2 TQi2MiBH +`BK2 b+2M2 #v Tk M/ H2i Sk `2T`2b2Mi i?2 HBM;2`BM; iBK2 i?2`2 Q7 i?2 +Q``2bTQM/BM; `2T`2b2MiiBp2 Q7 i?2 +QHH2;2 mi?Q`BivX h?2 T`Q##BHBiv /Bbi`B#miBQM Q7 σ Bb Qs M/ i?i Q7 S Bb Qc X bbmKBM; i?2 TQBMi T`Q+2bb2b Q7 bim/2Mib M/ Q7 i?2 +QTb iQ #2 ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λs > 0 M/ λc > 0- +QKTmi2 i?2 p2`;2 MmK#2` Q7 bim/2Mib +m;?i T2` mMBi Q7 iBK2X

jX3X 1s1_*Aa1a

RRj

1t2`+Bb2 jX3XR9X h?B`/ JQK2Mib Q7  a?Qi LQBb2 G2i N #2 M ?TT QM Rd rBi? BMi2MbBiv λ > 0X G2i {X(t)}t∈R #2 i?2 b?Qi MQBb2 +QMbi`m+i2/ QM Bi rBi? M BKTmHb2 7mM+iBQM h : Rd → R i?i Bb #QmM/2/ M/ rBi? +QKT+i bmTTQ`i UMmHH QmibB/2  #QmM/2/ bm#b2i Q7 Rd V,  X(t) := h(t − s) N (ds) . R

G2i t1 - t2 - t3 ∈ R X d

UV *QKTmi2 i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 i?2 p2+iQ` (X(t1 ), X(t2 ), X(t3 ))X U#V *QKTmi2 E [X(t1 )X(t2 )X(t3 )]X 1t2`+Bb2 jX3XR8X SQBMib BM  _M/QK AMi2`pH G2i N #2 M ?TT QM R+ rBi? BMi2MbBiv λ > 0X G2i Z1 M/ Z2 #2 irQ MQM@M2;iBp2 `2H `M/QK p`B#H2b bm+? i?i Z1 ≤ Z2 X :Bp2 i?2 T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 `M/QK p`B#H2 X = N ((Z1 , Z2 ])- bbmKBM; i?i Z1 M/ Z2 `2 BM/2T2M/2Mi Q7 NX 1t2`+Bb2 jX3XReX GTH+2 h`Mb7Q`K Q7  ?QKQ;2M2Qmb *Qt S`Q+2bb G2i N #2  *Qt T`Q+2bb QM Rm rBi? +QMbiMi BMi2MbBiv T`Q+2bb- i?i Bb ν(dx) := Λ m (dx)- r?2`2 m Bb i?2 G2#2b;m2 K2bm`2 QM R m M/ Λ Bb  MQM@M2;iBp2 `M/QK p`B#H2 rBi? GTH+2 i`Mb7Q`K LΛ (t) := E e−tΛ X *QKTmi2 Bib GTH+2 i`Mb7Q`KX 1t2`+Bb2 jX3XRdX pQB/M+2 S`Q##BHBiv Q7  *Qt S`Q+2bb G2i N #2  *Qt T`Q+2bb QM Rm rBi? `M/QK BMi2MbBiv K2bm`2 νX G2i N  #2  SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2bm`2 ν  2[mH iQ i?2 K2M Q7 ν Ui?i Bb- 7Q` HH C ∈ B(Rm )- ν  (C) = E [ν(C)]VX 6Bt Mv bm#b2i C ∈ B(Rm )X a?Qr i?i i?2 T`Q##BHBiv i?i i?2`2 Bb MQ TQBMi Q7 N BM C Bb ;`2i2` i?M Q` 2[mH iQ i?2 +Q``2bTQM/BM; T`Q##BHBiv 7Q` N  X 1t2`+Bb2 jX3XR3X GBM2 Q7 aB;?i- hF2 R *QMbB/2`  SQBbbQM N QM R2 rBi? MQM@iQKB+ M/ HQ+HHv }MBi2 BMi2MbBiv K2bm`2 νX h?2`2 Bb  dz`M/QK b?T2Ǵ +2Mi2`2/ `QmM/ 2+? Q7 Bib TQBMibX G2i i?2 ;2M2`B+ b?T2 S #2 dz/Bbi`B#mi2/ ++Q`/BM; iQ bQK2 T`Q##BHBiv /Bbi`B#miBQM QS ǴX LQr +QMbB/2` irQ `#Bi``v TQBMib A- BX q2 bv i?i A M/ B +M +QKKmMB+i2 B7 i?2 HBM2 +QMM2+iBM; A M/ B /Q2b MQi BMi2`b2+i Mv Q7 i?2 2tBbiBM; b?T2b `QmM/ i?2 TQBMib Q7 i?2 TQBMi T`Q+2bb U7Q` HH n ≥ 1- i?2 dz2tBbiBM; b?T2 `QmM/Ǵ Xn ∈ N Bb Xn + Sn - i?i Bb Sn i`MbHi2/ #v Xn - r?2`2 Sn Bb /Bbi`B#mi2/ ++Q`/BM; iQ QS VX q2 bbmK2 i?i {Sn }n∈N Bb M BB/ b2[m2M+2 BM/2T2M/2Mi Q7 N X *QKTmi2 i?2 T`Q##BHBiv i?i A M/ B +M bm++2bb7mHHv +QKKmMB+i2X Smb? i?2 +QKTmiiBQMb r?2M N Bb M ?TT Q7 BMi2MbBiv λ- i?2 `M/QK b?T2 Bb  +B`+H2 Q7 `/Bmb a- M/ i?2 /BbiM+2 7`QK A iQ B Bb 2[mH iQ lX 1t2`+Bb2 jX3XRNX GBM2 Q7 aB;?i- hF2 k h?2 bBimiBQM Bb i?i Q7 1t2`+Bb2 jX3XR3X h?2`2 Bb  i`p2HH2` ;QBM; 7`QK B iQ  i?B`/ TQBMi CX q?i Bb i?2 T`Q##BHBiv i?i ?2 rBHH ?p2 A BM bB;?i HH i?2 iBK2 /m`BM; ?Bb i`p2H\

RR9

*>Sh1_ jX aShAG SPAaaPL S_P*1aa1a

1t2`+Bb2 jX3XkyX *2HH S?QM2b M/ "b2 aiiBQMb G2i N1 M/ N2 #2 irQ BM/2T2M/2Mi SQBbbQM T`Q+2bb2b QM Rm rBi? `2bT2+iBp2 }MBi2 BMi2MbBiv K2bm`2b ν1 M/ ν2 X *QKTmi2 i?2 p2`;2 MmK#2` Q7 TQBMib Q7 N1 i?i b22 MQ TQBMi Q7 N2 rBi?BM /BbiM+2 aX Uh?2 iBiH2 Q7 i?Bb 2tKTH2 `272`b iQ i?2 bBimiBQM r?2`2 i?2 TQBMib Q7 N2 `2 i?2 HQ+iBQMb Q7 #b2 biiBQMb BM  rB`2H2bb +QKKmMB+iBQMb M2irQ`F M/ i?Qb2 Q7 N1 `2 i?2 HQ+iBQMb Q7 +2HH T?QM2bX h?2`27Q`2 i?2 [mMiBiv UV Bb i?2 p2`;2 MmK#2` Q7 +2HH T?QM2b i?i +MMQi #2 +QMM2+i2/ iQ  #b2 biiBQMXV 1t2`+Bb2 jX3XkRX _M/QK GBM2b G2i L #2 i?2 SQBbbQM `M/QK HBM2 KQ/2H Q7 a2+iBQM jXdX G2i D0 := D(r0 , θ0 ) #2 0 i?2 TQBMi T`Q+2bb QM D0 M `#Bi``v }t2/ `M/QK HBM2 Q7 R2 M/ /2MQi2 #v N 7Q`K2/ #v i?2 BMi2`b2+iBQMb Q7 i?Bb bi`B;?i HBM2 rBi? i?2 HBM2 T`Q+2bbX a?Qr i?i N0 Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv  | sin(θ0 − θ)| G(dθ) . λ0 = [0,π)

*?Ti2` 9 _2M2rH M/ _2;2M2`iBp2 S`Q+2bb2b 6`QK i?2 Ki?2KiB+H TQBMi Q7 pB2r- `2M2rH i?2Q`v Bb +QM+2`M2/ rBi? i?2 `2M2rH 2[miBQM  f (t − s) dF (s) ,

f (t) = g(t) + [0,t]

r?2`2 F Bb i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7  }MBi2 K2bm`2 QM i?2 TQbB@ iBp2 `2H HBM2X Aib KBM +QM+2`M Bb i?2 bvKTiQiB+ #2?pBQ` Q7 i?2 bQHmiBQM f Ui?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 r?B+? Bb MQi  `2H Bbbm2 mM/2` KBH/ +QM/BiBQMb- b r2 b?HH b22VX PM+2 2K#2//2/ BM i?2 7`K2rQ`F Q7 TQBMi T`Q+2bb2b- `2M2rH i?2Q`vM/ BM T`iB+mH` "H+Fr2HHǶb i?2Q`2K-R #2+QK2b  7mM/K2MiH iQQH Q7 T`Q##BH@ Biv i?2Q`v- T`iB+mH`Hv mb27mH BM i?2 bim/v Q7 `2;2M2`iBp2 T`Q+2bb2b-k  H`;2 M/ BKTQ`iMi +Hbb Q7 biQ+?biB+ T`Q+2bb2b r?B+? BM+Hm/2b 7Q` BMbiM+2 i?2 `2+m``2Mi +QMiBMmQmb@iBK2 ?K+b M/ i?2 b2KB@J`FQp T`Q+2bb2bX

9XR

_2M2rH SQBMi T`Q+2bb2b

*QMbB/2` M BB/ b2[m2M+2 {Sn }n≥1 Q7 MQM@M2;iBp2 `M/QK p`B#H2b rBi? +QKKQM +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F (x) := P (Sn ≤ x) . h?Bb +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Bb +HH2/ /272+iBp2 r?2M F (∞) := P (S1 < ∞) < 1- M/ T`QT2` r?2M F (∞) = 1X h?2 mMBMi2`2biBM; +b2 r?2`2 P (S1 = 0) = 1 Bb ?2M+27Q`i? 2HBKBMi2/X h?2 #Qp2 b2[m2M+2 Bb +HH2/ i?2 BMi2`@`2M2rH b2[m2M+2X h?2 bbQ+Bi2/ `2M2rH b2[m2M+2 {Tn }n≥0 Bb /2}M2/ #v Tn := Tn−1 + Sn

(n ≥ 1) ,

r?2`2 i?2 BMBiBH /2Hv T0 Bb  7BMBi2 MQM@M2;iBp2 `M/QK p`B#H2 BM/2T2M/2Mi Q7 i?2 BMi2`@`2M2rH b2[m2M+2X h?2 iBK2 Tn Bb +HH2/  `2M2rH iBK2- Q` M 2p2MiX q?2M T0 = 0- i?2 `2M2rH b2[m2M+2 Bb +HH2/ mM/2Hv2/X h?2 biQ+?biB+ T`Q+2bb R k

("H+Fr2HH- RN93)X (aKBi?- RN88)X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_4

RR8

RRe

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a N ([0, t]) :=



(t ≥ 0)

1{Tn ≤t}

n≥0

Bb i?2 +QmMiBM; T`Q+2bb Q7 i?2 `2M2rH b2[m2M+2c N ([0, t]) +QmMib i?2 MmK#2` Q7 2p2Mib BM i?2 +HQb2/ BMi2`pH [0, t]X LQi2 i?i T0 ≥ 0 Ui?Bb +QMp2MiBQM /Bz2`b 7`QK i?2 mbmH QM2- QMHv 7Q` i?Bb +?Ti2`V M/ i?i i?2 TQBMi i 0 Bb +QmMi2/ r?2M i?2`2 Bb QM2X *H2`Hv- i?2 `M/QK 7mM+iBQM t → N ([0, t]) Bb HKQbi bm`2Hv `B;?i@+QMiBMmQmb M/ ?b  HBKBi QM i?2 H27i 7Q` 2+? t > 0- MK2Hv N [0, t)X

0

T1

T0

T2

T3

t

h?2Q`2K 9XRXR 6Q` HH t ≥ 0- E[N ([0, t])] < ∞X AM T`iB+mH`- HKQbi bm`2HvN ([0, t]) < ∞ 7Q` HH t ≥ 0X S`QQ7X Ai bm{+2b iQ +QMbB/2` i?2 mM/2Hv2/ +b2 U1t2`+Bb2 9XdXRVX "v J`FQpǶb BM2[mHBiv P (Tn ≤ t) = P (e−Tn ≥ e−t ) ≤ et E e−Tn . "mi bBM+2 i?2 `2M2rH b2[m2M+2 Bb BB/n    n E e−Sk = αn , E e−Tn = E e− k=1 Sk = k=1





rBi? α = E e−S1 < 1 bBM+2 P (S1 = 0) < 1X h?2`27Q`2  E [N ([0, t])] − 1 = E =









1{Tn ≤t} =

n≥1

n≥1

P (Tn ≤ t) ≤ e

n≥1

E 1{Tn ≤t}

t



αn < ∞ .

n≥1

 h?2 7Q`r`/ `2+m``2M+2 {A(t)}t≥0 M/ i?2 #+Fr`/ `2+m``2M+2 {B(t)}t≥0 `2 /2}M2/ b 7QHHQrbX "Qi? T`Q+2bb2b `2 `B;?i@+QMiBMmQmb rBi? H27i@?M/ HBKBibX 6Q` n ≥ 0- i?2v ?p2 HBM2` i`D2+iQ`B2b BM (Tn , Tn+1 ) rBi? `2bT2+iBp2 bHQT2b −1 M/ +1- M/ i  `2M2rH TQBMi Tn A(Tn ) = Tn+1 − Tn , A(Tn+1 −) = 0 , B(Tn ) = 0 , B(Tn+1 −) = Tn+1 − Tn . 6Q` 0 ≤ t < T0 - A(t) = T0 − t M/ B(t) = tX

9XRX _1L1qG SPALh S_P*1aa1a

0

T1

T0

RRd

T2

t

T3

h?2 7Q`r`/ `2+m``2M+2 iBK2 T`Q+2bb

0

T1

T0

T2

T3

t

h?2 #+Fr`/ `2+m``2M+2 iBK2 T`Q+2bb

.2}MBiBQM 9XRXk h?2 7mM+iBQM R : R+ → R+ /2}M2/ #v R(t) := E[N ([0, t])], r?2`2 N Bb i?2 +QmMiBM; T`Q+2bb Q7 i?2 mM/2Hv2/ `2M2rH b2[m2M+2- Bb +HH2/ i?2 `2M2rH 7mM+iBQMX h?2 `2M2rH 7mM+iBQM Bb `B;?i@+QMiBMmQmb U1t2`+Bb2 9XdXjV M/ MQM@/2+`2bBM;X h?2`27Q`2- QM2 +M bbQ+Bi2 rBi? Bi  mMB[m2 K2bm`2 μR QM R+ bm+? i?i μR ([0, a]) = R(a)X h?Bb K2bm`2- +HH2/ i?2 `2M2rH K2bm`2- rBHH bQK2iBK2b #2 /2MQi2/ #v R- i?2 +QMi2ti pQB/BM; +QM7mbBQM #2ir22M i?2 K2bm`2 M/ Bib +mKm@ HiBp2 /Bbi`B#miBQM 7mM+iBQMX LQi2 i?i μR ({0}) = R(0) = 1X 1tKTH2 9XRXj, h?2 SQBbbQM S`Q+2bbX *QMbB/2` i?2 +b2 Q7 2tTQM2MiBH BMi2`@ 2p2Mi iBK2b, F (t) = 1−e−λt Ut ≥ 0VX h?2 mM/2Hv2/ `2M2rH T`Q+2bb Bb i?2M M ?TT rBi? BMi2MbBiv λ iQ r?B+?  TQBMi i iBK2 0 Bb //2/X h?2`27Q`2- R(t) = 1 + λtX Ai rBHH #2 +QMp2MB2Mi iQ 2tT`2bb i?2 `2M2rH 7mM+iBQM BM i2`Kb Q7 i?2 +QKKQM +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 `M/QK p`B#H2b Sn X 6Q` i?Bb- Q#b2`p2 i?i BM i?2 mM/2Hv2/ +b2 Tn := S1 + · · · + Sn Bb i?2 bmK Q7 n BM/2T2M/2Mi `M/QK p`B#H2b rBi? +QKKQM +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F M/ i?2`27Q`2 P (Tn ≤ t) = F ∗n (t) , r?2`2 F ∗n Bb i?2 n@7QH/ +QMpQHmiBQM Q7 F - /2}M2/ `2+m`bBp2Hv #v  ∗0 ∗n F (t) = 1[0,∞) (t), F (t) = F ∗(n−1) (t − s) dF (s) (n ≥ 1) .

U9XRV

U9XkV

[0,t]

Uh?2 `QH2 Q7 0 BM i?2 BMi2;`iBQM Qp2` [0, t] Bb K/2 T`2+Bb2 #v i?2 7QHHQrBM; 2[mHBiv,   ϕ(s) dF (s) = ϕ(0)F (0) + ϕ(s) dF (s).) [0,t]

q`BiBM; i?2 `2M2rH 7mM+iBQM b

(0,t]

RR3

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a E[N ([0, t])] = E[1 +



1{Tn ≤t} ] = 1 +

n≥1

r2 Q#iBM i?2 2tT`2bbBQM, R(t) =



P (Tn ≤ t) ,

n≥1

∞ 

F ∗n (t) .

U9XjV

n=0

h?2Q`2K 9XRX9 P (S1 < ∞) < 1

⇐⇒

P (N ([0, ∞)) < ∞) = 1

⇐⇒

E[N ([0, ∞))] < ∞ .

S`QQ7X Ai bm{+2b iQ T`Qp2 i?2 i?2Q`2K BM i?2 mM/2Hv2/ +b2X 6Q` HH k ≥ 1P (N ([0, ∞)) = k) = P (S1 < ∞, . . . , Sk−1 < ∞, Sk = ∞) = P (S1 < ∞) . . . P (Sk−1 < ∞)P (Sk = ∞) = F (∞)k−1 (1 − F (∞)), M/ P (N ([0, ∞)) < ∞) =

∞ 

F (∞)k−1 (1 − F (∞)) .

k=1

AM T`iB+mH`- P (N ([0, ∞)) < ∞) = 1 B7 F (∞) < 1 M/ P (N ([0, ∞) < ∞) = 0 B7 F (∞) = 1X HbQ- B7 F (∞) < 1E[N ([0, ∞))] =

∞ 

kF (∞)k−1 (1 − F (∞)) =

k=1

1 < ∞, 1 − F (∞)

r?2`2b B7 F (∞) = 1- E[N ([0, ∞))] = ∞X



 `2M2rH T`Q+2bb U/2Hv2/ Q` MQiV Bb +HH2/ `2+m``2Mi r?2M P (S1 < ∞) = 1 UF Bb T`QT2`V- M/ i`MbB2Mi r?2M P (S1 < ∞) < 1 UF Bb /272+iBp2VX h?2 7QHHQrBM; `2bmHi Bb +HH2/ i?2 2H2K2Mi`v `2M2rH i?2Q`2KX h?2Q`2K 9XRX8 q2 ?p2 lim

t→∞

N ([0, t]) 1 = t E[S1 ]

M/ lim

t→∞

S@XbX,

E[N ([0, t])] 1 = . t E[S1 ]

U9X9V

U9X8V

S`QQ7X 6Q` i?2 T`QQ7 Q7 U9X9V b22 1t2`+Bb2 9XdXkX S`QQ7 Q7 U9X8V, h?2 i`MbB2Mi +b2 ≤ E[Nt(∞)] - bBM+2 BM i?Bb +b2 E[S1 ] = ∞ 7QHHQrb 7`QK i?2 Q#pBQmb #QmM/ E[N ([0,t])] t M/ E [N (∞)] < ∞X 6Q` i?2 `2+m``2Mi +b2-  T`QQ7 Bb `2[mB`2/ Ui?2 +QM/BiBQMb Q7 i?2 /QKBMi2/ +QMp2`;2M+2 i?2Q`2K i?i rQmH/ ;m`Mi22 i?i U9X9V BKTHB2b U9X8V `2 MQi biBb}2/VX >Qr2p2`- #v 6iQmǶb H2KK

9XRX _1L1qG SPALh S_P*1aa1a lim inf E t→∞

RRN



N ([0, t]) N ([0, t]) 1 ≥ E lim inf = t→∞ t t E[S1 ]

M/ i?2`27Q`2 Bi bm{+2b iQ b?Qr i?i lim supt→∞ E[ N ([0,t]) ] ≤ E[S1 1 ] X .2}M2 7Q` }MBi2 t c>0 T0 := T0 , T1 := T0 + S1 ∧ c, T2 := T1 + S2 ∧ c, . . .

r?2`2 Sn := Sn ∧ c Un ≥ 1V- M/ H2i N  ([0, t]) := n≥0 1Tn ≤t X aBM+2 N  ([0, t]) ≥ N ([0, t]) 7Q` HH t ≥ 0lim sup t→∞

P#b2`p2 i?i 9XdXdV-

S1

E[N ([0, t])] E[N  ([0, t])] ≤ lim sup . t t t→∞

 + · · · + SN  ([0,t]) ≤ t + c M/ i?2`27Q`2- #v qH/Ƕb H2KK U1t2`+Bb2

 E[S1 ]E[N  ([0, t])] = E[S1 + · · · + SN  ([0,t]) ] ≤ t + c ,

bQ i?i lim sup t→∞

E[N  ([0, t])] ≤ lim sup t t→∞



 1  c 1 1 + = .  E[S1 ] t E[S1 ]

h?2`27Q`2- 7Q` HH c > 0lim sup t→∞

E[N ([0, t])] 1 ≤ . t E[S1 ∧ c]

aBM+2 limc↑∞ E[S1 ∧ c] = E[S1 ]- r2 }MHHv Q#iBM i?2 /2bB`2/ BM2[mHBiv lim sup t→∞

E[N ([0, t])] 1 ≤ . t E[S1 ] 

G2i F : R+ → R+ #2  ;2M2`HBx2/ +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM QM R+ i?i Bb- F (x) = c G(x) r?2`2 c > 0 M/ G Bb i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7  MQM@M2;iBp2 `2H `M/QK p`B#H2 i?i Bb T`QT2` UG(∞) = 1VX h?2 _2M2rH 1[miBQM h?2 #bB+ Q#D2+i Q7 `2M2rH i?2Q`v Bb i?2 `2M2rH 2[miBQM f = g + f ∗ F, i?i Bb- #v /2}MBiBQM Q7 i?2 +QMpQHmiBQM bvK#QH ∗ f (t) = g(t) + f (t − s) dF (s)

(t ≥ 0) ,

U9XeV

[0,t]

r?2`2 g : R+ → R Bb  K2bm`#H2 7mM+iBQM +HH2/ i?2 /iX A7 F (∞) = 1 QM2 `272`b iQ i?2 `2M2rH 2[miBQM b  T`QT2` `2M2rH 2[miBQMQ` Dmbi  `2M2rH 2[miBQMX h?2 `2M2rH 2[miBQM Bb +HH2/ /272+iBp2 B7 F (∞) < 1 M/ 2t+2bbBp2 B7 F (∞) > 1X

Rky

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

h?2Q`2K 9XRXe h?2 `2M2rH 7mM+iBQM R biBb}2b i?2 bQ@+HH2/ 7mM/K2MiH `2@ M2rH 2[miBQM R = 1 + R ∗ F. U9XdV S`QQ7X "v U9XjV     ∗n R∗F = ∗F = F (F ∗n ∗ F ) = F ∗n = R − F ∗0 = R − 1 . n≥0

n≥0

n≥1

 h?2 7QHHQrBM; bBKTH2 i2+?MB+H `2bmHi rBHH #2 M22/2/ Hi2` QMX G2KK 9XRXd 6Q` HH t ≥ b- R ([t − b, t]) ≤ (1 − F (b))−1 X S`QQ7X "v U9XdV M/ 7Q` t ≥ b  F (t − s) dR(s) = (1 − F (t − s)) dR(s) 1 = R(t) − [0,t] [0,t]  (1 − F (t − s)) dR(s) ≥ (1 − F (b))R ([t − b, t]) . ≥ [t−b,t]

 h?2 }`bi 2tKTH2 72im`2b i?2 #bB+ K2i?Q/ 7Q` Q#iBMBM; `2M2rH 2[miBQMbX 1tKTH2 9XRX3, GB72iBK2 Q7  h`MbB2Mi _2M2rH S`Q+2bb- hF2 RX h?2 HB72iBK2 Q7  `2M2rH b2[m2M+2 Bb i?2 `M/QK p`B#H2 L := sup{Tn ; Tn < ∞}. *H2`Hv B7 i?2 `2M2rH T`Q+2bb Bb `2+m``2Mi- L Bb HKQbi bm`2Hv BM}MBi2X q2 i?2`27Q`2 +QMbB/2` i?2 i`MbB2Mi +b2- 7Q` r?B+? L Bb HKQbi bm`2Hv }MBi2X AM i?2 mM/2Hv2/ +b2- i?2 7mM+iBQM f (t) = P (L > t) biBb}2b i?2 `2M2rH 2[miBQM  f (t − s) dF (s). f (t) = F (∞) − F (t) + [0,t]

S`QQ7X .2}M2 Sn = Sn+1 (n ≥ 1) M/ H2i {Tn }n≥0 #2 i?2 bbQ+Bi2/ mM/2Hv2/ `2M2rH T`Q+2bb r?Qb2 HB72iBK2 Bb /2MQi2/ #v LX no more event S1 T0 = 0

S2 = Sˆ1 T1

S3 = Sˆ2

T2

T3 = L ˆ L

t

9XRX _1L1qG SPALh S_P*1aa1a

RkR

*H2`Hv L M/ L ?p2 i?2 bK2 /Bbi`B#miBQMX HbQ 1{L>t} = 1{tt} + 1{t≥T1 } 1{L>t} . LQr- QM {t ≥ T1 }, {L > t} ≡ {L > t − T1 } M/ i?2`27Q`2 . 1{L>t} = 1{tt} + 1{t≥T1 } 1{L>t−T  1} hFBM; 2tT2+iiBQMbP (L > t) = P (L > t, t < T1 ) + P (L > t − T1 , T1 ≤ t). aBM+2 L M/ T1 `2 BM/2T2M/2Mi UL /2T2M/b QMHv QM S2 , S3 , . . .V  P (L > t − s) dF (s) = P (L > t − s) dF (s) , P (L > t − T1 , T1 ≤ t) = [0,t]

[0,t]

r?2`2 r2 ?p2 mb2/ i?2 7+i i?i L M/ L ?p2 i?2 bK2 /Bbi`B#miBQMX HbQP (L > t, t < T1 ) = P (t < T1 , T1 < ∞) = P (t < T1 < ∞) = F (∞) − F (t) . 

1tKTH2 9XRXN, h?2 _BbF JQ/2H- hF2 kX h?Bb 2tKTH2 Bb  +QMiBMmiBQM Q7 1tKTH2 kXRX9X q2 }M/  `2M2rH 2[miBQM 7Q` i?2 T`Q##BHBiv Q7 `mBM +Q``2bTQM/BM; iQ M BMBiBH +TBiH u, Ψ(u) := P (u + X(t) < 0 7Q` bQK2 t > 0) .

U9X3V

h?Bb 7mM+iBQM Bb MQM@BM+`2bBM;X Ai Bb +QMp2MB2Mi iQ rQ`F rBi? i?2 MQM@`mBM T`Q#@ #BHBiv Φ(u) := 1 − Ψ(u)X P7 +Qm`b2- Φ(u) = 0 B7 u ≤ 0X A7 i?2 TQBMi T`Q+2bb N Bb biiBQM`v rBi? p2`;2 `i2 λ- i?2 p2`;2 T`Q}i Q7 i?2 BMbm`M+2 +QKTMv i iBK2 t Bb E[X(t)] = (c − λμ)t . b 2tT2+i2/- BMbm`M+2 +QKTMB2b T`272` i?i c − λμ > 0 Q`- 2[mBpH2MiHv- i?i ρ :=

c − λμ c = − 1 > 0, λμ λμ

U9XNV

r?2`2 ρ Bb +HH2/ i?2 b72iv HQ/BM;X AM 7+i- #v i?2 bi`QM; Hr Q7 H`;2 MmK#2`bB7 i?2 b72iv HQ/BM; Bb M2;iBp2- i?2 T`Q##BHBiv Q7 `mBM Bb 1 BM/2T2M/2MiHv Q7 i?2 BMBiBH +TBiHX h?2Q`2K 9XRXRy A7 N Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb λ u Φ(u) = Φ(0) + Φ(u − z)(1 − G(z))dz . c 0

U9XRyV

Rkk

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

S`QQ7Xj amTTQb2 u ≥ 0X aBM+2 `mBM +MMQi Q++m` i  iBK2 < S1   Φ(u + cs − z)λe−λs ds dG(z) Φ(u) = Φ(u + cT1 − Z1 ) = (0,∞) (0,∞)   ∞  Φ(u + cs − z) dG(z) λe−λs ds = 0 (0,u+cs]    λ λu/c u Φ(x − z) dG(z) e−λx/c dx . = e c 0 (0,x] h?2 `B;?i@?M/ bB/2 Bb /Bz2`2MiB#H2- M/ /Bz2`2MiBiBQM H2/b iQ i?2 BMi2;`Q@ /Bz2`2MiBH 2[miBQM  λ λ Φ(u − z) dG(z). U9XRRV Φ (u) = Φ(u) − c c (0,u] h?2`27Q`2Φ(t) − Φ(0) = =

λ c



λ c



t

Φ(u) du + 0

λ c

 t Φ(u − z) d(1 − G(z)) du 0

(0,u]

t

Φ(u) du

 t  u λ  + Φ (u − z)(1 − G(z)) dz du Φ(0)(1 − G(u)) − Φ(u) + c 0 0   t  t  t λ λ Φ (u − z) du (1 − G(z)) dz = Φ(0) (1 − G(u)) du + c c 0 0 z  t  λ t λ (1 − G(z)(Φ(t − z) − Φ(0)) dz , = Φ(0) (1 − G(u)) du + c c 0 0 0

i?i Bb- U9XRyVX



aBKTH2 `;mK2Mib U1t2`+Bb2 9XdXNV b?Qr i?i BM i?2 +b2 Q7 TQbBiBp2 b72iv HQ/BM;- Φ(∞) = 1X G2iiBM; u ↑ ∞ BM U9XRyV- r2 ?p2 #v KQMQiQM2 +QMp2`;2M+2 i?i Φ(∞) = Φ(0) + λc Φ(∞) M/ i?2`27Q`2 i?2 T`Q##BHBiv Q7 `mBM rBi? x2`Q BMBiBH +TBiH Bb λμ . Ψ(0) = c h?2`27Q`2- B7 i?2 b72iv HQ/BM; Bb TQbBiBp2- r2 Q#iBM U9XRyVX  λμ λ u + (1 − Ψ(u − z))(1 − G(z)) dz c c 0    ∞  u λ =1− μ− (1 − G(z))dz + Ψ(u − z)(1 − G(z)) dz , c u 0

1 − Ψ(u) = 1 −

j

(62HH2`- RNdR)- TX R3jX

9XRX _1L1qG SPALh S_P*1aa1a

Rkj

i?i BbΨ(u) =

λ c





λ c

(1 − G(z))dz +

u



u

Ψ(u − z)(1 − G(z)) dz .

U9XRkV

0

1tKTH2 9XRXRR, h?2 GQiFėoQHi2`` SQTmHiBQM JQ/2HX h?Bb KQ/2H 72im`2b  TQTmHiBQM Q7 rQK2MX  rQKM Q7 ;2 a ;Bp2b #B`i? iQ ;B`Hb i i?2 `i2 λ(a) Ui?i Bb-  rQKM Q7 ;2 a rBHH ?p2 QM p2`;2 λ(a) da /m;?i2`b BM M BM}MBi2bBKH iBK2 BMi2`pH (a, a + da) BM ?2` HB72iBK2VX  rQKM Q7 ;2 a Bb HBp2 i iBK2 a + t rBi? T`Q##BHBiv p(a, t)X i i?2 Q`B;BM Q7 iBK2 i?2`2 `2 f0 (a) da rQK2M Q7 ;2 #2ir22M a M/ a + daX h?2 #B`i? `i2 f (t) i iBK2 t ≥ 0 Bb i?2 bmK Q7 i?2 #B`i? `i2 r(t) i iBK2 t /m2 iQ rQK2M #Q`M 7i2` iBK2 0 M/ Q7 i?2 #B`i? `i2 g(t) /m2 iQ rQK2M #Q`M #27Q`2 iBK2 0X aBM+2  rQKM Q7 ;2 a i iBK2 0 Bb a + t v2`b QH/ i iBK2 t ∞ g(t) = f0 (a)p(a, t)λ(a + t) da . 0

qQK2M #Q`M i iBK2 t − s ≥ 0 +QMi`B#mi2 #v f (t − s)p(0, s)λ(s) iQ i?2 #B`i? `i2 i iBK2 t M/ i?2`27Q`2  t r(t) = f (t − s)p(0, s)λ(s) ds . 0

h?2`27Q`2



t

f (t − s)p(0, s)λ(s) ds .

f (t) = g(t) + 0

h?Bb Bb  `2M2rH 2[miBQM U+HH2/ GQiFǶb 2[miBQMV rBi? /i g M/ +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM  t F (t) := p(0, s)λ(s) ds. 0



LQi2 i?i



p(0, s)λ(s) ds

F (∞) = 0

Bb i?2 p2`;2 MmK#2` Q7 /m;?i2`b 7`QK  ;Bp2M KQi?2`- i?i Bb- i?2 `2T`Q/m+iBQM `i2X _2K`F 9XRXRk h?2 #Qp2 2tKTH2 +QMM2+ib `2M2rH i?2Q`v iQ i?2 i?2Q`v Q7 oQHi2`` BMi2;`H 2[miBQMb Q7 i?2 b2+QM/ ivT2- i?i Bb- Q7 i?2 7Q`K  x ϕ(x) = f (x) + λ K(x, y)ϕ(y) dy , 0

r?2`2 i?2 F2`M2H K biBb}2b i?2 +QM/BiBQM K(x, y) ≡ 0 B7 y > x .

Rk9

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

h?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 bm+? M 2[miBQM +M #2 T`Qp2/ mM/2` bm{+B2MiHv ;2M2`H bbmKTiBQMbX PM2 Q7 i?2K Bb i?i #Qi? K M/ f #2 HQ+HHv L2 QM R+ × R+ M/ R+ `2bT2+iBp2HvX h?2 bQHmiBQM +M #2 2tTHB+BiHv ;Bp2M BM i?Bb +b2 b  x H(x, y, λ)f (y) dy , ϕ(x) = f (x) − λ 0

r?2`2 H Bb i?2 `2bQHp2Mi F2`M2H H(x, y, λ) :=

∞ 

λn Kn+1 (x, y)

n=0

M/ r?2`2 i?2 Bi2`i2/ F2`M2Hb {Kn }n≥1 `2 /2}M2/ #v i?2 `2+m``2M+2 7Q`KmH  x Kn+1 (x, y) = K(x, z)Kn (z, y) dz (n ≥ 1) 0

bi`iBM; rBi? K1 = KX AM i?2 +b2 Q7 BMi2`2bi BM i?Bb b2+iBQM- K Bb Q7 i?2 dz6HimM;Ǵ ivT2- i?i Bb K(x, y) = k(x − y)- 7Q` r?B+? i?2 2H2K2Mi`v i?2Q`v Q7 BMi2;`H 2[miBQMb ?b KQ`2 iQ bvX9 AM bQK2 +b2b- 2tTHB+Bi bQHmiBQMb +M #2 Q#iBM2/X _2M2rH i?2Q`v /2Hb rBi?  KQ`2 ;2M2`H ivT2 Q7 BMi2;`H 2[miBQMb- M/ Bib ;QHb `2 /Bz2`2Mi- +2Mi2`2/ KQbiHv QM i?2 bvKTiQiB+ #2?pBQm` Q7 i?2B` bQHmiBQMbX 1tKTH2 9XRXRj, h?2 1H2K2Mi`v _2M2rH h?2Q`2K, *Q``2+iBQM h2`K- hF2 RX "v i?2 2H2K2Mi`v `2M2rH i?2Q`2K- limt↑∞ R(t) = E[S1 1 ] X AM pB2r t Q7 Q#iBMBM; }M2` bvKTiQiB+ `2bmHib Ub22 1tKTH2 9XkXR9 #2HQrV- r2 b?HH bim/v i?2 7mM+iBQM t f (t) := R(t) − · E[S1 ] AM i?2 +b2 r?2`2 E[S1 ] < ∞- f biBb}2b i?2 `2M2rH 2[miBQM rBi? /i  ∞ 1 (1 − F (x))dx , g(t) := E[S1 ] t b r2 MQr T`Q+22/ iQ b?QrX G2i E[S1 ] := mX "v U9XdV 1 (f ∗ F )(t) = (R ∗ F )(t) − (t − s) dF (s) m [0,t]  1 (t − s) dF (s) = R(t) − 1 − m [0,t]     t 1 t − 1− + (t − s) dF (s) = R(t) − m m m [0,t]  ∞ 1 (1 − F (x))dx , = f (t) − m t r?2`2 i?2 Hbi 2[mHBiv Bb Q#iBM2/ #v i?2 7QHHQrBM; +QKTmiiBQMbX AMi2;`iBQM #v T`ib Uh?2Q`2K RXeXjV ;Bp2b 9

a22 7Q` BMbiM+2 (h`B+QKB- RN8d- RN38)- RXNX

9XRX _1L1qG SPALh S_P*1aa1a

Rk8





t(1 − F (t)) =

(1 − F (s)) ds − [0,t]

s dF (s) , [0,t]

∞ M/ i?2`27Q`2- bBM+2 m = 0 (1 − F (s)) ds   1 ∞ 1 ∞ 1 t (1 − F (s)) ds = (1 − F (s)) ds − (1 − F (s)) ds m t m 0 m 0     1 t 1 =1− t(1 − F (t)) + (1 − F (s)) ds = 1 − s dF (s) m 0 m [0,t]     1 t 1 t+ + (s − t) dF (s) = 1 − (t − s) dF (s) . =1− m m m [0,t] [0,t]

aQHmiBQM Q7 i?2 _2M2rH 1[miBQM M 2tT`2bbBQM Q7 i?2 bQHmiBQM Q7 i?2 `2M2rH 2[miBQM BM i2`Kb Q7 i?2 `2M2rH 7mM+iBQM Bb 2bv iQ Q#iBMX _2+HH i?2 7QHHQrBM; /2}MBiBQM,  7mM+iBQM g : R+ → R Bb +HH2/ HQ+HHv #QmM/2/ B7 7Q` HH a ≥ 0- supt∈[0,a] |g(t)| < ∞X h?2Q`2K 9XRXR9 A7 F (∞) ≤ 1 M/ B7 i?2 /i 7mM+iBQM g : R+ → R Bb HQ+HHv #QmM/2/- i?2 `2M2rH 2[miBQM U9XeV /KBib  mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM f : R+ → R ;Bp2M #v f = g ∗ R- i?i Bb f (t) = g(t − s)dR(s) . U9XRjV [0,t]

S`QQ7X h?2 7mM+iBQM f = g ∗ R Bb BM/22/ HQ+HHv #QmM/2/ bBM+2 g Bb HQ+HHv #QmM/2/ M/ R(t) Bb }MBi2 7Q` HH tX HbQ f ∗ F = (g ∗ R) ∗ F = g ∗ (R ∗ F ) = g ∗ (R − 1) = g ∗ R − g = f − g . h?2`27Q`2 f Bb BM/22/  bQHmiBQM Q7 i?2 `2M2rH 2[miBQMX G2i f1 #2 MQi?2` HQ+HHv #QmM/2/ bQHmiBQM M/ H2i h := f − f1 X h?Bb Bb  HQ+HHv #QmM/2/ bQHmiBQM r?B+? biBb}2b h = h ∗ F X "v Bi2`iBQMh = h ∗ F ∗n . h?2`27Q`2- 7Q` HH t ≥ 0-



 |h(t)| ≤

sup |h(s)| F ∗n (t). s∈[0,t]

aBM+2 R(t) = |h(t)| ≡ 0X

n≥0

F ∗n (t) < ∞- r2 ?p2 limn→∞ F ∗n (t) = 0- r?B+? BKTHB2b i?i 

h?2 }`bi bvKTiQiB+ `2bmHi QM i?2 bQHmiBQM Q7 i?2 `2M2rH 2[miBQM +QM+2`Mb i?2 /272+iBp2 +b2,

Rke

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

h?2Q`2K 9XRXR8 A7 F Bb /272+iBp2 M/ B7 i?2 /i 7mM+iBQM g : R+ → R Bb #QmM/2/ M/ ?b  HBKBi g(∞) := limt→∞ g(t)- i?2 mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM Q7 i?2 `2M2rH 2[miBQM biBb}2b lim f (t) =

t→∞

g(∞) . 1 − F (∞)

S`QQ7X 6`QK T`2pBQmb +QKTmiiBQMb- r2 ?p2 E[N ([0, ∞))] = R(∞) = M/ i?2`27Q`2

g(∞) = 1 − F (∞)

1 , 1 − F (∞)

 g(∞) dR(s). [0,∞)



HbQ

g(t − s) dR(s) ,

f (t) = [0,t]

M/ i?2`27Q`2 g(∞) = f (t) − 1 − F (∞)

 (g(t − s)1{s≤t} − g(∞)) dR(s). [0,∞)

h?2 Hii2` BMi2;`M/ Bb #QmM/2/ BM #bQHmi2 pHm2 #v 2 × sup |g(t)|-  }MBi2 +QM@ biMiX  +QMbiMi 7mM+iBQM Bb BMi2;`#H2 rBi? `2bT2+i iQ i?2 `2M2rH K2bm`2 #2@ +mb2- BM i?2 /272+iBp2 +b2- i?2 iQiH Kbb Q7 i?2 `2M2rH K2bm`2 Bb R(∞) = E [N ([0, ∞))] < ∞X LQr 7Q` }t2/ s ≥ 0- limt→∞ (g(t − s)1{s≤t} − g(∞)) = 0X h?2`27Q`2- #v /QKBMi2/ +QMp2`;2M+2- i?2 BMi2;`H +QMp2`;2b iQ 0 b t → ∞X  aiiBQM`v _2M2rH S`Q+2bb2b "v  T`QT2` +?QB+2 Q7 i?2 BMBiBH /2Hv-  `2M2rH T`Q+2bb +M #2 K/2 biiBQM`vBM  b2Mb2 iQ #2 K/2 T`2+Bb2X *QMbB/2`  `2M2rH T`Q+2bb T0 = S0 , T1 = S0 + S1 , . . . , Tn = S0 + · · · + Sn r?2`2 0 ≤ S0 < ∞X G2i G #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 BMBiBH /2Hv S0 := T0 M/ bmTTQb2 i?i P (S1 < ∞) = 1 Ui?2 `2M2rH T`Q+2bb Bb T`QT2`VX b mbmH- 2t+Hm/2 i`BpBHBiB2b #v BKTQbBM; i?2 +QM/BiBQM P (S1 = 0) < 1X 6Q` t ≥ 0/2}M2 S0 (t) := TN ([0,t]) − t M/ Sn (t) := TN ([0,t])+n − TN ([0,t])+n−1 (n ≥ 1) .

U9XR9V

AM T`iB+mH`- Sn (0) = Sn 7Q` HH n ≥ 0X HbQ Q#b2`p2 i?i S0 (t) = A(t)- i?2 7Q`r`/ `2+m``2M+2 iBK2 i tX t S0

S1

S2

S3 S0 (t)

S4

S5

S1 (t)

S2 (t)

9XRX _1L1qG SPALh S_P*1aa1a

Rkd

.2}MBiBQM 9XRXRe h?2 /2Hv2/ `2M2rH T`Q+2bb Bb +HH2/ biiBQM`v B7 i?2 /Bbi`B@ #miBQM Q7 i?2 b2[m2M+2 S0 (t) , S1 (t) , S2 (t) , . . . Bb BM/2T2M/2Mi Q7 iBK2 t ≥ 0X Ai im`Mb Qmi i?i S0 (t) Bb BM/2T2M/2Mi Q7 {Sn (t)}n≥1 M/ i?i i?2 Hii2` b2[m2M+2 ?b i?2 bK2 /Bbi`B#miBQM b {Sn }n≥1 U1t2`+Bb2 9XdXRRVX h?2`27Q`2, G2KK 9XRXRd 6Q`  /2Hv2/ `2M2rH T`Q+2bb iQ #2 biiBQM`v Bi Bb M2+2bb`v M/ bm{+B2Mi i?i 7Q` HH t ≥ 0- i?2 /Bbi`B#miBQM Q7 S0 (t) = A(t) #2 i?2 bK2 b i?i Q7 S0 X G2KK 9XRXR3 A7 i?2 /2Hv2/ `2M2rH T`Q+2bb Bb biiBQM`v- i?2M M2+2bb`BHv E[S1 ] < ∞ M/ E[N ([0, t])] = E[St 1 ] X S`QQ7X h?2 K2bm`2 M QM R+ /2}M2/ #v M (C) := E[N (C)] Bb i`MbHiBQM@ BMp`BMi M/ i?2`27Q`2  KmHiBTH2 Q7 i?2 G2#2b;m2 K2bm`2 Uh?2Q`2K XRX3V- i?i Bb- M (C) = K (C) 7Q` bQK2 +QMbiMi K- r?B+? Bb }MBi2 UM Bb HQ+HHv }MBi2V M/ TQbBiBp2 Ui?2 `2M2rH T`Q+2bb Bb MQi 2KTivVX "v i?2 2H2K2Mi`v `2M2rH i?2Q`2KK = lim t↑∞

M/ i?2`27Q`2

1 E[S1 ]

E[N ((0, t])] 1 = , t E[S1 ] 

> 0X

G2KK 9XRXRN A7 i?2 /2Hv2/ `2M2rH T`Q+2bb Bb biiBQM`v- i?2M M2+2bb`BHv E[S1 ] < ∞ M/ i?2 /Bbi`B#miBQM Q7 i?2 BMBiBH /2Hv T0 Bb  x 1 F0 (x) := (1 − F (y)) dy , U9XR8V E[S1 ] 0 +HH2/ i?2 biiBQM`v 7Q`r`/ `2+m``2M+2 iBK2 /Bbi`B#miBQMX S`QQ7X h?2 }MBi2M2bb Q7 E[S1 ] rb T`Qp2/ BM i?2 T`2pBQmb H2KKX 6Q` HH u ∈ RHH t ≥ 0- i?2 7QHHQrBM; 2[miBQM  t   d iuA(s) iuA(t) iuA(0) iuA(Tn ) iuA(Tn −) e e 1{Tn ≤t} + =e + −e ds e ds 0 n≥0 Bb Q#iBM2/ #v HQQFBM; i r?i ?TT2Mb i i?2 2p2Mi iBK2b M/ #2ir22M i?2 2p2Mi d iuA(s) iBK2b Ub22 1[MX RXk9VX P#b2`pBM; i?i ds e = −iueiuA(s) - A(0) = S0 - A(Tn −) = 0- A(Tn ) = Sn+1 - r2 i?2`27Q`2 ?p2  t   eiuA(t) = eiuS0 + eiuSn+1 − 1 1{Tn ≤t} − iu eiuA(s) ds . n≥0

0

h?2`27Q`2- iFBM; BMiQ ++QmMi i?2 BM/2T2M/2M+2 Q7 Sn+1 M/ Tn -

Rk3

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a E eiuA(t) = E eiuS0 + E eiuS1 − 1 E[N ((0, t])] − iu



t

E eiuA(s) ds .

0



"v i?2 bbmK2/ biiBQM`Biv- E[N ((0, t])] = E[St 1 ] M/ E eiuA(t) = E eiuS0 - M/ i?2`27Q`2 E eiuS1 − 1 − iuE eiuS0 = 0 , E[S1 ] i?i Bb-

E eiuS1 − 1 . E eiuS0 = iu E[S1 ]

"mi i?2 `B;?i@?M/ bB/2 Bb i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 F0 - b i?2 7QHHQrBM; +QK@ TmiiBQM b?Qrb,  ∞  ∞ 1 1 eiux (1 − F (x)) dx = eiux P (S1 > x) dx E[S1 ] 0 E[S1 ] 0  ∞ 1 eiux E 1{S1 >x} dx = E[S1 ] 0  ∞

1 iux = E e 1{S1 >x} dx E[S1 ] 0  S1 iuS1



e −1 1 1 = E E . eiux dx = E[S1 ] E[S1 ] iu 0  h?2Q`2K 9XRXky 6Q`  /2Hv2/ `2M2rH T`Q+2bb iQ #2 biiBQM`v- Bi Bb M2+2bb`v M/ bm{+B2Mi i?i E[S1 ] < ∞ M/ i?i P (T0 ≤ x) = F0 (x) , r?2`2 F0 Bb i?2 biiBQM`v 7Q`r`/ `2+m``2M+2 iBK2 /Bbi`B#miBQM U9XR8VX S`QQ7X h?2 T`QQ7 Q7 M2+2bbBiv Bb +QMiBM2/ BM G2KKb 9XRXR3 M/ 9XRXRNX 6Q` bm7@ }+B2M+v- r2 }`bi b?Qr i?i RF0 (t) := E[N ([0, t]))] =

t E[S1 ]

Ui?2 MQiiBQM 2KT?bBx2b i?2 `QH2 Q7 i?2 BMBiBH /2Hv rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F0 VX q2 ?p2    RF0 (t) = E 1{Tn ≤t} = P (Tn ≤ t) = F0 (t)+(F0 ∗F )(t)+(F0 ∗F ∗2 )(t)+· · · , n≥0

n≥0

i?i Bb- RF0 = F0 + RF0 ∗ F X h?2`27Q`2- #v h?2Q`2K 9XRXR9- RF0 Bb i?2 mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM Q7 i?2 `2M2rH 2[miBQM f = F0 + f ∗ F X Ai i?2M bm{+2b iQ b?Qr i?i f (t) = E[St 1 ] Bb BM/22/  bQHmiBQMX hQ p2`B7v i?Bb- Q#b2`p2 i?i 7Q` bm+? f-

9XkX h>1 _1L1qG h>1P_1J 1 (f ∗ F )(t) = E[S1 ] M/ i?2`27Q`2

RkN  (t − s) dF (s) [0,t]

  t 1 1 − t dF (s) + s dF (s) m E[S1 ] [0,t] E[S1 ] [0,t]  1 t (1 − F (t)) + s dF (s) . = E[S1 ] E[S1 ] [0,t]

f (t) − (f ∗ F )(t) =

Ai `2KBMb iQ b?Qr i?i i?2 `B;?i@?M/ bB/2 Q7 i?2 #Qp2 2[mHBiv Bb F0 (t)X AMi2;`@ iBQM #v T`ib /Q2b Bi,    t  1 1 t(1 − F (t)) + (1 − F (s)) ds = s dF (s) . F0 (t) = E[S1 ] 0 E[S1 ] [0,t] >pBM; T`Qp2/ i?i E[N ([0, t]))] = E[St 1 ] - r2 `2 HKQbi /QM2X 6`QK +QKTmi@ iBQMb BM i?2 T`QQ7 Q7 G2KK 9XRXRN- r2 2ti`+i i?2 B/2MiBiv  t t E eiuA(s) ds . − iu E eiuA(t) = E eiuS0 + E eiuS1 − 1 E[S1 ] 0 iuA(t) h?2 7mM+iBQM z(t) := E e Bb i?2`27Q`2  bQHmiBQM Q7 i?2 Q`/BM`v /Bz2`2MiBH 2[miBQM dz 1 = −iuz + E eiuS1 − 1 dt E[S1 ] rBi? BMBiBH +QM/BiBQM z(0) = E eiuS0 = E[S1 1 ] E eiuS1 − 1 - r?Qb2 mMB[m2 bQHmiBQM Bb E[S1 ] iuS1 E eiuA(t) = E eiuS0 = E e −1 . m h?2`27Q`2- 7Q` HH t ≥ 0- S0 (t) U= A(t)V ?b i?2 bK2 /Bbi`B#miBQM b S0 X h?2 +QM+HmbBQM i?2M 7QHHQrb 7`QK G2KK 9XRXRdX 

9Xk

h?2 _2M2rH h?2Q`2K

_2M2rH i?2Q`v /2Hb KBMHv rBi? i?2 HBKBiBM; #2?pBQm` Q7 i?2 bQHmiBQM Q7 i?2 `2M2rH 2[miBQMX h?2 i?2Q`v Bb `i?2` bBKTH2 r?2M i?2 `2M2rH T`Q+2bb Bb i`MbB2Mi Uh?2Q`2K 9XRXR8VX Ai #2+QK2b KQ`2 BMpQHp2/ BM i?2 `2+m``2Mi +b2X  p2`v bBKTH2 2tKTH2 rBHH ;Bp2 i?2 ~pQm` Q7 bm+?  `2bmHiX 1tKTH2 9XkXR, h?2 _2M2rH h?2Q`2K 7Q`  SQBbbQM S`Q+2bbX A7 {Tn }n≥1 Bb  SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ > 0- r2 FMQr Uh?2Q`2K 9XRXR8V i?i R(t) = 1 + λtX h?2`27Q`2- i?2 bQHmiBQM Q7 i?2 +Q``2bTQM/BM; `2M2rH 2[miBQM Bbr?2M i?2 /i g Bb MQM@M2;iBp2 t  t  t f (t) = g(t − s) R(ds) = λ g(t − s)ds = λ g(s)ds. 0

0

0

Rjy

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

>2`2 λ = 1/E[S1 ]X A7 r2 bmTTQb2 i?i g Bb BMi2;`#H2- i?2M ∞ lim f (t) =

t→∞

0

g(s) ds . E[S1 ]

()

.2}MBiBQM 9XkXk G2i F #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7  MQM@M2;iBp2 `2H `M/QK p`B#H2 S1 X "Qi? F M/ S1 `2 +HH2/ MQM@HiiB+2 B7 i?2`2 Bb MQ bi`B+iHv TQbBiBp2 `2H a bm+? i?i k≥0 P (S1 = ka) = 1X h?2 E2v _2M2rH h?2Q`2K Ai im`Mb Qmi i?i 7Q` MQM@HiiB+2 /Bbi`B#miBQMb UV Bb [mBi2 ;2M2`H- KQ/mHQ  KBH/ i2+?MB+H bbmKTiBQM QM i?2 /i g, i?Bb 7mM+iBQM ?b iQ #2 /B`2+iHv _B2KMM BMi2;`#H2X h?Bb Bb i?2 bBimiBQM +QMbB/2`2/ BM i?2 F2v `2M2rH i?2Q`2K Uh?2Q`2K 9XkXRj #2HQrVX q2 T`Q+22/ iQ ;Bp2 i?2 #bB+ 7+ib +QM+2`MBM; /B`2+i _B2KMM BMi2;`#BHBivX G2i g : R+ → R #2  MQMM2;iBp2 HQ+HHv #QmM/2/ 7mM+iBQMX .2}M2 7Q` 2+? b > 0 M/ 2+? t ≥ 0

g b (t) = sup{g(s); nb ≤ s < (n + 1)b} QM [nb, (n + 1)b) g b (t) = inf{g(s); nb ≤ s < (n + 1)b} QM [nb, (n + 1)b) .

h?2 7mM+iBQMb g b M/ g b `2 }MBi2 +QMbiMib QM i?2 BMi2`pHb [nb, (n + 1)b)- 7Q` HH n ∈ N- M/ i?mb G2#2b;m2 BMi2;`#H28 QM #QmM/2/ BMi2`pHbX .2}MBiBQM 9XkXj h?2 7mM+iBQM g ≥ 0 Bb bB/ iQ #2 _B2KMM BMi2;`#H2 U_BV  a QM i?2 #QmM/2/ BMi2`pH [0, a] B7 7Q` bQK2 UM/ i?2M 7Q` HHV b > 0- i?2 BMi2;`H 0 g b (t)dt Bb }MBi2- M/  a   a lim g b (t)dt − g b (t)dt = 0. U9XReV b↓0

0

0

Uh?2 7+i i?i- r?2M g b Bb BMi2;`#H2 7Q` bQK2 b > 0- i?2M g b Bb HbQ BMi2;`#H2 7Q` HH b > 0- 7QHHQrb 7`QK i?2 BM2[mHBiv g b (t) ≤

+n 

g b (t + kb),

k=−n

r?2`2 n = b /b XV 8 h?2 i?2Q`v Q7 i?2 _B2KMM BMi2;`H T`2/i2b i?i Q7 i?2 G2#2b;m2 BMi2;`HX q2 /Q MQi 7QHHQr i?2 ?BbiQ`B+H /2p2HQTK2Mi BM i?2 T`2b2Mi i`2iK2Mi Q7 _B2KMM BMi2;`Hb- M/ mb2 G2#2b;m2 BMi2;`iBQM i?2Q`v Ĝ BM T`iB+mH` i?2 TQr2`7mH G2#2b;m2Ƕb /QKBMi2/ +QMp2`;2M+2 i?2Q`2KĜ r?B+? HHQrb +QMbB/2`#Hv bBKTH2` `;mK2MibX

9XkX h>1 _1L1qG h>1P_1J

RjR

h?2Q`2K 9XkX9 G2i g #2  _B2KMM BMi2;`#H2 7mM+iBQM QM i?2 #QmM/2/ BMi2`pH [0, a]X h?2M, UBV g Bb #QmM/2/ M/ HKQbi 2p2`vr?2`2 +QMiBMmQmb QM [0, a] M/ UBBV i?2 HBKBi



a

g b (t)dt

lim b↓0

0

2tBbib M/ Bb }MBi2X h?Bb HBKBi Bb #v /2}MBiBQM i?2 _@BMi2;`H U_B2KMM BMi2@ ;`HV Q7 g QM [0, a]X Ai Bb /2MQi2/ #v  a  a R@ g(t) dt = lim g b (t)dt. b↓0

0

0

Ai +QBM+B/2b rBi? i?2 G2#2b;m2 BMi2;`H Q7 g QM [0, a]X S`QQ7X UBV "QmM/2/M2bb Q7 g Bb +H2` bBM+2 supx∈[0,a] g(x) ≤ b−1 }MBi2 #v bbmKTiBQMX

a 0

g b (t)dt- r?B+? Bb

q2 MQr b?Qr i?i i?2 b2i Q7 /Bb+QMiBMmBiv TQBMib Q7 g ?b  MmHH G2#2b;m2 K2bm`2X G2i g(x) = lim supy→x g(y)- M/ g(x) = lim inf y→x g(y)X "Qi? 7mM+iBQMb `2 K2bm`#H2X AM 7+i- KQ`2 Bb i`m2, g Bb mTT2` b2KB@+QMiBMmQmb- i?i Bb- 7Q` HH A ∈ R+ - i?2 b2i {x : g(x) ≥ A} Bb +HQb2/- r?BH2 g Bb HQr2` b2KB@+QMiBMmQmb- i?i Bb7Q` HH A ∈ R+ - i?2 b2i {x : g(x) ≤ A} Bb +HQb2/X q2 QKBi i?2 U2bvV T`QQ7 Q7 i?2b2 7+ibX h?2 b2i Q7 /Bb+QMiBMmBiv TQBMib Q7 g QM [0, a]- i?i Bb- {x : g(x) > g(x)}- Bb i?2`27Q`2 K2bm`#H2X amTTQb2 Bi Bb Q7 TQbBiBp2 G2#2b;m2 K2bm`2X h?2M- i?2`2 2tBbib M  > 0 bm+? i?i i?2 b2i {x : g(x) − g(x) > } Bb HbQ Q7 TQbBiBp2 G2#2b;m2 K2bm`2- bv δX aBM+2 7Q` HH b > 0- M/ bBM+2 7Q` HKQbi 2p2`v t ∈ [0, a] g b (t) ≥ g(t) ≥ g(t) ≥ g b (t), Bi 7QHHQrb i?i 7Q` HH b > 0 a

 g b (t)dt −

0

0

a

g b (t)dt ≥ δ > 0,

r?B+? +QMi`/B+ib i?2 bbmKTiBQM Q7 _B2KMM BMi2;`#BHBiv U9XReVX UBBV i  +QMiBMmBiv TQBMi x Q7 g- Bi ?QH/b i?i lim g b (x) = g(x) = g(x). b↓0

"v UBV- i?Bb +QMp2`;2M+2 ?QH/b HKQbi 2p2`vr?2`2X "v /QKBMi2/ +QMp2`;2M+2 UrBi? /QKBMiBM; 7mM+iBQM g 1 (x) + g 1 (x − 1) + g 1 (x + 1)V a  a lim g b (t)dt = g(t)dt . b↓0

0

0



Rjk

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

.2}MBiBQM 9XkX8 h?2 7mM+iBQM g ≥ 0 Bb bB/ iQ #2 _B2KMM BMi2;`#H2 QM [0, ∞) B7  a lim R@ g(t) dt a↑∞

0

2tBbib M/ i?2 HBKBi Bb i?2M- #v /2}MBiBQM- Bib _B2KMM BMi2;`H QM [0, ∞)X _2K`F 9XkXe  KDQ` `2bmHi Q7 _B2KMMǶb BMi2;`iBQM i?2Q`v Bb i?i i?2 _B2@ KMM BMi2;`H QM i?2 }MBi2 BMi2`pH [0, a] Q7  7mM+iBQM 2tBbib B7 M/ QMHv B7 i?Bb 7mM+iBQM Bb HKQbi 2p2`vr?2`2 +QMiBMmQmb M/ #QmM/2/ QM i?Bb BMi2`pH- i?i BbS`QT2`iv UBV BM i?2 #Qp2 T`QTQbBiBQM Bb MQi QMHv M2+2bb`v- #mi HbQ bm{+B2Mi 7Q` _B2KMM@BMi2;`#BHBivX h?Bb `2bmHi Bb MQi M22/2/ BM i?Bb #QQFX q2 MQr im`M iQ i?2 /2}MBiBQM Q7 i?2 /B`2+i _B2KMM BMi2;`HX dz.B`2+iǴ K2Mb i?i i?Bb BMi2;`H QM [0, ∞) Bb MQi /2}M2/ b  HBKBi Q7 BMi2;`Hb Qp2` }MBi2 BMi2`pHb#mi dz/B`2+iHvǴ QM [0, ∞)X .2}MBiBQM  ∞ 9XkXd h?2 7mM+iBQM g ≥ 0 Bb bB/ iQ #2 /B`2+iHv _B2KMM BMi2;`#H2 U/_BV B7 0 g b (t)dt < ∞ 7Q` bQK2 UM/ i?2M 7Q` HHV b > 0- M/ B7 



lim b↓0



0





g b (t)dt − 0

g b (t)dt

= 0.

6`QK i?2 /2}MBiBQMb- Bi Bb +H2` i?i 7Q` 7mM+iBQMb pMBb?BM; QmibB/2  #QmM/2/ BMi2`pH- i?2 MQiBQMb Q7 _B2KMM BMi2;`#BHBiv M/ Q7 /B`2+i _B2KMM BMi2;`#BH@ Biv `2 i?2 bK2X HbQ- i?2 7QHHQrBM; MHQ;m2 Q7 h?2Q`2K 9XkX9 ?QH/b 7Q` /B`2+i _B2KMM BMi2;`#BHBiv, h?2Q`2K 9XkX3 G2i g ≥ 0 #2 /_BX h?2M, UBV g Bb #QmM/2/- M/ HKQbi 2p2`vr?2`2 +QMiBMmQmb QM R+ X UBBV h?2 HBKBi





g b (t)dt

lim b↓0

0

2tBbib M/ Bb }MBi2X h?Bb HBKBi Bb- #v /2}MBiBQM- i?2 dR@BMi2;`H U/B`2+i _B2@ KMM BMi2;`HV Q7 g QM R+ - M/ Bb /2MQi2/ #v  ∞  ∞ dR@ g(t) dt = lim g b (t)dt. 0

b↓0

0

Ai +QBM+B/2b rBi? i?2 G2#2b;m2 BMi2;`H Q7 g QM R+ X h?2 T`QQ7 Bb B/2MiB+H iQ i?i Q7 h?2Q`2K 9XkX9X h?2 7QHHQrBM; 2tKTH2 72im`2b  7mM+iBQM i?i Bb _B2KMM BMi2;`#H2- #mi MQi /B`2+iHv _B2KMM BMi2;`#H2X

9XkX h>1 _1L1qG h>1P_1J

Rjj

1tKTH2 9XkXN,  *QmMi2`2tKTH2X G2i {an }n≥1 M/ {b n }n≥1 #2 b2[m2M+2b Q7 TQbBiBp2 `2H MmK#2`b bm+? i?i 1/2 > a > a > · · · 1 2 n≥1 bn = ∞ M/

a b < ∞X G2i g #2 MmHH QmibB/2 i?2 mMBQM Q7 i?2 BMi2`pHb [n − an , n + an ]n n n≥1 n ≥ 1- M/ bm+? i?i 7Q` HH n ≥ 1- g(n − an ) = g(n + an ) = 0 M/ g(n) = bn - M/ g Bb HBM2` BM i?2 BMi2`pHb [n − an , n] M/ [n, n + an ]X h?2M- g Bb _B2KMM BMi2;`#H2,  ∞  g(t) dt = an b n < ∞ . R@ 0

n≥1

Ai Bb ?Qr2p2` MQi /B`2+iHv _B2KMM BMi2;`#H2 bBM+2

∞ 0

g¯b (t)dt = ∞ 7Q` HH b > 0X

h?2`2 2tBbi ?Qr2p2`  72r `2bbm`BM; `2bmHib, h?2Q`2K 9XkXRy UV A7 g Bb /B`2+iHv _B2KMM BMi2;`#H2- Bi Bb _B2KMM BMi2;`#H2 QM [0, ∞) M/  ∞  ∞ R@ g(t)dt = dR@ g(t)dt . 0

0

U#V LQM@M2;iBp2 MQM@BM+`2bBM; 7mM+iBQMb `2 /B`2+iHv _B2KMM BMi2;`#H2 B7 M/ QMHv B7 i?2v `2 _B2KMM BMi2;`#H2 QM [0, ∞)X U+V  MQM@M2;iBp2  ∞ 7mM+iBQM i?i Bb _B2KMM BMi2;`#H2 QM HH }MBi2 BMi2`pHb- M/ bm+? i?i 0 g¯1 (t)dt < ∞- Bb /B`2+iHv _B2KMM BMi2;`#H2X AM T`iB+mH`-  ∞ MQM@M2;iBp2 HKQbi 2p2`vr?2`2 +QMiBMmQmb 7mM+iBQM bm+? i?i 0 g¯1 (t)dt < ∞ Bb /B`2+iHv _B2KMM BMi2;`#H2X U/V  MQM@M2;iBp2 7mM+iBQM i?i Bb _B2KMM BMi2;`#H2 M/ #QmM/2/ #Qp2 #v  /B`2+iHv _B2KMM@BMi2;`#H2 7mM+iBQM Bb /B`2+iHv _B2KMM BMi2;`#H2X S`QQ7X UV aBM+2 g Bb /B`2+iHv _B2KMM BMi2;`#H2 ∞  a 0 = lim (¯ gb (t) − g b (t))dt ≥ lim (¯ gb (t) − g b (t))dt, b↓0

b↓0

0

0

BKTHvBM; _B2KMM@BMi2;`#BHBiv QM [0, a]X 6Q` HH a > 0- `2+HHBM; i?i g Bb UG2#2b;m2V BMi2;`#H2 QM [0, +∞)& & ∞ &  ∞ &  a  a  ∞ & & & & &=& &= &dR− g(t)dt − _@ g(t)dt g(t)dt − g(t)dt g(t)dt. & & & & 0

0

0

0

a

h?2 `B;?i@?M/ bB/2 i2M/b iQ x2`Q b a → ∞ #v /QKBMi2/ +QMp2`;2M+2X U#V h?2 M2+2bbBiv 7QHHQrb 7`QK UVX AM pB2r Q7 T`QpBM; bm{+B2M+v- bmTTQb2 i?i i?2 MQM@M2;iBp2 MQM@BM+`2bBM; 7mM+iBQM g Bb _B2KMM BMi2;`#H2 QM [0, +∞)X Ai Bb BM T`iB+mH` UG2#2b;m2V BMi2;`#H2 QM [0, a] 7Q` HH }MBi2 a > 0- M/ i?2 BMi2;`H

Rj9

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

a

g(t)dt /KBib  }MBi2 HBKBi b a → ∞X h?2`27Q`2- g Bb UG2#2b;m2V BMi2;`#H2 QM 0 R+ - #v KQMQiQM2 +QMp2`;2M+2X aBM+2 Bi Bb MQM@BM+`2bBM;- 7Q` HH b > 0  ∞  ∞  g¯b (t)dt = bg(nb) ≤ bg(0) + g(t)dt < ∞ . 0

6m`i?2`KQ`2 ∞

0

n≥0

 g¯b (t)dt −

0

0



g b (t)dt =



bg(nb) −



bg(nb) = bg(0).

n>0

n≥0

h?2 Hii2` i2`K pMBb?2b b b → 0- 2bi#HBb?BM; i?i g Bb /B`2+iHv _B2KMM BMi2@ ;`#H2X ∞ U+V 6Bt ε > 0 M/ b2H2+i a > 0 bm+? i?i a g¯1 (t)dt ≤ εX 6Q` HH b ∈ (0, 1]g b (t) ≤ g¯b (t) ≤ g¯1 (t − 1) + g¯1 (t) + g¯1 (t + 1). Ai 7QHHQrb i?i  ∞  g¯b (t)dt − 0

∞ 0

 g b (t)dt ≤

a+1

0



≤ 0

a+1

  g¯b (t)dt − g b (t) dt + 3





g¯1 (t)dt a

 g¯b (t)dt − g b (t) dt + 3ε .



b g Bb bbmK2/ _B2KMM BMi2;`#H2 QM [0, a + 1]- i?2 `B;?iKQbi BMi2;`H i2M/b iQ x2`Q b b → 0X aBM+2 ε > 0 Bb `#Bi``v- r2 +QM+Hm/2 i?i i?2 H27i@?M/ bB/2 ;Q2b iQ x2`Q b b → 0X >2M+2- g Bb /B`2+iHv _B2KMM BMi2;`#H2X U/V 6QHHQrb 7`QK U+V bBM+2 g Bb _B2KMM BMi2;`#H2 QM }MBi2 BMi2`pHb- M/ +HHBM; z i?2 #QmM/BM; 7mM+iBQM- r2 ?p2- bBM+2 g¯1 ≤ z¯1  ∞  ∞ g¯1 (t) dt ≤ z¯1 (t) dt, 0

0

 }MBi2 [mMiBiv #2+mb2 z Bb /B`2+iHv _B2KMM BMi2;`#H2 #v bbmKTiBQMX



h?2 7QHHQrBM; Bb  7`2[m2MiHv 2M+QmMi2`2/ 2tKTH2 Q7  /B`2+iHv _B2KMM@ BMi2;`#H2 7mM+iBQM, 1tKTH2 9XkXRR, hBH /Bbi`B#miBQM Q7 M BMi2;`#H2 p`B#H2X G2i F #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 M BMi2;`#H2 MQM@M2;iBp2 `M/QK p`B#H2 S1 X h?2M 1 − F Bb /B`2+iHv _B2KMM BMi2;`#H2X h?Bb 7QHHQrb 7`QK U#V Q7 h?2Q`2K ∞ 9XkXRy M/ 0 (1 − F (t)) dt = E[S1 ] < ∞X _2K`F 9XkXRk 1tKTH2 9X9X8 #2HQr 72im`2b  bT2+i+mH` 2tKTH2 b?QrBM; i?i i?2 /B`2+i BMi2;`#BHBiv +QM/BiBQM +MMQi #2 /BbT2Mb2/ rBi? BM ;2M2`HX h?2Q@ `2K 9XkXRy #Qp2 M/ h?2Q`2K 9X9Xe ;Bp2 T`+iB+H rvb iQ T`Qp2 /B`2+i _B2KMM BMi2;`#BHBivX

9XkX h>1 _1L1qG h>1P_1J

Rj8

h?2 E2v _2M2rH h?2Q`2K h?2 `2M2rH T`Q+2bb2b +QMbB/2`2/ 7`QK MQr QM MQr `2 i?Qb2 rBi?  `2M2rH /Bb@ i`B#miBQM i?i Bb MQM@HiiB+2 U.2}MBiBQM 9XkXkVX h?2Q`2K 9XkXRj G2i F #2  MQM@HiiB+2 /Bbi`B#miBQM 7mM+iBQM bm+? i?i F (∞) = 1 UrBi? TQbbB#Hv BM}MBi2 K2MV M/ H2i R #2 i?2 bbQ+Bi2/ `2M2rH 7mM+iBQMX h?2M, UαV "H+Fr2HHǶb i?2Q`2K,e 7Q` HH τ ≥ 0lim{R(t + τ ) − R(t)} = t↑∞

τ . E[S1 ]

U9XRdV

UβV E2v `2M2rH i?2Q`2K, B7 g : R+ → R Bb  MQM@M2;iBp2 /B`2+iHv _B2KMM@ BMi2;`#H2 7mM+iBQM ∞ 1 lim(R ∗ g)(t) = g(y) dy . U9XR3V t↑∞ E[S1 ] 0 AM 7+i- UαV M/ UβV `2 2[mBpH2MiX S`QQ7X UαV q2 b?HH /KBi Bi 7Q` i?2 iBK2 #2BM;X HH 2tBbiBM; T`QQ7b `2 bQK2r?i i2+?@ MB+HX  T`QQ7- #b2/ QM i?2 bQ@+HH2/ dz+QmTHBM; K2i?Q/Ǵ- Bb ;Bp2M Hi2` Ubi`iBM; rBi? h?2Q`2K 9XjXRV r?2M E[S1 ] < ∞X UβV _2+HH i?i r?2M g Bb HQ+HHv #QmM/2/- f = R ∗ g Bb i?2 mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM Q7 i?2 `2M2rH 2[miBQM  f (t) = g(t) + f (t − s) dF (s). [0,t]

ah1S RX *b2 g(t) = 1[(n−1)b,nb) (t)X h?2M f (t) = R(t − (n − 1)b) − R(t − nb)M/ i?2 `2bmHi Bb Dmbi "H+Fr2HHǶb i?2Q`2KX

ah1S kX *b2 g(t) = n≥1 cn 1[(n−1)b,nb) (t)- r?2`2 cn ≥ 0- n≥1 cn < ∞- M/ b Bb bm+? i?i F (b) < 1X h?2M  f (t) = cn (R(t − (n − 1)b) − R(t − nb)). n≥1

"v G2KK 9XRXdsup(R(t − (n − 1)b) − R(t − nb)) ≤ (1 − F (b))−1 < ∞. t≥0

AM T`iB+mH`- #v /QKBMi2/ +QMp2`;2M+2e

("H+Fr2HH- RN93)X

Rje

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a 



b 1 = cn (R(t − (n − 1)b) − R(t − nb)) = cn lim t↑∞ E[S ] E[S 1 1] n≥1 n≥1





g(y) dy . 0

ah1S jX A7 g Bb /B`2+iHv _B2KMM BMi2;`#H2- i?2 7mM+iBQMb g¯b M/ g b T`2pBQmbHv   /2}M2/ `2 Q7 i?2 ivT2 +QMbB/2`2/ BM ai2T k bBM+2 g b ≤ g¯b < ∞X "mi g b ≤ g ≤ g¯b M/ i?2`27Q`2   1 g b (s) ds = lim g b (t − s) R(ds) t↑∞ E[S1 ]  ≤ lim inf g(t − s)R(ds) t↑∞  ≤ lim sup g(t − s) R(ds) t↑∞   1 ≤ lim g¯b (t − s) R(ds) = g¯b (s) ds. t↑∞ E[S1 ] h?2 `2bmHi 7QHHQrb #v H2iiBM; b i2M/ iQ 0X q2 b?Qr2/ i?i UαV BKTHB2b UβVX h?2 +QMp2`b2 BKTHB+iBQM 7QHHQrb #v +?QQbBM;  g(t) := 1[0,τ ] (t)X 1tKTH2 9XkXR9, 1H2K2Mi`v _2M2rH h?2Q`2K, *Q``2+iBQM h2`KhF2 kX AM i?2 `2+m``2Mi +b2- i?2 2H2K2Mi`v `2M2rH i?2Q`2K i2HHb mb i?i lim

t→∞

R(t) 1 = . t E[S1 ]

AM Q`/2` iQ Q#iBM KQ`2 BM7Q`KiBQM QM i?2 bvKTiQiB+ #2?pBQm` Q7 i?2 `2M2rH 7mM+iBQM- r2 b?HH bim/v i?2 #2?pBQm` Q7 f (t) := R(t) −

t E[S1 ]

b t ;Q2b iQ ∞ BM i?2 MQM@HiiB+2 +b2 r?2M S1 ?b }MBi2 }`bi M/ b2+QM/ KQK2MibX *HHBM; σ 2 i?2 +QKKQM p`BM+2 Q7 i?2 BMi2`@`2M2rH iBK2b M/ H2iiBM; m := E[S1 ]r2 ?p2   t 1 E[S1 ]2 + σ 2 lim R(t) − = . U9XRNV t↑∞ E[S1 ] 2 E[S1 ]2 S`QQ7X _2+HH 7`QK 1tKTH2 9XRXRj i?i f biBb}2b i?2 `2M2rH 2[miBQM f = g + F ∗ f rBi? /i  ∞ 1 g(t) = (1 − F (x)) dx . E[S1 ] t h?2 7mM+iBQM g Bb Q7 i?2 7Q`K 1 − F0 r?2`2 F0 Bb i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+@ iBQM Q7  MQM@M2;iBp2 p`B#H2 i?i Bb BMi2;`#H2X Ai Bb i?2`27Q`2 /B`2+iHv _B2KMM BMi2;`#H2 U1tKTH2 9XkXRRVX h?2 F2v `2M2rH i?2Q`2K i?2M ;Bp2b    1 ∞ t lim R(t) − = g(s) ds. t→∞ m m 0

9XkX h>1 _1L1qG h>1P_1J

Rjd

"mi 1 m



  ∞  ∞ 1 (1 − F (x)) dx ds m2 0   ∞ s x 1 = 2 (1 − F (x))ds dx m 0 0  ∞  1 1 ∞ 2 1 x(1 − F (x)) dx = 2 x dF (x), = 2 m 0 m 2 0



g(s)ds = 0

?2M+2 i?2 `2bmHiX US`QQ7 Q7 i?2 Hbi 2[mHBiv, 1 2 E X =E 2



X





x dx = E 0









1{x t) = P (eαS1 > eαt ) ≤ e−αt E[eαS1 ]. _2K`F 9XkXRd *H2`Hv- 7`QK U9XkjV- BM i?2 MQM@HiiB+2 /272+iBp2 +b2 M/ bbmK@ BM; i?2 2tBbi2M+2 Q7 bm+? α- i?2 bQHmiBQM Q7 i?2 `2M2rH 2[miBQM /2+vb 2tTQM2M@ iBHHv 7bi b t → ∞- r?2`2b BM i?2 MQM@HiiB+2 2t+2bbBp2 +b2 U7Q` r?B+? α Hrvb 2tBbibV i?2 bQHmiBQM Q7 i?2 `2M2rH 2[miBQM 2tTHQ/2b 2tTQM2MiBHHv 7bi b t → ∞X 1tKTH2 9XkXR3, bvKTiQiB+b BM i?2 h`MbB2Mi *b2X amTTQb2 i?i F Bb /272+iBp2 UF (∞) < 1V M/ i?i i?2`2 2tBbib M α UM2+2bb`BHv > 0V biBb7vBM; U9XkkVX "v h?2Q`2K 9XRXR8- r?2M i?2 /i g Bb #QmM/2/ M/ bm+? i?i i?2`2 2tBbib g(∞) = limt→∞ g(t)- i?2 mMB[m2 bQHmiBQM f Q7 i?2 `2M2rH 2[miBQM f = g + f ∗ F biBb}2b g(∞) lim f (t) = . U9Xk9V t↑∞ 1 − F (∞) qBi? i?2 ?2HT Q7 i?2 /272+iBp2 `2M2rH i?2Q`2K //BiBQMH BM7Q`KiBQM +QM+2`MBM; i?2 bvKTiQiB+ #2?pBQ` Q7 f +M #2 Q#iBM2/X AM 7+i- B7 i?2 7mM+iBQM g1 /2}M2/ #v F (t) − F (∞) g1 (t) = g(t) − g(∞) + g(∞) 1 − F (∞)

R9y

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

Bb bm+? i?i i?2 7mM+iBQM t → g1 (t) = eαt g1 (t) Bb /B`2+iHv _B2KMM BMi2;`#H2- i?2M   g(∞) lim eαt f (t) − =C, U9Xk8V t→∞ 1 − F (∞) r?2`2

∞ C=

0

eαt [g(t) − g(∞)]dt − ∞ teαt dF (t) 0

S`QQ7X .2}M2 f1 (t) := f (t) −

g(∞) α

.

g(∞) . 1 − F (∞)

ai`B;?i7Q`r`/ +QKTmiiBQMb mbBM; i?2 B/2MiBiv R ∗ F = R − 1 b?Qr i?i f1 = R∗g1 X h?2`27Q`2 f1 Bb  bQHmiBQM Q7 i?2 U/272+iBp2V `2M2rH 2[miBQM f1 = g1 +f1 ∗F X aBM+2 g1 (t) = eαt g1 (t) Bb bbmK2/ iQ #2 /B`2+iHv _B2KMM BMi2;`#H2 ∞ αt e g1 (t)dt αt . lim e f1 (t) = 0∞ αt t→∞ te dF (t) 0 "mi

 0





 dF (s) dt 0 (t,∞)   ∞  t 1 αs e ds dF (t) = (1 − F (∞)), = α 0 0 

eαt (F (∞) − F (t))dt =



eαt

7`QK r?B+? i?2 #Qp2 2tT`2bbBQM 7Q` C 7QHHQrbX



1tKTH2 9XkXRN, h?2 _BbF JQ/2H- hF2 jX h?Bb 2tKTH2 Bb  +QMiBMmiBQM Q7 1tKTH2 9XRXNX _2+HH 1[MX U9XRkV BM i?2 +b2 r?2`2 i?2 b72iv HQ/BM; Bb TQbBiBp2M/ r`Bi2 i?Bb 2[miBQM BM i?2 7Q`K   u λ ∞ Ψ(u) = (1 − G(z)) dz + Ψ(u − z) dF (z) , c u 0 r?2`2 F (x) :=

λ c



x

(1 − G(z)) dz . 0

∞ < 1 mM/2` i?2 h?Bb `2M2rH 2[miBQM Bb /272+iBp2 bBM+2 λc 0 (1 − G(z)) dz = λμ c TQbBiBp2 b72iv HQ/BM; +QM/BiBQMX bbmK2 i?2 2tBbi2M+2 Q7 α > 0 bm+? i?i  λ x αz e (1 − G(z)) dz = 1. c 0 "v i?2 /272+iBp2 `2M2rH i?2Q`2K- B7   λ ∞ αu ∞ e (1 − G(z)) dz du < ∞ , c 0 u

9XkX h>1 _1L1qG h>1P_1J i?2M

R9R

 ∞ αu  ∞ e (1 − G(z)) dz du lim e Ψ(u) = C := 0  ∞ uαz u↑∞ ze (1 − G(z)) dz 0 αu

Q`- 2[mBpH2MiHv-

Ψ(u) = Ce−αu + o(u) .

*b2 Q7  .272+iBp2 am#2tTQM2MiBH .Bbi`B#miBQM 6Q` t ≥ 0- γ ∈ (0, ∞)- H2i ∞ 

Rγ (t) :=

γ n F ∗n (t) ,

n=0

r?2`2 F ∗n Bb i?2 n@i? +QMpQHmiBQM Q7 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F Q7  MQM@M2;iBp2 `M/QK p`B#H2 UrBi? i?2 mbmH +QMp2MiBQM F ∗0 (t) ≡ 1VX A7 γF (0) < 1- i?2M i?2 7mM+iBQM Rγ - +HH2/ i?2 r2B;?i2/ `2M2rH 7mM+iBQM- iF2b }MBi2 pHm2b QM R+ M/ Bb HQ+HHv #QmM/2/X S`QQ7X G2i {Sn }n≥1 #2 M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b rBi? +QKKQM +mKm@ HiBp2 /Bbi`B#miBQM 7mM+iBQM F X h?2M- 7Q` θ ≥ 0  γ n F ∗n (t) = γ n P (S1 + · · · Sn ≤ t) = γ n P e−θ(S1 +···Sn ) ≥ e−θt  n ≤ γ n eθt E e−θ(S1 +···Sn ) = eθt γe−θS1 . aBM+2 γF (0) < 1- i?2`2 2tBbib  θ > 0 bm+? i?i γE e−θS1 < 1 M/ i?2`27Q`2  −1 Rγ (t) ≤ eθt 1 − E e−θS1 .  "v biM/`/ `;mK2Mib- 7Q` Mv γ ∈ R+ - Rγ biBb}2b i?2 `2M2rH 2[miBQM  Rγ (t) = 1 + γ Rγ (t − s) F (ds) , [0,t]

M/ i?2 /272+iBp2 `2M2rH 2[miBQM

 f (t − s) F (ds)

f (t) = g(t) + γ [0,t]

/KBib 7Q` mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM  g(t − s) dRγ (s) . f (t) = g ∗ Rγ := [0,t]

A7 γ = 1- i?2 +HbbB+H `2M2rH i?2Q`2K TTHB2bX 6Q` i?2 +b2 γ = 1- r2 ?p2  /272+iBp2 Uγ < 1V Q` 2t+2bbBp2 Uγ > 1V `2M2rH 2[miBQMX q2 ?p2 b22M ?Qr iQ

R9k

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

`2/m+2 i?2 bvKTiQiB+ bim/v Q7 bm+? mMT`QT2` `2M2rH 2[miBQMb BM i?2 +b2 QM2 +M }M/ α biBb7vBM; U9XkkVX q2 ?p2 HbQ b22M i?i r?2`2b bm+? α Hrvb 2tBbib BM i?2 2t+2bbBp2 +b2- Bib 2tBbi2M+2 Bb MQi Hrvb ;`Mi2/ BM i?2 /272+iBp2 +b2X h?2 bvKTiQiB+ bim/v Bb ?Qr2p2` TQbbB#H2 BM i H2bi QM2 BKTQ`iMi +b2- KQ`2 T`2+Bb2Hv r?2M F Bb  bm#2tTQM2MiBH +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQMX .2}MBiBQM 9XkXky Ud V  /Bbi`B#miBQM 7mM+iBQM F Bb bB/ iQ #2 bm#2tTQM2MiBH Ui?Bb Bb /2MQi2/ #v F ∈ SV B7 1 − F ∗2 (t) lim = 2. U9XkeV t↑∞ 1 − F (t) h?i Bb- B7 S1 M/ S2 `2 BM/2T2M/2Mi `M/QK p`B#H2b rBi? i?2 +QKKQM /Bbi`B@ #miBQM F P (S1 + S2 > t) lim = 2. t↑∞ P (S1 > t) 1tKTH2 9XkXkR, 1tKTH2bX h?2 `M/QK p`B#H2b r?Qb2 /Bbi`B#miBQMb ?p2 iBHb rBi? i?2 7QHHQrBM; bvKTiQiB+b `2 bm#2tTQM2MiBH, 1 − F (t) ∼ t−k

(k > 0)

−tβ

∼e

(0 < β < 1) .

h?2 KBM T`QT2`iB2b Q7 bm#2tTQM2MiBH /Bbi`B#miBQMb Q7 /B`2+i BMi2`2bi BM i?Bb #QQF rBHH #2 ;Bp2M rBi?Qmi T`QQ7X3 6B`bi Q7 HH-  bm#2tTQM2MiBH /Bbi`B#miBQM F Bb ?2pv@ iBH2/- Q` KQ`2 T`2+Bb2Hv- Bib iBH Bb dzbm#2tTQM2MiBHǴ BM i?2 b2Mb2 i?i 7Q` HH α > 0lim eαt (1 − F (t)) = ∞ . t↑∞

AM T`iB+mH`- i?2`2 Bb MQ α biBb7vBM; U9XkkVX G2KK 9XkXkk A7 F ∈ S- 7Q` HH A ∈ (0, ∞)lim t↑∞

1 − F (t − A) = 1. 1 − F (t)

G2KK 9XkXkj A7 F ∈ S- 7Q` HH n ≥ 11 − F ∗n (t) = n. t↑∞ 1 − F (t)

lim

h?2 M2ti `2bmHi Bb FMQrM b E2bi2MǶb #QmM/, d 3

(*?BbivFQp- RNe9)X a22 7Q` BMbiM+2 (6Qbb- EQ`b?mMQp M/ w+?`v- kyRR)X

9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha

R9j

G2KK 9XkXk9 A7 F ∈ S- ;Bp2M Mv ε > 0- i?2`2 2tBbib  D = D(ε) < ∞ bm+? i?i 7Q` HH n ≥ M/ t ≥ 01 − F ∗n (t) ≤ D(1 + ε)n . 1 − F (t) >pBM; +QHH2+i2/ i?2b2 T`2HBKBM`v H2KKb- i?2 KBM `2bmHi Q7 i?Bb bm#b2+iBQM +M #2 bii2/ M/ T`Qp2/, h?2Q`2K 9XkXk8 UN V A7 F ∈ S M/ 0 < γ < 1(1 − γ)−1 − Rγ (t) γ = . t↑∞ 1 − F (t) (1 − γ)2

lim

S`QQ7X

(1 − γ)−1 − Rγ (t)  n 1 − F ∗n (t) = . γ 1 − F (t) 1 − F (t) n=0 ∞

∗n

(t) ≤ D(γ(1 + ε))n - M/ qBi? ε bm+? i?i γ(1 + ε) < 1- #v E2bi2MǶb #QmM/- γ n 1−F 1−F (t) i?2`27Q`2- QM2 Kv iF2 i?2 HBKBi b t ↑ ∞ i2`K #v i2`K BM i?2 `B;?i@?M/ bB/2 iQ Q#iBM- mbBM; G2KK 9XkXkj-

lim t↑∞

∞  n=0

1 − F ∗n (t)  n 1 − F ∗n (t)  n γ = = γ lim nγ = . t↑∞ 1 − F (t) 1 − F (t) (1 − γ)2 n=0 n=0 ∞

γn





9Xj

"H+Fr2HHǶb h?2Q`2K M/ Bib _2}M2K2Mib

q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 "H+Fr2HHǶb i?2Q`2K BM i?2 MQM@HiiB+2 +b2 M/ mM/2` i?2 //BiBQMH +QM/BiBQM E[S1 ] < ∞X h?2 *QmTHBM; S`QQ7 Q7 "H+Fr2HHǶb h?2Q`2K q2 }`bi ;Bp2 i?`22 2[mBpH2Mi 7Q`KmHiBQMb Q7 "H+Fr2HHǶb i?2Q`2KX G2KK 9XjXR *QMbB/2`  `2M2rH T`Q+2bb rBi? BMBiBH /2Hv T0 UHKQbi bm`2Hv }MBi2V /Bbi`B#mi2/ ++Q`/BM; iQ GX G2i  x F0 (x) = μ−1 (1 − F (y)) dy U9XkdV 0





M/ RG (t) = E



1{Tn ≤t} .

n≥0 N

(i?`2v M/ L2v- RNdk)- h?2Q`2K AoX9Xj- TX R8yX

R99

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

UBV 6Q` HH a > 0- limt↑∞ (RG (t + a) − RG (t)) = μ−1 a U"H+Fr2HHǶb i?2Q`2KVX UBBV 6Q` HH x ≥ 0- limt↑∞ P (B(t) ≤ x) = F0 (x)X UBBBV 6Q` HH x ≥ 0- limt↑∞ P (A(t) ≤ x) = F0 (x)X S`QQ7X UBBV ⇔ UBBBVX Cmbi Q#b2`p2 i?i 7Q` t ≥ 0- x ≥ 0P (A(t) ≤ x) = P (N [0, t + x] − N ([0, t]) ≥ 1) = P (B(t + x) ≤ x). UBV ⇒ UBBVX q?2M T0 ≡ 0- i?2 7mM+iBQM t → P (B(t) ≤ x) biBb}2b  `2M2rH 2[miBQM rBi? /i g(t) = (1 − F (t))1[0,x] (t)X h?Bb 7mM+iBQM Bb /B`2+iHv _B2KMM BMi2;`#H2- M/ i?2`27Q`2 #v i?2 F2v `2M2rH i?2Q`2K U +QMb2[m2M+2 Q7 "H+Fr2HHǶb i?2Q`2KV  ∞

lim P (B(t) ≤ x) = μ−1 t↑∞

g(t)dt = F0 (x). 0

h?2 +b2 Q7  MQM@MmHH M/ T`QT2` BMBiBH /2Hv 7QHHQrb #v i?2 mbmH `;mK2Mi Ub22 i?2 T`QQ7 Q7 h?2Q`2K 9X9X9VX UBBBV ⇒ UBVX .2}M2 Gt (x) = P (A(t) ≤ x) M/ Q#b2`p2 i?i RG (t + a) − RG (t) = (Gt ∗ R)(a)  a = R(a − s)Gt (ds) 0 a = Gt (a − s)R(ds) 0

M/ i?i bBM+2 #v ?vTQi?2bBb limt↑∞ Gt (x) = F0 (x)- r2 ?p2 #v /QKBMi2/ +QMp2`@ ;2M+2 UR ;Bp2b }MBi2 Kbb iQ #QmM/2/ BMi2`pHbV  a  a lim Gt (a − s)R(ds) = F0 (a − s)R(ds) = μ−1 a. t↑∞

0

0

 h?2`27Q`2- 7Q` i?2 T`QQ7 Q7 "H+Fr2HHǶb i?2Q`2K- Bi Bb 2MQm;? iQ T`Qp2 UBBBV Q7 G2KK 9XjXRX q2 /Q i?Bb BM i?2 +b2 r?2`2 μ < ∞XRy >2`2 Bb i?2 +QmTHBM; `;mK2MiXRR *QMbB/2` irQ BM/2T2M/2Mi `2M2rH b2[m2M+2b rBi? i?2 bK2 BMi2```BpH /Bb@ i`B#miBQM F X h?2 }`bi QM2 Bb mM/2Hv2/, S0 = 0- S1 - S2 - ĘM/ i?2 b2+QM/ QM2 Bb biiBQM`v, S0 - S1 - S2 X AM T`iB+mH`- i?2 /Bbi`B#miBQM Q7 S0 Bb F0 ;Bp2M #v U9XkdVX *QMbi`m+i  `2M2rH b2[m2M+2 {Sn∗ }n≥1 b 7QHHQrbX hF2 Sn∗ = Sn mMiBH i?2 }`bi iBK2 r?2`2 irQ TQBMib Q7 i?2 iBH/2/ M/ mMiBH/2/ T`Q+2bb2b `2 ε@+HQb2- r?2`2 ε Ry RR

6Q` i?2 2ti2MbBQM iQ μ = ∞- b22 7Q` BMbiM+2 (GBM/pHH- RNNk)- TTX deĜddX (GBM/pHH- RNdd)X

9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha

R98

Bb }t2/X UAM i?Bb +b2 r2 bv i?i ε@+QmTHBM; rb bm++2bb7mH- r?B+? Bb MQi ;`Mi2/ BM ;2M2`HX h?2 i2+?MB+H T`i Q7 i?2 T`QQ7 Q7 "H+Fr2HHǶb i?2Q`2K Bb iQ b?Qr i?i ε@+QmTHBM; Bb +imHHv `2HBx#H2 rBi? T`Q##BHBiv R r?2M i?2 BMi2`pH /Bbi`B#miBQM Bb MQM@HiiB+2XV h?2M 7QHHQr i?2 iBH/2/ T`Q+2bbX 6Q` BMbiM+2- bmTTQb2 i?i T5 M/ ∗ T3 `2 i  /BbiM+2 H2bb i?M εX h?2M Sn∗ = Sn 7Q` n = 1, 2, 3, 4, 5- M/ S5+k = S3+k 7Q` k ≥ 1X .2MQi2 #v T = T ε i?2 }`bi TQBMi Q7 i?2 rB/2iBH/2/ T`Q+2bb r?B+? Bb ε@+HQb2 iQ  TQBMi Q7 i?2 mMrB/2iBH/2/ T`Q+2bb UBM i?2 2tKTH2 T = T3 VX T5 S1 = S1∗

S2 = S2∗

S3 = S3∗ S4 = S4∗

S5 = S5∗ ≤

S˜0

S˜1

S˜2

S˜3 S˜6∗

S˜7∗

T˜3

G2KK 9XjXk qBi? i?2 bbmKTiBQMb Q7 h?2Q`2K 9XkXRj- B7 ε@+QmTHBM; ?TT2Mb HKQbi bm`2Hv- i?i Bb- B7 P (T < ∞) = 1- i?2M "H+Fr2HHǶb i?2Q`2K Bb T`Qp2/X  S`QQ7X 6Q` bBKTH2` MQiiBQM- H2i T := Tε X G2i {A(t)}t≥0 M/ {A(t)} t≥0 #2 i?2 `2+m``2M+2 iBK2b +Q``2bTQM/BM; iQ i?2 mM/2Hv2/ bi``2/ `2M2rH T`Q+2bb M/ i?2  + D) r?2`2 UbiiBQM`vV iBH/2/ `2M2rH T`Q+2bbX 6Q` HH t ≥ T - r2 ?p2 A(t) = A(t |D| ≤ εX G2i f #2  +QMiBMmQmb 7mM+iBQM #QmM/2/ #v 1- M/ /2}M2 & & &   && . Mε (t) = sup &f (A(t + s)) − f (A(t)) |s|≤ε

LQi2 i?i limε↓0 E [Mε (0)] = 0X LQr&  & & & & & &   && &E f (A(t)) − f (A(t)) & ≤ E &f (A(t)) − f (A(t)) & &  &  && 1{t < T } = E &(f (A(t)) − f (A(t)) & &  &  && 1{t ≥ T } + E &(f (A(t)) − f (A(t)) ≤ 2P (T > t) + E [Mε (t)] = 2P (T > t) + E [Mε (0)] ,  .2/m+2 7`QK i?Bb i?i- bBM+2 r?2`2 7QHHQrb 7`QK biiBQM`Biv Q7 AX   i?2 Hbi  2[mHBiv   E f (A(t)) = E f (S0 )   lim E [f (A(t))] = E f (S0 ) . t↑∞

AM Qi?2` rQ`/b- bBM+2 f Bb M `#Bi``v +QMiBMmQmb 7mM+iBQM #QmM/2/ #v 1- A(t) i2M/b BM /Bbi`B#miBQM iQ S0 b t ↑ ∞X AM T`iB+mH`- bBM+2 i?2 /Bbi`B#miBQM F0 Q7 S0 Bb +QMiBMmQmb- 7Q` HH x ∈ R+ -

R9e

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a lim P (A(t) ≤ x) = F0 (x). t↑∞

h?2 +QM+HmbBQM 7QHHQrb 7`QK UBBBV Q7 G2KK 9XjXRX



AM Q`/2` iQ T`Qp2 @+QmTHBM;- r2 }`bi 2tKBM2 i?2 `QH2 Q7 i?2 MQM@HiiB+2 b@ bmKTiBQMX _2+HH i?i  TQBMi x Bb bB/ iQ #2 BM i?2 bmTTQ`i Q7 i?2 /Bbi`B#miBQM 7mM+iBQM F B7 F (x + ) − F (x − ) > 0 7Q` HH  > 0X h?2 b2i Q7 HH bm+? TQBMib Bb +HH2/ i?2 bmTTQ`i Q7 F M/ Bb /2MQi2/ #v bmTT(F )X h?2 F2v BKTHB+iBQM Q7 i?2 MQM@HiiB+2 bbmKTiBQM Bb i?2 7QHHQrBM;, G2KK 9XjXj G2i F #2  MQM@HiiB+2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQMX G2i G /2@ MQi2 i?2 b2i Q7 }MBi2 HBM2` +QK#BMiBQMb Q7 2H2K2Mib Q7 supp(F ) rBi? +Q2{+B2Mib BM N- i?i Bb   n   G= gi ; g1 , . . . , gn ∈ supp(F ) . U9Xk3V i=1

n∈N

h?2M G Bb bvKTiQiB+HHv /2Mb2 BM R+ - i?i Bb lim d(x, G) = 0,

x→∞

r?2`2 d(x, G) = inf g∈G |x − g|X P#b2`p2 i?i i?2 b2i G b /2}M2/ #v U9Xk3V Bb i?2 mMBQM Q7 i?2 bmTTQ`ib Q7 i?2 +mKmHiBp2 /Bbi`B#miBQM F ∗n Un ∈ NV Q`- 2[mBpH2MiHv- i?2 bmTTQ`i Q7 i?2

7mM+iBQMb ∗n `2M2rH 7mM+iBQM R = n∈N F bbQ+Bi2/ iQ F X S`QQ7X G2iiBM; μ :=

inf

g,h∈G,g>h

{g − h} ,

U9XkNV

r2 }`bi T`Qp2 i?i μ = 0X amTTQb2 BM pB2r Q7 +QMi`/B+iBQM i?i μ > 0X h?2 BM}KmK BM U9XkNV Bb i?2M M2+2bb`BHv iiBM2/- 7Q` Qi?2`rBb2 i?2`2 rQmH/ 2tBbi b2[m2M+2b gn hn BM G bm+? i?i gn − hn > gn+1 − hn+1 - M/ gn − hn → μ b n → ∞X h?2M7Q` n H`;2 2MQm;?- gn − hn < μ + μ/2X *QMb2[m2MiHv- H2iiBM; g := gn + hn+1 M/ h = hn + gn+1 - Bi ?QH/b i?i g − h = (gn − hn ) − (gn+1 − hn+1 ) ∈ (0, μ/2). h?Bb Bb  +QMi`/B+iBQM- BM pB2r Q7 i?2 /2}MBiBQM Q7 μ M/ i?2 7+i i?i g, h ∈ GX h?2`2 Kmbi i?2`27Q`2 2tBbi g, h ∈ G bm+? i?i g − h = μX aBM+2 F Bb MQM@HiiB+2i?2`2 2tBbib  z ∈ bmTT(F ) bm+? i?i- 7Q` bQK2 k ∈ Nkμ < z < (k + 1)μ. .2}M2 i?2M g  := z + kh M/ h := kgX "Qi? g  M/ h #2HQM; iQ GX 6m`i?2`KQ`2g  − h = z − kμ ∈ (0, m)- ;BM  +QMi`/B+iBQMX L2+2bb`BHv i?2M- μ = 0X h?2`27Q`2- 7Q` Mv  > 0- i?2`2 2tBbi g, h ∈ G bm+? i?i g − h ∈ (0, )X *QMbB/2` i?2 bm#b2i G  Q7 G +QMbBbiBM; Q7 i?2 2H2K2Mib kg + h Uk, ∈ NVX q2 `;m2 i?i

9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha

R9d

limx→∞ d(x, G  ) ≤ X AM/22/- H2i m = h/ X G2i x > mhX q`Bi2 x = nh + r- rBi? n ∈ N- n ≥ m- M/ r ∈ [0, h)X G2i k ∈ N #2 bm+? i?i (n − k)h + kg ≤ x < (n − k)h + kg + (g − h). L2+2bb`BHv k ≤ m bBM+2 r < hX h?2 i2`K (n − k)h + kg i?mb #2HQM;b iQ G  - b n − k ≥ 0X 6m`i?2`KQ`2- Bi Bb i KQbi  T`i 7`QK x- #v i?2 TB` Q7 BM2[mHBiB2b /BbTHv2/ #Qp2X Ai 7QHHQrb i?i lim sup d(x, G) ≤ lim sup d(x, G  ) ≤ . x→∞

x→∞

b  Bb `#Bi``v- i?Bb +QM+Hm/2b i?2 T`QQ7 Q7 i?2 i?2Q`2KX



PM2 bvb i?i @+QmTHBM; ?QH/b 7Q` `2M2rH T`Q+2bb2b rBi? BMi2`@`2M2rH +mKm@ HiBp2 /Bbi`B#miBQM 7mM+iBQM F B7 7Q`  > 0 M/ }t2/ BMBiBH /2Hvb t1 , t1 - QM2 +M +QMbi`m+i DQBMiHv irQ `2M2rH T`Q+2bb2b rBi? i?2 +Q``2bTQM/BM; /2Hvb bm+? i?irBi? T`Q##BHBiv R- i?2`2 `2 BM/B+2b m, n bm+? i?i i?2 +Q``2bTQM/BM; `2M2rH iBK2b Tm - Tn `2 H2bb i?M  T`iX G2KK 9XjX9 qBi? i?2 bbmKTiBQMb Q7 h?2Q`2K 9XkXRj- ε@+QmTHBM; ?TT2Mb H@ KQbi bm`2HvX S`QQ7X G2i

Zi := min{Tj − Ti ; Tj − Ti ≥ 0}

6Q` }t2/ ε > 0- H2i h?2M

(i ≥ 0) .

Ai := {Zj < ε 7Q` bQK2 j ≥ i} .

A0 ⊇ · · · ⊇ A ⊇ · · · ⊇ ∩∞ i=0 Ai = A∞ := {Zi < ε i.o.} .

aBM+2 i?2 b2[m2M+2 {Ti+n − Ti }n≥1 ?b  /Bbi`B#miBQM BM/2T2M/2Mi Q7 i M/ Bb BM@  Bb biiBQM`v- i?2 b2[m2M+2 {Zi }i≥0 Bb  ≡ {Tn }n≥0 - M/ bBM+2 N /2T2M/2Mi Q7 N HbQ biiBQM`vX h?2`27Q`2 i?2 2p2Mib Ai Ui ≥ 0V ?p2 i?2 bK2 T`Q##BHBiv- M/ BM T`iB+mH` P (A0 ) = P (A∞ ) . *QM/BiBQMHHv QM S0 - i?2 2p2Mi A∞ Bb M 2t+?M;2#H2 2p2Mi Q7 i?2 bvKK2i`B+ b2@ [m2M+2 {(Sn , Sn )}n≥1 X h?2`27Q`2- #v i?2 >2rBiiĜap;2 y@R Hr Uh?2Q`2K XRXkdV7Q` HH t > 0 P (A∞ | S0 = t) = 0 Q` 1 . () G2KK 9XjXj ;m`Mi22b i?i 7Q` bm{+B2MiHv H`;2 u M/ }t2/ εP (u − t < Tj − S0 < u − t + ε 7Q` bQK2 j) > 0 . h?2`27Q`2 P (A0 | S0 = t) > 0 7Q` HH t ≥ 0 M/ BM T`iB+mH`  ∞ P (A0 ) = λ P (A0 | S0 = t)(1 − F (t)) dt > 0 , 0

R93

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

r?B+? BKTHB2b bBM+2 P (A0 ) = P (A∞ )



P (A∞ ) = λ

P (A∞ | S0 = t)(1 − F (t)) dt > 0 .

0

AM pB2r Q7 UV- i?Bb BKTHB2b i?i P (A∞ | S0 = t) = 1 7Q` HH t bm+? i?i F (t) < 1 . h?2`27Q`2 P (A∞ ) = 1 = P (A0 )- bQ i?i P (Zi < ε 7Q` bQK2 i) = 1X



h?2 aT`2/@Qmi *b2 h?2 /B`2+i _B2KMM BMi2;`#BHBiv +QM/BiBQM 7Q` i?2 /i 7mM+iBQM +M #2 `2TH+2/ #v  r2F2` `2[mB`2K2Mi B7 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 BMi2`@`2M2rH iBK2b Bb bT`2/@Qmi Ui?2 T`2+Bb2 bii2K2Mi Bb ;Bp2M BM h?2Q`2K 9XjXd #2HQrVX .2}MBiBQM 9XjX8 h?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F Bb bB/ iQ #2 bT`2/@Qmi B7 i?2`2 2tBbib M BMi2;2` n0 bm+? i?i i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F ∗n0 ?b M #bQHmi2Hv +QMiBMmQmb +QKTQM2Mi- i?i BbF ∗n0 (t) = Σ(t) + A(t) r?2`2 Σ Bb i?2 /Bbi`B#miBQM 7mM+iBQM Q7  /272+iBp2 `M/QK p`B#H2 Uσ := Σ(∞) < 1V M/  t

A(t) =

a(s)ds 0

7Q` bQK2 MQM@M2;iBp2 K2bm`#H2 7mM+iBQM a : R+ → RX h?2`2 `2 Q7 +Qm`b2 BM}MBi2Hv KMv i`BTH2b n0 , Σ, a rBi? i?2 #Qp2 T`QT2`iB2b b bQQM b i?2`2 Bb QM2X q2 b?HH KF2 mb2 Q7 i?Bb iQ T`Qp2 i?i- rBi?Qmi HQbb Q7 ;2M2`HBiv- a +M #2 iF2M #QmM/2/- +QMiBMmQmb- rBi? +QKT+i bmTTQ`i M/ bm+? i?i a(0) = 0X S`QQ7X 6B`bi r2 b?Qr i?i a +M #2 bbmK2/ #QmM/2/ BMi2;`#H2 M/ +QMiBMmQmbX A7 MQi- `2TH+2 n0 #v 2n0 - M/ mb2 i?2 7+i i?i F ∗2n0 ?b M #bQHmi2Hv +QMiBMmQmb +QKTQM2Mi rBi?  /2MbBiv a ∗ a- M/ i?i B7 a Bb #QmM/2/ M/ BMi2;`#H2- i?2M a ∗ a Bb #QmM/2/ BMi2;`#H2 M/ +QMiBMmQmbX hQ b?Qr i?Bb Hbi bb2`iBQM Q#b2`p2 i?i Mv #QmM/2/ M/ BMi2;`#H2 7mM+iBQM g +M #2 TT`QtBKi2/ #v  b2[m2M+2 Q7 #QmM/2/ M/ BMi2;`#H2 +QMiBMmQmb 7mM+iBQMb gn bm+? i?i ||g − gn ||L1 → 0X "v /QKBMi2/ +QMp2`;2M+2- g ∗ gn Bb +QMiBMmQmbX aBM+2 ||g ∗ gn − g ∗ g||L∞ ≤ ||g||L∞ × ||g − gn ||L1 i?2M g ∗ g Bb +QMiBMmQmb b mMB7Q`K HBKBi Q7 +QMiBMmQmb 7mM+iBQMbX  q2 ?p2 i?2 7QHHQrBM; /2+QKTQbBiBQM,Rk Rk

(aiQM2- RNej)X

9XjX "G*Eq1GGǶa h>1P_1J L. Aha _16AL1J1Lha

R9N

h?2Q`2K 9XjXe amTTQb2 i?i i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 i?2 `2@ M2rH T`Q+2bb Bb bT`2/@Qmi rBi? K2M mX G2i R #2 i?2 `2M2rH 7mM+iBQM Q7 i?2 mM/2Hv2/ `2M2rH T`Q+2bb- M/ H2i RG = G ∗ R #2 i?2 `2M2rH 7mM+iBQM Q7 i?2 /2Hv2/ `2M2rH T`Q+2bb rBi? }MBi2 /2Hv Q7 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM GX h?2M RG = R1G + R2G U9XjyV r?2`2 i?2 7mM+iBQMb BM i?2 `B;?i@?M/ bB/2 `2 /Bbi`B#miBQM 7mM+iBQMb- R2G (∞) < ∞M/ R1G Bb #bQHmi2Hv +QMiBMmQmb- rBi?  /2MbBiv u1G biBb7vBM; lim u1G (t) = t↑∞

1 . E[S1 ]

U9XjRV

S`QQ7X AM i?2 +QKTmiiBQMb #2HQr- i?2 bvK#QH 7Q`  /Bbi`B#miBQM 7mM+iBQM Bb HbQ mb2/ iQ `2T`2b2Mi i?2 bbQ+Bi2/ K2bm`2XRj AM T`iB+mH`- i?2 .B`+ K2bm`2 i 0 #2BM; /2MQi2/ #v ε0 - r2 ?p2- rBi? i?Bb +QMp2MiBQM- F ∗0 = ε0 - M/ i?2 7mM/K2MiH `2M2rH 2[miBQM R = 1 + R ∗ F `2/b R = ε0 + R ∗ F X .2}M2 i?2 K2bm`2b   F ∗nn0 M/ ρ := F ∗n . R3 := 0≤n≤n0 −1

n≥0

P#b2`p2 i?i i?2 iQiH Kbb Q7 ρ Bb }MBi2- 2[mH iQ n0 - M/ i?i R = ρ ∗ R3 .

U9XjkV

HbQ- R3 Bb i?2 `2M2rH K2bm`2 Q7 F ∗n0 M/ i?2`27Q`2 biBb}2b i?2 `2M2rH 2[miBQM R3 = ε0 + F ∗n0 ∗ R3 = ε0 + (Σ + A) ∗ R3 M/ i?2`27Q`2

R3 ∗ (ε0 − Σ) = ε0 + A ∗ R3 .

h?2 K2bm`2 Rσ :=



U9XjjV

Σ∗n

n≥0

#2BM; i?2 `2M2rH K2bm`2 Q7  /272+iBp2 `2M2rH T`Q+2bb ?b }MBi2 iQiH Kbb 2[mH iQ (1 − σ)−1 - M/ (ε0 − Σ) ∗ Rσ = ε0 . h?2 UmMB[m2V bQHmiBQM Q7 U9XjjV BM R3 Bb R3 = R σ + A ∗ R σ ∗ R 3 . AMb2`iBM; i?Bb 2tT`2bbBQM BMiQ U9XjkV ;Bp2b R = ρ ∗ Rσ + A ∗ ρ ∗ R σ ∗ R3 , M/ }MHHv RG = R1G + R2G , Rj

h?2 +QMi2ti rBHH /BbbBTi2 i?2 +QM7mbBQMX

R8y

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

r?2`2

R1G := A ∗ G ∗ ρ ∗ Rσ ∗ R3 M/ R2G := G ∗ ρ ∗ Rσ .

aBM+2 ρ- Rσ M/ G ?p2 }MBi2 iQiH Kbb2b- R2G HbQ ?b }MBi2 iQiH KbbX b  +QMpQHmiBQM Q7 M #bQHmi2Hv +QMiBMmQmb K2bm`2 A rBi?  HQ+HHv }MBi2 K2bm`2 G ∗ ρ ∗ Rσ ∗ R3 - R1G ?b  HQ+HHv BMi2;`#H2 /2MbBiv u1G := H ∗ R3 ∗ a ,

U9Xj9V

r?2`2 H := G ∗ ρ ∗ Rσ ?b i?2 U}MBi2V iQiH Kbb H(∞) = 1 × n0 × (1 − σ)−1 . Ai `2KBMb iQ T`Qp2 U9XjRVX 6Q` i?Bb r2 TTHv i?2 F2v `2M2rH i?2Q`2K iQ i?2 /B`2+iHv _B2KMM BMi2;`#H2 7mM+iBQM a iQ Q#iBM ∞ 1 1 lim(R3 ∗ a)(t) = (1 − σ), a(t) dt = t↑∞ n0 E[S1 ] 0 n0 E[S1 ] r?2`2 r2 ?p2 Q#b2`p2/ i?i R3 Bb i?2 `2M2rH 7mM+iBQM bbQ+Bi2/ iQ i?2 T`QT2` M/ MQM@HiiB+2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F ∗n0 rBi? K2M n0 E[S1 ]X h?2 K2bm`2 H BM U9Xj9V ?pBM; }MBi2 iQiH Kbb 2[mH iQ 1 × n0 × (1 − σ)−1 - Bi 7QHHQrb i?i- #v /QKBMi2/ +QMp2`;2M+2lim u1G (t) = lim(H ∗ R3 ∗ a)(t) = H(∞) lim(R3 ∗ a)(t) t↑∞

t↑∞

t↑∞

1 1 (1 − σ) = . = H(∞) n0 E[S1 ] E[S1 ]  *Q`QHH`v 9XjXd UR9 V amTTQb2 i?i i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 i?2 `2M2rH T`Q+2bb Bb bT`2/@Qmi rBi? }MBi2 K2M E[S1 ]X G2i R #2 i?2 `2M2rH 7mM+iBQM Q7 i?2 mM/2Hv2/ `2M2rH T`Q+2bb- M/ H2i RG = G ∗ R #2 i?2 `2M2rH 7mM+iBQM Q7 i?2 /2Hv2/ `2M2rH T`Q+2bb rBi? }MBi2 /2Hv rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM GX G2i h : R+ → R #2  MQM@M2;iBp2 #QmM/2/ BMi2;`#H2 7mM+iBQM bm+? i?i limt↑∞ h(t) = 0X h?2M & &  ∞ & & 1 & g(x) dx&& = 0 . U9Xj8V lim sup &RG ∗ g(t) − t↑∞ 0≤g≤h E[S1 ] 0 S`QQ7X aBM+2 R1G ?b }MBi2 Kbb-



lim sup |(R1G ∗ g)(t)| ≤ lim R1G ∗ h(t) = lim t↑∞ 0≤g≤h

t↑∞

t↑∞

R+

h(t − s)1{s≤t} R1G (ds) = 0 ,

#v /QKBMi2/ +QMp2`;2M+2X Ai i?2`27Q`2 `2KBMb iQ b?Qr i?i & t &  ∞ & & 1 lim sup && u1G (t − x)g(x) dx − g(x) dx&& = 0 . t↑∞ 0≤g≤h E[S1 ] 0 0 R9

(`Db- LmKK2HBM M/ hr22/B2- RNd3)X

9X9X _1:1L1_hAo1 S_P*1aa1a "mi

R8R

& t &  ∞ & & 1 sup && u1G (t − x)g(x) dx − g(x) dx&& E[S1 ] 0 0≤g≤h 0 &  ∞& & 1 && & ≤ &u1G (t − x) − E[S1 ] & h(x) dx . 0

h?2 7mM+iBQM |u1G (t) − E[S1 1 ] | i2M/b iQ 0 b t ↑ ∞- M/ i?2`27Q`2 i?2`2 2tBbib  C < ∞ bm+? i?i 7Q` bm{+B2MiHv H`;2 t- bv t ≥ T - Bi Bb #QmM/2/ #v CX q`Bi2 7Q` t ≥ T - rBi? u1G (x) = 0 B7 x < 0&  ∞& & & &u1G (t − x) − 1 & h(x) dx & & E[S ] 1 0 & &  t−T &  ∞& & & & & &u1G (t − x) − 1 & h(x) dx + &u1G (t − x) − 1 & h(x) dx = & & & E[S1 ] E[S1 ] & 0 t−T & &  t−T &  T& & & & & &u1G (t − x) − 1 & h(x) dx + &u1G (x) − 1 & h(t − x) dx . = & & & E[S1 ] E[S1 ] & 0 0 "v /QKBMi2/ +QMp2`;2M+2- #Qi? BMi2;`Hb BM i?2 & `B;?i@?M/ bB/2 &i2M/ iQ 0 b & & t ↑ ∞X h?2 }`bi QM2 #2+mb2 i?2 7mM+iBQM x → &u1G (t − x) − E[S1 1 ] & 1{x≤t−T } h(x) Bb #QmM/2/ #v C iBK2b i?2 BMi2;`#H2 7mM+iBQM h M/ i2M/b iQ 0 b t → ∞X h?2 & & & & b2+QM/ #2+mb2 QM [0, T ] i?2 7mM+iBQM x → &u1G (x) − E[S1 1 ] & Bb BMi2;`#H2 U`2+HH i?i R1G Bb  `2M2rH 7mM+iBQM M/ i?2`27Q`2 HQ+HHv }MBi2V M/ limt↑∞ h(t − x) = 0X  h?2 bBimiBQM r?2`2 2t+i +QmTHBM; Q++m`b Bb MQi 2t+2TiBQMHX AM 7+i,R8 h?2Q`2K 9XjX3  MQM@HiiB+2 `2M2rH T`Q+2bb rBi? E[S1 ] < ∞ /KBib 2t+i +Qm@ THBM; B7 M/ QMHv B7 F Bb bT`2/@QmiX

9X9

_2;2M2`iBp2 S`Q+2bb2b

G2i (E, E) #2  K2bm`#H2 bT+2X .2}MBiBQM 9X9XR G2i {X(t)}t≥0 #2  K2bm`#H2 E@pHm2/ biQ+?biB+ T`Q+2bb M/ H2i {Tn }n≥0 #2  T`QT2` `2+m``2Mi `2M2rH T`Q+2bb- TQbbB#Hv /2Hv2/ U`2+HH- ?Qr2p2`i?i i?2 BMBiBH /2Hv T0 Bb Hrvb bbmK2/ }MBi2VX h?2 T`Q+2bb {X(t)}t≥0 Bb bB/ iQ #2 `2;2M2`iBp2 rBi? `2bT2+i iQ {Tn }n≥0 B7 7Q` HH n ≥ 0UV i?2 /Bbi`B#miBQM Q7 i?2 TQbi@Tn T`Q+2bb Sn+1 , Sn+2 , . . . , {X(t + Tn )}t≥0 Bb BM@ /2T2M/2Mi Q7 n ≥ 0- M/ U#V i?2 TQbi@Tn T`Q+2bb Bb BM/2T2M/2Mi Q7 T0 , . . . , Tn X h?2 iBK2b Tn `2 +HH2/ `2;2M2`iBQM iBK2b Q7 i?2 `2;2M2`iBp2 T`Q+2bbX R8

a22 (bKmbb2M- RN3d)- *?Ti2` oA- h?K kXjX

R8k

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

1tKTH2 9X9Xk, *QMiBMmQmb@iBK2 J`FQp *?BMbX G2i {X(t)}t≥0 #2  `2@ +m``2Mi +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J`FQp +?BM iFBM; Bib pHm2b BM i?2 bii2 bT+2 E = NX amTTQb2 i?i Bi bi`ib 7`QK bii2 0 i iBK2 t = 0X "v i?2 bi`QM; J`FQp T`QT2`iv- {X(t)}t≥0 Bb `2;2M2`iBp2 rBi? `2bT2+i iQ i?2 b2[m2M+2 {Tn }n≥0 r?2`2 Tn Bb i?2 n@i? iBK2 Q7 pBbBi iQ bii2 0 Q7 i?2 +?BMX _2;2M2`iBp2 T`Q+2bb2b `2 i?2 KBM bQm`+2b Q7 `2M2rH 2[miBQMbX h?2Q`2K 9X9Xj G2i {X(t)}t≥0 M/ {Tn }n≥0 #2 b BM .2}MBiBQM 9X9XR 2t+2Ti 7Q` i?2 //BiBQMH bbmKTiBQM T0 ≡ 0 UmM/2Hv2/ `2M2rH T`Q+2bbV M/ H2i h : E → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX h?2 7mM+iBQM f : R+ → R /2}M2/ #v f (t) := E [h(X(t))] biBb}2b i?2 `2M2rH 2[miBQM rBi? /i g(t) = E h(X(t))1{t t)X "v h?2Q`2K 9XkXRy- T`i U#V- Bi Bb i?2`27Q`2 2MQm;? iQ T`Qp2 i?i Bi Bb _B2KMM BMi2;`#H2 QM }MBi2 BMi2`pHbX AM 7+i- b r2 MQr b?Qr- Bi Bb HKQbi

R89

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

2p2`vr?2`2 +QMiBMmQmb Ur?B+? BKTHB2b i?i Bi Bb _B2KMM BMi2;`#H2 QM }MBi2 BM@ i2`pHb bBM+2 Bi Bb #QmM/2/VX aBM+2 |K(t, A) − K(s, A)| ≤ P ({X(t) ∈ A, t < S1 } # {X(s) ∈ A, s < S1 }), Bi bm{+2b iQ b?Qr i?i i?2 Hii2` i2M/b iQ 0 b s → t ∈ / D- r?2`2 D Bb i?2 mMBQM Q7 i?2 b2i Q7 }t2/ /Bb+QMiBMmBiB2b Q7 i?2 T`Q+2bb UbbmK2/ +QmMi#H2V M/ Q7 i?2 U+QmMi#H2V b2i Q7 /Bb+QMiBMmBiB2b Q7 F X q2 i?2`27Q`2 ?p2 iQ T`Qp2 i?i P ({X(t) ∈ A, t < S1 } ∩ {X(s) ∈ A, s < S1 }) → 0 M/ P ({X(s) ∈ A, s < S1 } ∩ {X(t) ∈ A, t < S1 }) → 0X q2 iF2 +`2 Q7 i?2 }`bi QM2- i?2 Qi?2` QM2 #2BM; bBKBH`X q2 ?p2 P ({X(t) ∈ A, t < S1 } ∩ {X(s) ∈ A, s < S1 }) = P ({X(t) ∈ A, t < S1 , S1 ≤ s} ∪ {X(t) ∈ A, t < S1 , X(s) ∈ / A}) ≤ P ({X(t) ∈ A, t < S1 , S1 ≤ s}) + P ({X(t) ∈ A, t < S1 , X(s) ∈ / A}). h?2 }`bi i2`K QM i?2 `B;?i@?M/ bB/2 Bb #QmM/2/ #Qp2 #v F (s∨t)−F (s∧t)- r?B+? i2M/b iQ 0 bBM+2 t ∈ / D Bb  TQBMi Q7 +QMiBMmBiv Q7 F X h?2 b2+QM/ i2`K Bb #QmM/2/ #Qp2 #v P ({X(t) ∈ A, X(s) ∈ / A})X aBM+2 t Bb MQi  }t2/ /Bb+QMiBMmBiv TQBMi Q7 i?2 T`Q+2bb- X(s) → X(t)X h?2`27Q`2- bBM+2 A Bb QT2M- lims→t 1A (X(t)1Ac (X(s)) = 0r?B+? ;Bp2b lims→t P ({X(t) ∈ A, X(s) ∈ / A}) = 0X  _2K`F 9X9Xd h?2 ?vTQi?2bBb i?i {X(t)}t≥0 ?b }t2/ /Bb+QMiBMmBiB2b 7Q`KBM; i KQbi  +QmMi#H2 b2i Bb Hrvb biBb}2/ B7 i?2 i`D2+iQ`B2b Q7 i?Bb T`Q+2bb `2 HKQbi bm`2Hv BM D([0, ∞))- i?2 b2i Q7 7mM+iBQMb i?i `2 `B;?i@+QMiBMmQmb QM [0, ∞) M/ ?p2 H27i@?M/ HBKBib QM (0, ∞)X AM i?2 7QHHQrBM; 2tKTH2b- /B`2+i _B2KMM BMi2;`#BHBiv Q7 i?2 /i g Bb  +QM@ b2[m2M+2 Q7 h?2Q`2K 9X9XeX 1tKTH2 9X9X3, h?2 6mM/K2MiH _2HB#BHBiv h?2Q`2KX *QMbB/2` i?2 bBimiBQM Q7  `2M2rH T`Q+2bb 7Q` r?B+? i?2 BMi2``2M2rH b2[m2M+2 {Sn }n≥1 Bb Q7 i?2 7Q`K Sn = Un +Vn - r?2`2 {Un }n≥1 M/ {Vn }n≥1 `2 BM/2T2M/2Mi BB/ b2[m2M+2bM/ /2}M2  1 B7 t ∈ (Tn , Tn + Un ], X(t) = 0 B7 t ∈ (Tn + Un , Tn+1 + Un + Vn ]. h?2 BMi2`T`2iiBQM Q7 i?2 T`Q+2bb {X(t)}t≥0 BM i2`Kb Q7 `2HB#BHBiv Bb i?i X(t) = 1 r?2M i iBK2 t  ;Bp2M K+?BM2 Bb +m``2MiHv BM rQ`FBM; +QM/BiBQM- r?2`2b B7 X(t) = 0 Bi Bb BM `2TB`X G2i a M/ b #2 i?2 `2bT2+iBp2 K2Mb Q7 U1 M/ V1 X AM i?2 `2+m``2Mi MQM@HiiB+2 +b2- r2 ?p2 i?i lim P (X(t) = 1) =

t→∞

a . a+b

9X9X _1:1L1_hAo1 S_P*1aa1a

R88

Ai bm{+2b iQ TTHv aKBi?Ƕb `2;2M2`iBp2 7Q`KmH U9X9yV rBi? A = {1}- M/ iQ ∞ Q#b2`p2 i?i E[S1 ] = E[U1 ] + E[V1 ] = a + b M/ 0 P (X(s) = 1, s < S1 ) ds = ∞ a P (U1 > s) ds = E[U1 ] = aX AM  `2HB#BHBiv +QMi2ti- a+b `2T`2b2Mib i?2 pBH@ 0 #BHBiv Q7  ;Bp2M K+?BM2 rBi? K2M HB72iBK2 a M/ K2M `2TB` iBK2 bX 1tKTH2 9X9XN, 6Q`r`/ M/ "+Fr`/ _2+m``2M+2 hBK2bX G2i {Tn }n≥0 #2 M mM/2Hv2/ UT0 = 0V `2M2rH T`Q+2bbX *H2`Hv i?2 7Q`r`/ M/ #+Fr`/ `2+m``2M+2 iBK2b `2 `2;2M2`iBp2 rBi? `2bT2+i iQ i?2 `2M2rH T`Q+2bb {Tn }n≥0 X 6`QK aKBi?Ƕb `2;2M2`iBp2 7Q`KmH U9X9yV- BM i?2 MQM@HiiB+2 +b2  1 ∞ lim P (A(t) > x) = P (A(s) > x, s < S1 ) ds . t→∞ m 0 aBM+2 P (A(s) > x, s < S1 ) = P (S1 > s + x) = 1 − F (s + x)- r2 ?p2  ∞  ∞ P (A(s) > x, s < S1 ) ds = (1 − F (s + x)) ds 0 0 ∞ = (1 − F (s)) ds , x

M/ i?2`27Q`2 lim P (A(t) > x) =

t→∞

1 m





(1 − F (s)) ds .

U9X9kV

x

aBKBH` `;mK2Mib vB2H/ 7Q` i?2 #+Fr`/ `2+m``2M+2 iBK2  1 ∞ (1 − F (s))ds. lim P (B(t) > y) = t→∞ m y

U9X9jV

Uh?Bb iBK2- i?2 /i 7mM+iBQM Bb (1 − F (t))1{t>y} XV PM2 +M T`Qp2 /B`2+iHv i?2 /B`2+i _B2KMM BMi2;`#BHBiv Q7 i?2 /i 7mM+iBQMb Q7 i?Bb 2tKTH2- Q` mb2 h?2Q`2K 9X9XeX 1tKTH2 9X9XRy, h?2 "mb S`/QtX h?2 bmK A(t) + B(t) Bb i?2 BMi2`@2p2Mi BMi2`pH `QmM/ iBK2 tX AMi2`T`2iBM; t b i?2 iBK2 i r?B+? vQm ``Bp2 i  #mb biQT- M/ i?2 b2[m2M+2 {Tn }n≥1 b i?2 b2[m2M+2 Q7 iBK2b i r?B+? #mb2b ``Bp2 i UM/ BKK2/Bi2Hv /2T`i 7`QKV i?2 #mb biQT- A(t) Bb vQm` rBiBM; iBK2X A7 t Bb H`;2 2MQm;?- QM2 +M- BM pB2r Q7 U9X9kV- bbmK2 i?i A(t) Bb /Bbi`B#mi2/ b  `M/QK p`B#H2 A rBi? i?2 /Bbi`B#miBQM  1 ∞ P (A > x) = (1 − F (s)) ds . U9X99V m x aBKBH`Hv- i?2 iBK2 B(t) #v r?B+? vQm KBbb2/ i?2 T`2pBQmb #mb Bb TT`QtBKi2Hvr?2M t Bb H`;2- /Bbi`B#mi2/ b  `M/QK p`B#H2 B rBi? i?2 bK2 /Bbi`B#miBQM b AX h?2 #mb T`/Qt +M #2 bii2/ BM b2p2`H rvbX PM2 Q7 i?2K Bb, i?2 K2M iBK2 BMi2`pH #2ir22M i?2 #mb vQm KBbb2/ M/ i?2 #mb vQm rBHH +i+? Bb bvKTiQiB+HHv b t → ∞ 2[mH iQ E[A + B] = 2E[A]- M/ Bb BM ;2M2`H /Bz2`2Mi 7`QK i?2 K2M Q7 i?2 ivTB+H BMi2`pH E[S1 ] #2ir22M irQ bm++2bbBp2 #mb2bX

R8e

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

aKBi?Ƕb `2;2M2`iBp2 i?2Q`2K rBHH MQr #2 `2pBbBi2/ BM i?2 bT`2/@Qmi +b2X *QM@ bB/2`  K2bm`#H2 biQ+?biB+ T`Q+2bb {X(t)}t∈R+ rBi? pHm2b BM bQK2 K2bm`#H2 bT+2 (E, E) M/ `2;2M2`iBp2 rBi? `2bT2+i iQ  `2M2rH T`Q+2bb rBi? bT`2/@Qmi /Bbi`B#miBQM F Q7 }MBi2 K2MX h?2 7mM+iBQM h(t) = P (S1 > t) ?b i?2 T`QT2`iB2b `2@ [mB`2/ BM *Q`QHH`v 9XjXdX 6Q` Mv A ∈ E- i?2 7mM+iBQM gA : t → P (X(t) ∈ A, S1 > t) Bb #QmM/2/ #v h- M/ i?2`27Q`2 & &  ∞ & & 1 & gA (x) dx&& = 0 , lim sup &RG ∗ gA (t) − t↑∞ A∈E E[S1 ] 0 i?i Bb iQ bv lim sup |P (X(t) ∈ A) − Q∗ (A)| = 0 , t↑∞ A∈E

r?2`2 Q∗ (A) :=

1 E0 E[S1 ]



S1

1A (X(s)) ds ,

0

r?2`2 E0 /2MQi2b 2tT2+iiBQM rBi? `2bT2+i iQ i?2 mM/2Hv2/ `2;2M2`iBp2 T`Q+2bbX h?2`27Q`2- b t → ∞- i?2 /Bbi`B#miBQM Q7 X(t) +QMp2`;2b BM p`BiBQM iQ Q∗ X G2i MQr {X(t)}t∈R #2  biQ+?biB+ T`Q+2bb iFBM; Bib pHm2b BM  K2i`B+ bT+2 E- ?pBM; `B;?i@+QMiBMmQmb Ti?b M/ #2BM; `2;2M2`iBp2 `2HiBp2 iQ i?2 UTQbbB#Hv /2Hv2/V `2M2rH b2[m2M+2 {Tn }n≥0 rBi? MQM@HiiB+2 M/ }MBi2 K2M BMi2`@2p2Mi /Bbi`B#miBQM μX G2i P0 M/ E0 bvK#QHBx2 `2bT2+iBp2Hv i?2 T`Q##BHBiv M/ i?2 2t@ T2+iiBQM +Q``2bTQM/BM; iQ i?2 mM/2Hv2/ p2`bBQM Q7 i?2 `2M2rH b2[m2M+2X h?2M  S1

1 1A (X(s)) ds P ∗ (A) := E0 μ 0 /2}M2b  T`Q##BHBiv K2bm`2 QM (E, B(E))X h?2Q`2K 9X9XRR lM/2` i?2 #Qp2 +QM/BiBQMb- X(t) +QMp2`;2b BM /Bbi`B#miBQM iQ P ∗ b t ↑ ∞X S`QQ7X q2 Kmbi b?Qr i?i 7Q` HH #QmM/2/ Ubv- #v 1V +QMiBMmQmb 7mM+iBQMb h : E → R

 S1 1 h(X(s)) ds . U9X98V lim E [h(X(t))] = E0 t↑∞ μ 0 "v i?2 mbmH `2M2rH `;mK2Mi U+QM/BiBQMBM; QM T0 - r?Qb2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Bb /2MQi2/ #v FT0 V f (t − s) dFT0 (s) , () E [h(X(t))] = E h(X(t))1{tSh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

S`QQ7X h?2 7+i i?i R∗g Bb r2HH /2}M2/- HQ+HHv #QmM/2/- M/ biBb}2b i?2 `2M2rH 2[miBQM Bb T`Qp2/ BM i?2 bK2 rv b BM i?2 mMBp`Bi2 +b2X G2i MQr f M/ f #2 irQ p2+iQ`b Q7 HQ+HHv #QmM/2/ 7mM+iBQMb biBb7vBM; i?2 `2M2rH 2[miBQM- M/ H2i h := f − fX h?2M h = R ∗ h- M/ Bi2`iBp2Hv- h = F ∗n ∗ h 7Q` HH n ≥ 1- bQ i?i |h| ≤ F ∗n ∗ |h|X G2i supi |hi (t)| ≤ M (a) < ∞ QM [0, a]- bv M (a) = 1- rBi?Qmi HQbb Q7 ;2M2`HBiv- bQ i?i |h| ≤ F ∗n ∗ 1 QM [0, a] M/ i?2`27Q`2- QM [0, a] Fij∗n (t) = Pi (S1 + · · · + Sn ≤ t) , |hi (t)| ≤ j∈E

 [mMiBiv i?i i2M/b iQ 0 b n ↑ ∞ bBM+2 T∞ = ∞ mM/2` i?2 T`2pBHBM; +QM/BiBQMb Ui?2 2K#2//2/ +?BM Bb B``2/m+B#H2 `2+m``2MiVX h?2`27Q`2 h ≡ 0 QM HH [0, a] Ua ∈ R+ V M/ i?2`27Q`2 QM R+ X  Ai 7QHHQrb 7`QK U9X9dV i?i B7 gi Bb /B`2+iHv _B2KMM BMi2;`#H2  νj ∞ gj (s) ds , Rij ∗ gj (t) → μ 0 M/ 7`QK i?Bb- BM i?2 +b2 r?2`2 E Bb }MBi2fi (t) =



Rij ∗ gj (t) →

j∈E

1 νj μ j∈E





gj (s) ds . 0

h?2 +b2 r?2M E Bb BM}MBi2 `2[mB`2b 7m`i?2` +QM/BiBQMb M/ rBHH MQi #2 i`2i2/ ?2`2XR3 AKT`QT2` JmHiBp`Bi2 _2M2rH 1[miBQMb h?2 #Qp2 `2bmHib +QM+2`M i?2 +b2 r?2`2 Q := {||Fij ||}i,j∈E Bb  biQ+?biB+ Ki`Bti?i Bb- i?2 i`MbBiBQM Ki`Bt Q7  ?QKQ;2M2Qmb J`FQp +?BM UMK2Hv- i?2 i`M@ bBiBQM Ki`Bt P Q7 i?2 2K#2//2/ J`FQp +?BMVX >Qr2p2` i?2 `2M2rH 2[miBQMb U9X9eV KF2 b2Mb2 2p2M B7 i?Bb Bb MQi i?2 +b2X q2 MQr ;Bp2 `2bmHibRN Q7 i?2 bK2 FBM/ b i?2 QM2b BM i?2 /272+iBp2 Q` 2t+2bbBp2 mMBp`Bi2 `2M2rH 7mM+iBQMb r?2M i?2 bii2 bT+2 Bb }MBi2X h?2 Ki`Bt Q Bb MQ HQM;2`  biQ+?biB+ Ki`Bt- #mi biBHH bbmK2/ B``2/m+B#H2X .2}M2 7Q` bQK2 `2H β i?2 Ki`Bt A := {aij }i,j∈E  ∞ aij := eβt dFij (t) . 0

bbmK2 i?i β +M #2 +?Qb2M bm+? i?i A ?b bT2+i`H `/Bmb 1X AM T`iB+mH`i?2`2 2tBbib irQ TQbBiBp2 p2+iQ`b ν M/ h bm+? i?i ν T A = ν M/ Ah = h . h?2 2tBbi2M+2 Q7 ν M/ h Bb 2Mbm`2/ #v i?2 S2``QMĜ6`ƺ#2MBmb i?2Q`2KX h?2 7QHHQr@ BM; 7+ib `2 2bvX 6B`bi i?2 Ki`Bt R3 RN

a22- 7Q` BMbiM+2- (ÎBMH`- RNd8)X (bKmbb2M M/ >2`BM;- RNdd)X

9XdX 1s1_*Aa1a

ReR  := Q



hj aij hi

 i,j∈E

Bb M UB``2/m+B#H2V biQ+?biB+ Ki`Bt /KBiiBM; i?2 BMp`BMi K2bm`2 ν ;Bp2M #v νi = νi hi X G2i hj Fij (t) := hi



t

eβs dFij (s) . 0

 = {||Fij ||}i,j∈E Bb B``2/m+B#H2 M/ h?Bb /2}M2b  b2KB@J`FQp F2`M2H 7Q` r?B+? Q `2+m``2MiX .2}MBM; fi (t) := eβt fi (t)/hi M/ gi := eβt gi (t)/hi , r2 b22- MHQ;QmbHv iQ i?2 mMBp`Bi2 +b2- i?i f = g + F ∗ f. h?2`27Q`2- B7 F Bb MQM@HiiB+2 M/ i?2 gi Ƕb `2 HQ+HHv #QmM/2/ M/ BMi2;`#H2 ∞ 1 νj gj (s) ds , lim fi (t) = t↑∞ μ  j∈E 0 i?i Bb hi lim e fi (t) = βt

t↑∞

j∈E

k,j∈E

νj

ν k hj

∞ 0∞ 0

eβs gj (s) ds seβs dFkj (s)

.

h?2 2tBbi2M+2 Q7 β Bb ;m`Mi22/- 7Q` BMbiM+2-ky r?2M i?2 bT2+i`H `/Bmb Q7 {||Fij ||}i,j∈E Bb bi`B+iHv H2bb i?i 1X

9Xd

1t2`+Bb2b

1t2`+Bb2 9XdXRX h?2Q`2K 9XRXR i`m2 BM i?2 mM/2Hv2/ +b2 S`Qp2 i?i b h?2Q`2K 9XRXR Bb i`m2 BM i?2 mM/2Hv2/ +b2- Bi Bb i?2M i`m2 BM i?2 /2Hv2/ +b2 U`2+HH, rBi? }MBi2 /2HvVX 1t2`+Bb2 9XdXkX h?2 bvKTiQiB+ +QmMiBM; `i2 S`Qp2 U9X9V- i?i BbN ([0, t]) 1 = - S@XbX lim t→∞ t E[S1 ]

1t2`+Bb2 9XdXjX _B;?i@+QMiBMmBiv Q7 i?2 `2M2rH 7mM+iBQM a?Qr i?i i?2 `2M2rH 7mM+iBQM R Bb `B;?i@+QMiBMmQmbX ky

a22 (bKmbb2M- RN3d)- S`Q#H2K kXj- +?TX sX

Rek

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

1t2`+Bb2 9XdX9X #Qmi i?2 /Bbi`B#miBQM Q7 N ([0, t]) AM i?2 mM/2Hv2/ +b2- +QKTmi2 P (N ([0, t]) = n) 7Q` n ≥ 1 BM i2`Kb i?2 βNQ7 +QM@ ([0,t]) pQHmiBQM Bi2`i2b Q7 i?2 BMi2`@`2M2rH /Bbi`B#miBQM M/ b?Qr i?i E e X} . :Bp2 M 2tT`2bbBQM Q7 i?2 +mKmHiBp2 /Bbi`B#miBQM Q7 T BM i2`Kb Q7 G- i?2 `2M2rH 7mM+iBQM R M/ i?2 pQB/M+2 7mM+iBQM v U/2}M2/ BM a2+iBQM RXjVX 1t2`+Bb2 9XdXeX 6Q`r`/ `2+m``2M+2 iBK2 q?i Bb i?2 HBKBi /Bbi`B#miBQM Q7 i?2 7Q`r`/ `2+m``2M+2 iBK2 Q7  `2M2rH T`Q+2bb UTQbbB#Hv /2Hv2/V r?2M S1 Bb /2i2`KBMBbiB+- 2[mH iQ a\ 1t2`+Bb2 9XdXdX qH/ǰb H2KK 7Q` `2M2rH T`Q+2bb2b G2i {Sn }n≥1 #2  `2M2rH b2[m2M+2 M/ H2i {N ([0, t])}t≥0 #2 i?2 +QmMiBM; T`Q+2bb Q7 i?2 +Q``2bTQM/BM; UTQbbB#Hv /2Hv2/V `2M2rH T`Q+2bbX bbmK2 i?i E [S1 ] < ∞X a?Qr i?i 7Q` HH t ≥ 0E[S1 + · · · + SN ([0,t]) ] = E[S1 ]E[N ([0, t])].

1t2`+Bb2 9XdX3X 1tT2+i2/ HB72iBK2 AM 1tKTH2 9XRX3- +QKTmi2 i?2 2tT2+iiBQM Q7 i?2 HB72iBK2 L BM i?2 i`MbB2Mi +b2X 1t2`+Bb2 9XdXNX a72iv HQ/ h?Bb 2t2`+Bb2 `272`b iQ 1tKTH2 kXRX9X S`Qp2 i?i BM i?2 +b2 Q7 TQbBiBp2 b72iv HQ/BM;- Φ(∞) = 1X 1t2`+Bb2 9XdXRyX h?2 #mb T`/Qt a22 1tKTH2 9X9XRy 7Q` i?2 +QMi2tiX *QMbB/2` M mM/2Hv2/ `2M2rH T`Q+2bb rBi? }MBi2 K2M BMi2`@`2M2rH iBK2 E[S1 ]X 6Q` t ≥ 0- +QMbB/2` i?2 BMi2`pH #2ir22M i?2 Hbi `2M2rH iBK2 #27Q`2 t M/ i?2 }`bi `2M2rH iBK2 7i2` tX a?Qr i?i BM i?2 SQBbbQM +b2 Ui?2 BMi2```BpH /Bbi`B#miBQM Bb 2tTQM2MiBHV i?2 K2M H2M;i? Q7 i?Bb BMi2`pH Bb bvKTiQiB+HHv 2E[S1 ]X a?Qr i?i Bi Bb 2[mH iQ E[S1 ] B7 M/ QMHv S1 Bb  +QMbiMiX 1t2`+Bb2 9XdXRRX 6B`bi 2p2Mi 7i2`  7Bt2/ iBK2 _272` iQ .2}MBiBQM 9XRXRe 7Q` i?2 MQiiBQMX S`Qp2 i?i S0 (t) Bb BM/2T2M/2Mi Q7 {Sn (t)}n≥1 M/ i?i i?2 Hii2` b2[m2M+2 ?b i?2 bK2 /Bbi`B#miBQM b {Sn }n≥1 X

9XdX 1s1_*Aa1a

Rej

1t2`+Bb2 9XdXRkX  HBKBi i?2Q`2K 7Q` +QMiBMmQmb@iBK2 ?K+b G2i {X(t)}t≥0 - #2  TQbBiBp2 `2+m``2Mi +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J`FQp +?BM iFBM; Bib pHm2b BM i?2 bii2 bT+2 E = NX G2i P0 M/ E0 /2MQi2 `2bT2+iBp2Hv T`Q##BHBiv M/ 2tT2+iiBQM ;Bp2M X(0) = 0X G2i T0 #2 i?2 `2im`M iBK2 iQ 0 UT0 := inf{t > 0 ; X(t) = 0, X(t−) = 0}VX _2+HH i?i BM i?2 TQbBiBp2 `2+m``2Mi +b2E0 [T0 ] < ∞X a?Qr i?i   T E0 0 0 1{X(s)=i} ds lim P (X(t) = i) = . t↑∞ E0 [T0 ]

1t2`+Bb2 9XdXRjX GQiFėoQHi2`` bvKTiQiB+b AM i?2 GQiFĜoQHi2`` KQ/2H- ;Bp2 i?2 /2iBHb +QM+2`MBM; i?2 bvKTiQiB+b Q7 i?2 #B`i? `i2 f BM i?2 +b2b F (∞) = 1 M/ F (∞) > 1X q?i +M vQm bv #Qmi i?2 /272+iBp2 +b2 F (∞) < 1\ 1t2`+Bb2 9XdXR9X "+Fr`/ M/ 7Q`r`/ `2+m``2M+2 T`Q+2bb2b _272` iQ .2}MBiBQM 9X9XNX *QKTmi2 limt→∞ P (A(t) > x, B(t) > y) 7Q` x, y ≥ 0X 1t2`+Bb2 9XdXR8X bvKTiQiB+ p`BM+2 Q7 N ((0, t]) 6Q`  T`QT2` `2M2rH T`Q+2bb rBi? M BMi2```BpH /Bbi`B#miBQM Q7 }MBi2 p`BM+2b?Qr i?i o`N ((0, t]) o`S1 lim = . t↑∞ t E[S1 ] 1t2`+Bb2 9XdXReX h?2 ;2 `2TH+2K2Mi TQHB+v q2 BMi2`T`2i i?2 `M/QK p`B#H2b S1 , S2 , . . . b i?2 HB72iBK2b Q7 K+?BM2b bm++2b@ bBp2Hv Tmi BMiQ b2`pB+2-  M2r K+?BM2 BKK2/Bi2Hv `2TH+BM;  7BH2/ QM2X Ai rBHH #2 bbmK2/ i?i E[S1 ] < ∞- M/ i?2`27Q`2- #v U9X9V- E[S1 1 ] Bb i?2 bvKTiQiB+ 7BH@ m`2 `i2 T2` mMBi iBK2X AM bQK2 bBimiBQMb- i?2 BM+QMp2MB2M+2 +mb2/ #v  7BHm`2 Bb iQQ BKTQ`iMi- M/ i?2 7BHm`2 `i2 Kmbi #2 +QMi`QHH2/X h?2 ;2 `2TH+2K2Mi TQHB+v bm;;2bib i?i M 2M;BM2 b?QmH/ #2 `2TH+2/ i 7BHm`2 iBK2 Q` i  }t2/ iBK2 T > 0r?B+?2p2` Q++m`b }`biX q?i Bb i?2 bvKTiQiB+ 7BHm`2 `i2\ U `2TH+2K2Mi Bb MQi +QMbB/2`2/ b  7BHm`2XV 1t2`+Bb2 9XdXRdX MQi?2` KBMi2MM+2 TQHB+v  ;Bp2M K+?BM2 +M #2 BM 2Bi?2` QM2 Q7 i?`22 bii2b, : U;QQ/V- J UBM KBMi2@ MM+2V- Q` _ UBM `2TB`VX Aib bm++2bbBp2 T2`BQ/b r?2`2 Bi Bb BM bii2 : U`2bTX- J- _V 7Q`K M BM/2T2M/2Mi M/ B/2MiB+HHv /Bbi`B#mi2/ b2[m2M+2 {Sn }n≥0 U`2bTX- {Un }n≥0 {Vn }n≥0 V rBi? }MBi2 K2MX HH i?2b2 b2[m2M+2b `2 bbmK2/ KmimHHv BM/2T2M/2MiX h?2 KBMi2MM+2 TQHB+v mb2b  MmK#2` T > 0X A7 i?2 K+?BM2 ?b ;2 T M/ ?b MQi 7BH2/- Bi ;Q2b iQ bii2 JX A7 Bi 7BHb #27Q`2 Bi ?b `2+?2/ ;2 T - Bi 2Mi2`b bii2 _X 6`QK bii2b J M/ _- i?2 M2ti bii2 Bb :X 6BM/ i?2 bi2/v bii2 T`Q##BHBiv i?i i?2 K+?BM2 Bb QT2`iBQMHX ULQi2 i?i dz;QQ/Ǵ /Q2b MQi K2M dzQT2`iBQMHǴX h?2 K+?BM2 +M #2 dz;QQ/Ǵ #mi- /m2 iQ i?2 QT2`iBQMb TQHB+v- BM KBMi2MM+2- M/

Re9

*>Sh1_ 9X _1L1qG L. _1:1L1_hAo1 S_P*1aa1a

i?2`27Q`2 MQi QT2`iBQMHX >Qr2p2`- 7i2`  T2`BQ/ Q7 KBMi2MM+2 Q` Q7 `2TB`- r2 +QMbB/2` i?i i?2 K+?BM2 bi`ib M2r- M/ 2Mi2`b  : T2`BQ/XV 1t2`+Bb2 9XdXR3X  irQ bii2 b2KB@J`FQp T`Q+2bb G2i   α 1−α P= , 1−β β r?2`2 α, β ∈ (0, 1)- M/ H2i G1 M/ G2 #2 irQ T`QT2` +mKmHiBp2 /Bbi`B#miBQM 7mM+@ iBQMbX G2i {X(t)}t≥0 #2 i?2 biQ+?biB+ T`Q+2bb 2pQHpBM; b 7QHHQrbX q?2M BM bii2 i Ui = 1, 2V Bi bivb i?2`2 7Q`  `M/QK iBK2 rBi? /Bbi`B#miBQM Gi Ui = 1, 2V 7i2` r?B+? Bi KQp2b iQ bii2 j UTQbbB#Hv i?2 bK2 bii2V rBi? i?2 T`Q##BHBiv pij Ui?2 Ui, jV@2Mi`v Q7 PVX h?2 bm++2bbBp2 bQDQm`M iBK2b UBM 2Bi?2` bii2V `2 BM/2T2M/2Mi ;Bp2M i?2 FMQrH2/;2 Q7 i?2 bii2 i?2 T`Q+2bb Bb BM Ui?2 `2/2` rBHH +H`B7v i?Bb BK@ T`2+Bb2 b2Mi2M+2VX q?i Bb i?2 bvKTiQiB+ /Bbi`B#miBQM Q7 i?2 T`Q+2bb- i?i Bb- r?i Bb limt↑∞ P (X(t) = 1)\

*?Ti2` 8 SQBMi S`Q+2bb2b rBi?  aiQ+?biB+ AMi2MbBiv G2i {Ft }t∈R #2 bQK2 ?BbiQ`v Q7  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb N QM R- i?i Bb-  MQM@/2+`2bBM; 7KBHv Q7 σ@}2H/b bm+? i?i 7Q` HH t ∈ R M/ HH a ≤ b ≤ tN ((a, b]) Bb Ft @K2bm`#H2X A7 Bi ?QH/b i?i 7Q` HH t ∈ Rlim h↓0

1 E[N (t, t + h] | Ft ] = λ(t) , h

P @XbX,

()

7Q` bQK2 MQM@M2;iBp2 HQ+HHv BMi2;`#H2 Ft @/Ti2/ biQ+?biB+ T`Q+2bb {λ(t)}t∈R i?2 Hii2` Bb +HH2/  biQ+?biB+ Ft @BMi2MbBiv Q7 N X h?Bb /2}MBiBQM BMpQHpBM;  HBK@ BiBM; /2`BpiBp2@ivT2 T`Q+2/m`2 Bb /pMi;2QmbHv `2TH+2/ #v  ;HQ#H /2}MBiBQM BM i2`Kb Q7 K`iBM;H2b i?i MQi QMHv HHQrb QM2 iQ +H`B7v  72r Ki?2KiB+H Bbbm2b #mi- KQ`2 BKTQ`iMiHv- Bb i i?2 Q`B;BM Q7  M2r T`/B;K R BM i?2 bim/v Q7 biQ+?biB+ bvbi2Kb /`Bp2M #v TQBMi T`Q+2bb2b M/ `2p2Hb  T`Q/m+iBp2 MHQ;v rBi? i?2 i?2Q`v Q7 biQ+?biB+ bvbi2Kb /`Bp2M #v r?Bi2 MQBb2k i?i rBHH #2 2tTHQBi2/ BM *?Ti2`b Ry M/ RkX

8XR

h?2 aKQQi?BM; 6Q`KmHb

6Q`  SQBbbQM T`Q+2bb N QM i?2 `2H HBM2 rBi? i?2 HQ+HHv BMi2;`#H2 BMi2MbBiv 7mM+iBQM λ(t)- Bi ?QH/b i?i 7Q` HH BMi2`pHb [c, d] ⊂ R E N (c, d] | FcN =



d

λ(s) ds c

Q`- 2[mBpH2MiHv bBM+2 i?2 `B;?i@?M/ bB/2 Bb  /2i2`KBMBbiB+ [mMiBiv

E N (c, d] |

FcN







d

λ(s) ds |

=E

FcN

.

c

h?Bb KQiBpi2b i?2 7QHHQrBM; /2}MBiBQM Q7 biQ+?biB+ BMi2MbBivX R k

("`ûKm/- RNdk)X h?Bb rBHH #2 #`B2~v 2tTHBM2/ BM a2+iBQM 8X3X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_5

Re8

Ree *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu .2}MBiBQM 8XRXR G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R- H2i {Ft }t∈R #2  ?BbiQ`v Q7 N M/ H2i {λ(t)}t∈R #2  MQM@M2;iBp2 XbX HQ+HHv BMi2;`#H2 `2H@ pHm2/ Ft @T`Q;`2bbBp2Hv K2bm`#H2 biQ+?biB+ T`Q+2bbX A7 7Q` HH a ∈ R- HH BMi2`pHb (c, d] ⊂ (a, ∞) M/ HH BMi2;2`b n ≥ 1  (a) d∧Tn

E[N (c ∧ Tn(a) , d ∧ Tn(a) ]|Fc ] = E

(a)

λ(s)ds | Fc ,

U8XRV

c∧Tn (a)

(a)

b2[m2M+2 Q7 Ft @biQTTBM; iBK2b bm+? i?i Tn > r?2`2 {Tn }n≥1 Bb  MQM@/2+`2bBM;   (a) (a) a- limn↑∞ Tn = ∞ M/ E N (a, Tn ] < ∞- N Bb i?2M bB/ iQ /KBi i?2 biQ+?biB+ (P, Ft )@BMi2MbBiv {λ(t)}t∈R X h?2 /2}MBiBQM Q7 biQ+?biB+ BMi2MbBiv QT2Mb  +QMM2+iBQM rBi? i?2 i?2Q`v Q7 K`iBM;H2bX AM 7+i- .2}MBiBQM 8XRXR Q7 biQ+?biB+ BMi2MbBiv BKTHB2b i?i  t M (t) := N (0, t] − λ(s) ds 0

Bb  HQ+H (P, Ft )@K`iBM;H2 U1t2`+Bb2 8XNXRV- +HH2/ i?2 7mM/K2MiH UHQ+HV Ft @ K`iBM;H2 Q7 i?2 TQBMi T`Q+2bb N X q?2M i?2 +?QB+2 Q7 T`Q##BHBiv P Bb +H2` 7`QK i?2 +QMi2ti- QM2 bvb dzi?2 Ft @BMi2MbBivǴ BMbi2/ Q7 dzi?2 (P, Ft V@BMi2MbBivǴX _2K`F 8XRXk h?2 `2[mB`2K2Mi Q7 Ft @T`Q;`2bbBp2M2bb Q7 i?2 biQ+?biB+ BMi2MbBiv t ;m`Mi22b i?i i?2 BMi2;`i2/ BMi2MbBiv T`Q+2bb { 0 λ(s) ds}t∈R Bb K2bm`#H2 M/ Ft @/Ti2/ U1t2`+Bb2 8XNX9VX _2K`F 8XRXj q?2M +QMbB/2`BM; TQBMi T`Q+2bb2b QM i?2 TQbBiBp2 ?H7@HBM2- i?2 BMi2`p2MiBQM Q7 i?2 aǶb Bb bmT2`~mQmb- M/ Bi bm{+2b iQ `2[mB`2 i?i U8XRV ?QH/b 7Q` a = 0X >Qr2p2`- i?2 bHB;?iHv KQ`2 +QKTHB+i2/ /2}MBiBQM ;Bp2M #Qp2 Bb M22/2/ BM Q`/2` iQ ?M/H2 TQBMi T`Q+2bb2b QM i?2 r?QH2 `2H HBM2- 2bT2+BHHv biiBQM`v TQBMi T`Q+2bb2bX _2K`F 8XRX9 h?2 `2bQM r?v `2[mB`2K2Mi U8XRV +MMQi #2 `2TH+2/ #v i?2 bBK@ TH2` QM2

 d

E [N (c, d] | Fc ] = E

λ(s) ds | Fc ,

(†)

c (a)

MQi BMpQHpBM; i?2 biQTTBM; iBK2b Tn - Bb i?i Bi Kv Q++m` i?i #Qi? bB/2b BM U†V `2 BM}MBi2- BM r?B+? +b2 i?2 BM7Q`KiBQM +QMiBM2/ BM U†V Bb MBHX h?Bb ?TT2Mb 7Q` BMbiM+2 r?2M N Bb  ?QKQ;2M2Qmb *Qt T`Q+2bb r?Qb2 `M/QK BMi2MbBiv Λ ?b M BM}MBi2 2tT2+iiBQM Ub22 1tKTH2 8XRX8 #2HQrVX 1tKTH2 8XRX8, SQBbbQM M/ *Qt S`Q+2bb2bX G2i N #2  *Qt T`Q+2bb QM ν R+ rBi? +QM/BiBQMH BMi2MbBiv K2bm`2  ν rBi? `2bT2+i iQ G ⊇ F Ub22 .2}MBiBQM RXRXRyV M/ bmTTQb2 i?i ν(C) := C λ(s) ds UC ∈ B(R+ )V r?2`2 {λ(t)}t≥0 Bb 

8XRX h>1 aJPPh>AL: 6P_JlGa

Red

HQ+HHv BMi2;`#H2 MQM@M2;iBp2 T`Q+2bbX h?2M N /KBib i?Bb T`Q+2bb b Ft @BMi2MbBivr?2`2 Ft := FtN ∨ G Ut ≥ 0V U1t2`+Bb2 8XNXjVX h?2Q`2K 8XRXRR #2HQr ;Bp2b  7Q`KmH i?i +M #2 +QMbB/2`2/ #Qi?  `2}M2@ K2Mi Q7 *KT#2HHǶb 7Q`KmH M/ M 2ti2MbBQM Q7 i?2 bKQQi?BM; 7Q`KmH 7Q` ?TTb Uh?2Q`2K kXkXRVX h?2 MQiBQM Q7 T`2/B+i#H2 T`Q+2bb rBHH #2 M22/2/X .2}MBiBQM 8XRXe G2i T = R Q` R+X G2i {Ft }t∈T #2  ?BbiQ`vX h?2 T`2/B+i#H2 σ@}2H/ P (F· ) QM T × Ω Bb i?2 σ@}2H/ ;2M2`i2/ #v i?2 +QHH2+iBQM Q7 b2ib (a, b] × A

([a, b] ⊂ T, A ∈ Fa ) ,

U8XkV

iQ r?B+? QM2 Kmbi //- BM i?2 +b2 r?2`2 T = R+ - i?2 b2ib {0} × A (A ∈ F0 )X  biQ+?biB+ T`Q+2bb {X(t)}t∈T iFBM; Bib pHm2b BM  K2bm`#H2 bT+2 (E, E) Bb +HH2/ M Ft @T`2/B+i#H2 T`Q+2bb B7 i?2 KTTBM; (t, ω) → X(t, ω) Bb P(F· )@K2bm`#H2X 6Q` b?Q`i- QM2 i?2M bvb, {X(t)}t∈T Bb BM P (F· )X _2K`F 8XRXd M Ft @T`2/B+i#H2 T`Q+2bb Bb Ft @T`Q;`2bbBp2 U1t2`+Bb2 8XNXeVX .2}MBiBQM 8XRX3 G2i T = R Q` R+ M/ H2i {Ft }t∈T #2  ?BbiQ`vX G2i (K, K) #2 bQK2 `#Bi``v K2bm`#H2 bT+2X G2i H : (T×Ω×K, P (F· )⊗K) → (R, B(R))X PM2 i?2M bvb i?i {H(t, z)}t∈T,z∈K Bb M Ft @T`2/B+i#H2 biQ+?biB+ T`Q+2bb BM/2t2/ #v KX 6Q` b?Q`i QM2 r`Bi2b, H ∈ P (F· ) ⊗ KX

1tKTH2 8XRXN, G27i@+QMiBMmBiv M/ S`2/B+i#BHBivX  +QKTH2t@pHm2/ biQ+?biB+ T`Q+2bb {X(t)}t∈R /Ti2/ iQ {Ft }t∈R M/ rBi? H27i@+QMiBMmQmb i`D2+iQ@ `B2b Bb Ft @T`2/B+i#H2X AM 7+i- #v H27i@+QMiBMmBiv- X(t, ω) = limn↑∞ Xn (t, ω)- r?2`2 Xn (t, ω) :=

n +n2 

X(k2−n , ω)1(k2−n ,(k+1)2−n ] (t) ,

k=−n2n

M/ bBM+2 X(k2−n ) Bb Fk2−n @K2bm`#H2- (t, ω) → Xn (t, ω) Bb P(F· )@K2bm`#H2X

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E H(t) N (dt) = E H(t)λ(t) dt , U8XjV R

R

i?2M {λ(t)}t∈R Bb  (P, Ft )@BMi2MbBiv Q7 N X UBBV A7 {λ(t)}t∈R Bb  (P, Ft )@BMi2MbBiv Q7 N - U8XjV ?QH/b i`m2 7Q` HH MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb2b {H(t)}t∈R X S`QQ7X UBV G2i a ∈ R M/ H2i  Tn(a)



:= inf t ≥ a ;



t

λ(s)ds ≥ n

.

a (a)

(a)

aBM+2 {λ(t)}t∈R Bb HQ+HHv BMi2;`#H2- limn↑∞ Tn = +∞X HbQ- 7Q` 2+? n ≥ 1- Tn Bb M Ft @biQTTBM; iBK2X G2i 7Q` [c, d] ⊆ [a, +∞) M/ C ∈ Fc H(t, ω) := 1C (ω)1(c,d] (t)1(a,Tn(a) (ω)] (t) .

h?Bb /2}M2b  H27i@+QMiBMmQmb Ft @/Ti2/ biQ+?biB+ T`Q+2bbX AMb2`iBM; i?Bb T`Q+2bb BMiQ U8XjV ;Bp2b  E[1C N (c ∧

Tn(a) , d



Tn(a) ]]





(a)

d∧Tn

= E 1C

(a)

λ(s)ds .

c∧Tn

aBM+2 C Bb `#Bi``v BM Fc - i?Bb Bb 2[mBpH2Mi iQ U8XRVX (a)

UBBV G2i a ∈ R M/ H2i Tn #2 b BM .2}MBiBQM 8XRXRX .2}M2 7Q` 2+? n ≥ 1 i?2 TQBMi T`Q+2bb Nn #v Nn (C) := N (C ∩ (a, Tn(a) ]) M/ H2i H #2 i?2 +QHH2+iBQM Q7 MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb2b {H(t)}t∈R bm+? i?i  (a) 

 Tn

H(t)Nn (dt) = E

E [a,+∞)

H(t)λ(t)dt .

U8X9V

a

H Bb  d@bvbi2K Q7 7mM+iBQMb U.2}MBiBQM XRX8V +QMiBMBM; i?2 +QMbiMi 1 b r2HH b i?2 7mM+iBQMb Q7 i?2 7Q`K 1D UD ∈ SV r?2`2 S := {C ×(c, d]; a < c < d < +∞, C ∈ Fc }X JQ`2Qp2`- S Bb  π@bvbi2K ;2M2`iBM; P (F· )X h?2`27Q`2- #v .vMFBMǶb i?2Q`2K Uh?2Q`2K XRX9V- H +QMiBMb HH i?2 MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb2bX G2iiBM; n ↑ +∞ BM U8X9V M/ i?2M a ↓ −∞ ;Bp2b i?2 `2bmHi #v KQMQiQM2 +QMp2`;2M+2X 

8XRX h>1 aJPPh>AL: 6P_JlGa

ReN

_2K`F 8XRXRk 6Q`KmH U8XjV Bb +HH2/ i?2 bKQQi?BM; 7Q`KmHX Ai bvb i?i i?2 dzMQM@bKQQi?Ǵ K2bm`2 P (dω) N (ω, dt) M/ i?2 dzbKQQi?Ǵ K2bm`2 P (dω)λ(t, ω) dt ;`22 QM P(F· )X 1tKTH2 8XRXRj, h`MbBiBQM hBK2b Q7  *QMiBMmQmb@iBK2 ?K+- hF2 RX _272` iQ i?2 +QMbi`m+iBQM BM a2+iBQM kX8 Q7  +QMiBMmQmb@iBK2 ?K+ {X(t)}t≥0 rBi? +QmMi#H2 bii2 bT+2 E M/ r?Qb2 BM}MBi2bBKH ;2M2`iQ` A = {qij }i,j∈E Bb bi#H2 M/ +QMb2`piBp2- M/ bmTTQb2 i?i i?Bb ?K+ Bb `2;mH` UMQ ++mKmHiBQM TQBMi Q7 i?2 i`MbBiBQM iBK2bVX h?Bb +QMbi`m+iBQM Bb #b2/ QM  7KBHv Nij Ui = jV Q7 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b rBi? `2bT2+iBp2 BMi2MbBiB2b qij Ui = jV ij ;Bp2M #v M/ i?2 i`MbBiBQM iBK2b 7`QK bii2 i iQ bii2 j 7Q`K  TQBMi T`Q+2bb N  ij (t) = Zi (s−) Nij (ds) , N (0,t]

r?2`2 Zi (t) := 1{X(t)=i} X "v h?2Q`2K kXkXR- 7Q` HH MQM@M2;iBp2 H27i@+QMiBMmQmb FtX @/Ti2/ T`Q+2bb2b {H(t)}t∈R 



ij (dt) = E E H(t)N H(t)Zi (t−) Nij (dt) R R



=E H(t)Zi (t−) qij dt = E H(t)Zi (t) qij dt . R

R

ij (t) = Zi (t) qij X ij /KBib i?2 F X @BMi2MbBiv λ h?2`27Q`2- #v UBV Q7 h?2Q`2K 8XRXRR- N t

1tiBM+iBQM M/ S2`bBbi2M+2 1tKTH2 8XRXR9, aiQ+?biB+ 6BHm`2 _i2X 6Q`  TQBMi T`Q+2bb N QM R M/ t Mv t ∈ R- H2i N−∞ ∪ ∅∞ t /2MQi2 i?2 TQBMi T`Q+2bb +QBM+B/BM; rBi? N QM (−∞, t] M/ rBi?Qmi TQBMib QM (t, +∞)X G2i N #2  bBKTH2 TQBMi T`Q+2bb QM R rBi? i?2 FtN @BMi2MbBiv {λ(t)}t∈R Q7 i?2 7Q`K t λ(t, ω) = v(t, (N−∞ ∪ ∅∞ t )(ω)) ,

r?2`2 v : R×M (R) → R+ Bb K2bm`#H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(R)⊗M(R) M/ B(R+ )X h?2M- 7Q` HH [a, b] ⊂ R  b  a ∞ P (N ((a, b]) = 0 | Fa ) = exp − v(s, N−∞ ∪ ∅a ) ds , a

M/ BM T`iB+mH`- H2iiBM; b ↑ ∞-

  P (N ((a, ∞)) = 0|Fa ) = exp −

∞ a

 a v(s, N−∞ ∪ ∅∞ a ) ds

S`QQ7X G2i Z(t) := 1{N ((a,t])=0} 7Q` t ≥ aX h?2M U1tKTH2 RXeXRyV-

.

U8X8V

Rdy *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu  Z(t) = 1 −

Z(s−)N (ds) . (a,t]

6Q` HH A ∈ FaN - #v h?2Q`2K 8XRXRR UbBM+2 (t, ω) → 1A (ω)Z(t−, ω) Bb P (F·N )@ K2bm`#H2V 

E[1A Z(t)] = P (A) − E 1A Z(s−)λ(s) ds (a,t]

 1A Z(s)λ(s) ds = P (A) − E (a,t]  t

s ∞ = P (A) − E 1A Z(s)v(s, N−∞ ∪ ∅s ) ds

a t a 1A Z(s)v(s, N−∞ ∪ ∅∞ ) ds = P (A) − E a

a t a E[Z(s) | Fa ]v(s, N−∞ ∪ ∅∞ ) ds . = P (A) − E 1A a a

h?2`27Q`2



t

E[Z(t) | Fa ] = 1 − a

M/ }MHHv

a E[Z(s) | Fa ]v(s, N−∞ ∪ ∅∞ a ) ds

  t  a v(s, N−∞ ∪ ∅∞ ) ds . E[Z(t) | Fa ] = exp − a a

 h?2 bK2 T`QQ7- i?Bb iBK2 rBi? Z(t) := 1{N ((Tn ,Tn +t])=0} Ut > 0V- Bb 2bBHv 2ti2M/2/ U1t2`+Bb2 8XNX3V iQ T`Qp2 i?i QM {Tn < ∞}  Tn +h  Tn ∞ P (N ((Tn , Tn + h]) = 0 | FTn ) = exp − v(s, N−∞ ∪ ∅Tn ) ds , Tn

i?i Bb P (Tn+1 − Tn >

h | FTNn )

  = exp −

Tn +h Tn

 Tn v(s, N−∞



∅∞ Tn ) ds

.

h?Bb ;Bp2b i?2 BMi2`T`2iiBQM Q7 biQ+?biB+ BMi2MbBiv b  biQ+?biB+ 7BHm`2 `i2X

1tKTH2 8XRXR8, hBH .Bbi`B#miBQM Q7 i?2 hBK2 iQ 1tiBM+iBQMX h?Bb 2tKTH2 +QMiBMm2b 1tKTH2 8XRXR9X Aib Tm`TQb2 Bb iQ ;Bp2 2biBKi2b Q7 i?2 iBH Q7 i?2 /Bbi`B#miBQM Q7 iBK2 T iQ 2tiBM+iBQM Q7 i?2 TQBMi T`Q+2bb i?2`2Q7X q2 ?p2 7Q` HH a ≥ a P (T ≥ a) = P (N ((a, ∞) > 0) = 1 − P (N ((a, ∞) = 0) , M/- #v U8X8V-

8XRX h>1 aJPPh>AL: 6P_JlGa

RdR



  t a v(s, N−∞ ∪ ∅∞ ) ds . P (N ((a, ∞) = 0) = E exp − a a

h?2`27Q`2- bBM+2

1 x − x2 ≤ 1 − e−x ≤ x (x ≥ 0) 2

r2 Q#iBM i?2 #QmM/b  ∞

a E v(s, N−∞ ∪ ∅∞ ) ds ≥ P (T ≥ a) a a

 ≥E



a

a v(s, N−∞



∅∞ a ) ds





−E a

2  a v(s, N−∞



∅∞ a ) ds

. U8XeV

h?2Q`2K XjXN ?b b  +Q`QHH`v i?2 7QHHQrBM; pi` Q7 i?2 "Q`2HĜ*Mi2HHB i?2Q`2KX h?2Q`2K 8XRXRe G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? i?2 Ft @BMi2MbBiv {λ(t)}t≥0 X h?2M  ∞ λ(s) ds < ∞ P @XbX N (∞) < ∞ ⇐⇒ 0

S`QQ7X TTHv h?2Q`2K XjXN U1t2`+Bb2 8XNXRjVX AM i?2 b2[m2H- QMHv  T`iBH `2bmHi rBHH #2 M22/2/,  ∞  P (N (0, ∞) = ∞) = 1 ⇐⇒ P λ(s) ds = ∞ = 1 . 0

6Q` b2H7@+QMiBM2/M2bb- Bi rBHH #2 T`Qp2/ rBi?Qmi `272`2M+2 iQ h?2Q`2K XjXNX ∞ X amTTQb2 i?i N (0, ∞) = ∞ XbX M/ i?i BiBb MQi i`m2 i?i 0 λ(s) ds = ∞ ∞ XbX h?2M- i?2`2 2tBbib  c < ∞ bm+? i?i P 0 λ(s) ds ≤ c > 0X "v i?2 bKQQi?BM; 7Q`KmH  ∞

 ∞

E 1{0s λ(u) du≤c} N (ds) = E 1{0s λ(u) du≤c} λ(s) ds . 0

0

h?2 `B;?i@?M/ bB/2 Bb #QmM/2/ #Qp2 #v c- r?2`2b i?2 H27i@?M/ bB/2 Bb #QmM/2/ #2HQr #v   E 1{0∞ λ(u) du≤c} N (0, ∞) = ∞ , ∞  +QMi`/B+iBQMX h?2`27Q`2 0 λ(s) ds = ∞ XbX ∞ "X amTTQb2 i?i 0 λ(s) ds = ∞ XbX M/ i?i Bi Bb MQi i`m2 i?i N (0, ∞) = ∞ XbX h?2M i?2`2 2tBbib  c < ∞ bm+? i?i P (N (0, ∞) ≤ c) > 0X "v i?2 bKQQi?BM; 7Q`KmH  ∞

 ∞

E 1{N ((0,s))≤c} N (ds) = E 1{N ((0,s))≤c} λ(s) ds 0

0

Rdk *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu Uri+? i?2 T`2Mi?2b2b 7Q` H27i@+QMiBMmBivVX h?2 H27i@?M/ bB/2 Bb #QmM/2/ #v c + 1r?2`2b i?2 `B;?i@?M/ bB/2 Bb #QmM/2/ #2HQr #v

 ∞ E 1{N (0,∞)≤c} λ(u) du = ∞ , 0

 +QMi`/B+iBQMX h?2`27Q`2 N (0, ∞) = ∞ XbX



aiQ+?biB+ AMi2MbBiv E2`M2Hb h?2 MQiBQM Q7 biQ+?biB+ BMi2MbBiv rBHH MQr #2 2ti2M/2/ iQ TQBMi T`Q+2bb2b QM R×K r?2`2 (K, K) Bb bQK2 K2bm`#H2 bT+2X h?2 KBM `QH2 Bb ii`B#mi2/ iQ i?2 }`bi UdziBK2ǴV +QQ`/BMi2 tBb RX G2i (Ω, F, P ) #2  T`Q##BHBiv bT+2- M/ H2i (K, K) #2  K2bm`#H2 bT+2X G2i λ : R × Ω × K → R+ #2  KTTBM; bm+? i?i • 7Q` HH ω ∈ Ω- HH t ∈ R- C → λ(ω, t, C) /2}M2b  σ@}MBi2 K2bm`2 QM (K, K)• 7Q` HH L ∈ K- i?2 KTTBM; (t, ω) → λ(t, ω, L) Bb K2bm`#H2X h?2M λ(·, ·) Bb +HH2/  biQ+?biB+ F2`M2H 7`QK (R × Ω, B(R) ⊗ F) iQ (K, K)X A7 KQ`2Qp2`- 7Q` bQK2 ?BbiQ`v {Ft }t∈R M/ 7Q` HH L ∈ K- i?2 biQ+?biB+ T`Q+2bb {λ(t, L)}t∈R Bb Ft @/Ti2/ U`2bTX Ft @T`Q;`2bbBp2- Ft @T`2/B+i#H2V- i?2 biQ+?biB+ F2`M2H λ Bb +HH2/ Ft @/Ti2/ U`2bTX Ft @T`Q;`2bbBp2- Ft @T`2/B+i#H2VX A7 i?2`2 2tBbib  b2[m2M+2 {Lk }k≥1 Q7 b2ib Q7 K BM+`2bBM; iQ K M/ bm+? i?i 7Q` HH #QmM/2/ b BMi2`pHb [a, b]- HH k ≥ 1- a λ(s, Lk )ds < ∞ XbX- i?2 F2`M2H λ Bb +HH2/ HQ+HHv BMi2;`#H2X G2i MQr N #2  TQBMi T`Q+2bb QM R × KX .2}M2 7Q` HH t ∈ R FtN := σ{N (A) ; A ∈ B(R) ⊗ K, A ⊆ (−∞, t] × K} .

U8XdV

h?2 7KBHv {FtN }t∈R Bb i?2 BMi2`MH ?BbiQ`v Q7 N M/ Mv ?BbiQ`v {Ft }t∈R bm+? i?i FtN ⊆ Ft 7Q` HH t ∈ R Bb +HH2/  ?BbiQ`v Q7 N X q2 b?HH HbQ mb2 i?2 MQiiBQM N L (C) := N (C × L)X .2}MBiBQM 8XRXRd  TQBMi T`Q+2bb N QM R × K Bb +HH2/ bBKTH2 M/ HQ+HHv }MBi2 B7 i?2`2 2tBbib  b2[m2M+2 {Lk }k≥1 BM K BM+`2bBM; iQ K M/ bm+? i?i 7Q` HH k ≥ 1N Lk Bb  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM RX .2}MBiBQM 8XRXR3 G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R × K M/ H2i {Ft }t∈R #2  ?BbiQ`v Q7 N X G2i λ(·, ·) #2 M Ft @/Ti2/ HQ+HHv BMi2;`#H2 biQ+?biB+ F2`M2H 7`QK (R × Ω, B ⊗ F ) iQ (K, K)X h?2 TQBMi T`Q+2bb N Bb bB/ iQ /KBi i?2 biQ+?biB+ (P, Ft )@BMi2MbBiv F2`M2H λ(·, ·) B7 7Q` HH [a, b] ⊂ R- HH L ∈ K M/ HH k ≥ 1- i?2 TQBMi T`Q+2bb N L∩Lk /KBib i?2 biQ+?biB+ (P, Ft V@BMi2MbBiv {λ(t, L∩Lk )}t∈R X q2 b?HH Q7i2M r`Bi2 dzi?2 Ft @BMi2MbBiv F2`M2H λ(t, dz)Ǵ BMbi2/ Q7 dzi?2 Ft @ BMi2MbBiv F2`M2H λ(·, ·)ǴX

8XRX h>1 aJPPh>AL: 6P_JlGa

Rdj

_2K`F 8XRXRN LQi2 i?i i?2 b2[m2M+2b {Lk }k≥1 Q7 i?2 /2}MBiBQM Q7  HQ+HHv BMi2;`#H2 F2`M2H M/ Q7 .2}MBiBQMb 8XRXRd M/ 8XRXR3 +M Hrvb #2 iF2M B/2MiB+H rBi?Qmi HQbb Q7 ;2M2`HBivX h?2 T`QQ7 Q7 i?2 M2ti `2bmHi Bb M 2bv /TiiBQM Q7 i?i Q7 h?2Q`2K 8XRXRRX h?2Q`2K 8XRXky G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R×KX Ai /KBib i?2 biQ+?biB+ Ft @BMi2MbBiv F2`M2H λ(·, ·) B7 M/ QMHv B7

 

  H(t, z)N (dt × dz) = E H(t, z)λ(t, dz)dt U8X3V E R×K

R×K

7Q` HH MQM@M2;iBp2 P(F· ) ⊗ K@K2bm`#H2 7mM+iBQMb H : R × Ω × K → RX 6Q` i?2 dzB7Ǵ T`i r2 M22/ iQ biBb7v U8X3V QMHv 7Q` HH MQM@M2;iBp2 B(R)⊗F ⊗K@K2bm`#H2 7mM+iBQMb i?i `2 /Ti2/ iQ {Ft }t∈R M/ H27i@+QMiBMmQmb BM i?2 t@p`B#H2X h?2 7Q`KmH U8X3V Bb +HH2/ i?2 bKQQi?BM; 7Q`KmH 7Q` iBK2@bT+2 TQBMi T`Q+2bb2bX M BKK2/Bi2 +Q`QHH`v Q7 h?2Q`2K 8XRXky +QM+2`Mb +QKTH2t@pHm2/ BMi2;`M/b, *Q`QHH`v 8XRXkR G2i N #2 b BM h?2Q`2K 8XRXky- M/ H2i H ∈ P(F· ) ⊗ K #2  +QKTH2t@pHm2/ 7mM+iBQM bm+? i?i i H2bi QM2 Q7 i?2 7QHHQrBM; bii2K2Mib Bb i`m2,

  |H(t, z)|N (dt × dz) < ∞ , E R×K

 

|H(t, z)|λ(t, dz)dt < ∞ .

E R

K

  h?2M   i?2 Qi?2` bii2K2Mi Bb HbQ i`m2- i?2 BMi2;`Hb R K H(t, z)N (dt × dz) M/ H(t, z)λ(t, dz)dt `2 r2HH /2}M2/ M/ }MBi2- M/ 2[mHBiv U8X3V ?QH/bX R K h?2 *b2 Q7 J`F2/ SQBMi S`Q+2bb2b aQ 7`- i?2 T`QD2+iBQM QM R Q7 N M22/ MQi #2  HQ+HHv }MBi2 TQBMi T`Q+2bb- i?i BbN (C) := N (C × K) Kv #2 BM}MBi2 7Q` bQK2 #QmM/2/ "Q`2H b2ib C U7Q` BMbiM+2 B7 N Bb M ?TT Q7 BMi2MbBiv 1VX h?Bb Bb r?v QM2 M22/b i?2 Lk ǶbX h?2 +b2 Q7  K`F2/ TQBMi T`Q+2bb Bb /Bz2`2Mi 7`QK i?Bb TQBMi Q7 pB2r- #mi Bi Bb M2p2`i?2H2bb  T`iB+mH` +b2 iQ r?B+? i?2 ;2M2`H `2bmHib TTHvX _2+HH i?2 7QHHQrBM; /2}MBiBQM, .2}MBiBQM 8XRXkk G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R- +HH2/ i?2 #b2- rBi? iBK2 b2[m2M+2 {Tn }n∈Z X G2i Z := {Zn }n∈Z #2  `M/QK b2[m2M+2 rBi? pHm2b BM bQK2 K2bm`#H2 bT+2 (K, K)X q2 i?2M bv i?i (N, Z) Bb  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb rBi? K`Fb BM K- M/ /2}M2 i?2 bbQ+Bi2/ HB7i2/ TQBMi T`Q+2bb NZ QM R × K #v  1A ((Tn , Zn )) (A ∈ B ⊗ K) . NZ (A) := n∈Z

Rd9 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu U_2K2K#2` i?2 +QMp2MiBQM ++Q`/BM; iQ r?B+? i?2 bmK 2ti2M/b QMHv iQ i?Qb2 BM/B+2b n 7Q` r?B+? |Tn | < ∞XV _2TH+BM; N #v NZ BM U8XdV- r2 Q#iBM i?2 /2}MBiBQM Q7 FtNZ - r?B+? r2 b?HH HbQ r`Bi2 b FtN,Z X .2}MBiBQM 8XRXkj G2i (N, Z) #2  bBKTH2 M/ HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb rBi? K`Fb BM K M/ rBi? bbQ+Bi2/ HB7i2/ TQBMi T`Q+2bb NZ QM R × KX G2i {Ft }t∈R #2  ?BbiQ`v Q7 NZ X M Ft @/Ti2/ biQ+?biB+ F2`M2H 7`QK (R × Ω, B(R) ⊗ F) iQ (K, K) Bb +HH2/  (P, Ft )@BMi2MbBiv F2`M2H Q7 (N, Z) B7 Bi Bb  (P, Ft )@BMi2MbBiv F2`M2H Q7 NZ X LQi2 i?i- BM i?Bb +b2- i?2`2 Bb MQ M22/ 7Q` i?2 b2[m2M+2 {Lk }k≥1 BM .2}MBiBQM 8XRXR3 bBM+2 i?2 TQBMi T`Q+2bb NZL QM R /2}M2/ #v NZL (C) := NZ (C × L) =



1C (Tn )1L (Zn )

(C ∈ B, L ∈ K)

n∈Z

Bb bBKTH2 M/ HQ+HHv }MBi2X G2i (N, Z) #2  bBKTH2 M/ HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb rBi? K`Fb BM K- rBi? bbQ+Bi2/ HB7i2/ TQBMi T`Q+2bb NZ QM R × K- M/ bmTTQb2 i?i Bi /KBib i?2 Ft @biQ+?biB+ F2`M2H λ(t, dz)X AM T`iB+mH`- i?2 biQ+?biB+ T`Q+2bb λ(t) := λ(t, K) Bb i?2 Ft @BMi2MbBiv Q7 N X q2 MQr /2}M2 Φ(t, C) := rBi? i?2 +QMp2MiBQM

0 0

λ(t, C) , λ(t, K)

= 0- bQ i?i λ(t, dz) = λ(t)Φ(t, dz).

h?2 K`F2/ TQBMi T`Q+2bb (N, Z) Bb i?2M bB/ iQ ?p2 i?2 HQ+H Ft @+?`+i2`BbiB+b (λ(t), Φ(t, dz))X A7 Ft = FtN,Z 7Q` HH t ≥ 0- i?2 BMi2`T`2iiBQM Q7 Φ Bb i?i QM {Tn < ∞}Φ(Tn , L) = P (Zn ∈ L | FTn −) Uh?2Q`2K 8XkX9 #2HQrV r?2`2 r2 `2+HH i?i FTn − := σ(Tk , Zk ; k ≤ n − 1) ∨ σ(Tn )X 1tKTH2 8XRXk9, SQBbbQM rBi? AM/2T2M/2Mi BB/ J`FbX A7 i?2 #b2 TQBMi T`Q+2bb N Bb  SQBbbQM T`Q+2bb rBi? U/2i2`KBMBbiB+V BMi2MbBiv λ(t)- M/ B7 i?2 b2@ [m2M+2 Q7 K`Fb Bb BB/ M/ BM/2T2M/2Mi Q7 N - (N, Z) /KBib i?2 FtN,Z @BMi2MbBiv F2`@ M2H λ(t)Q(dz) Q`- 2[mBpH2MiHv- i?2 HQ+H FtN,Z @+?`+i2`BbiB+b (λ(t), Q(dz))- r?2`2 Q Bb i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 K`Fb U1t2`+Bb2 8XNXdVX

8XRX h>1 aJPPh>AL: 6P_JlGa

Rd8

1tKTH2 8XRXk8, JmHiBp`Bi2 SQBMi S`Q+2bb2b- hF2 RX h?2 /2}MBiBQMb M/ `2bmHib #Qp2 rBHH MQr #2 bT2+BHBx2/ iQ KmHiBp`Bi2 TQBMi T`Q+2bb2b #v H2iiBM; K = E-  /2MmK2`#H2 b2iX 6Q` 2+? i ∈ E- /2}M2  TQBMi T`Q+2bb Ni QM R #v Ni (C) := N (C × {i}) 7Q` HH C ∈ B(R)- M/ H2i λi (t) := λ(t, {i})X h?Bb /2}M2b  KmHiBp`Bi2 TQBMi T`Q+2bb {Ni }i∈E - 2+? Ni ?pBM; i?2 Ft @BMi2MbBiv {λi (t)}t∈R r?2`2 {Ft }t∈R Bb  +QKKQM ?BbiQ`v Q7 i?2 Ni ǶbX

1tKTH2 8XRXke, MQi?2` HQQF i i?2 GQiFėoQHi2`` TQTmHiBQM KQ/2HX M pi` Q7 i?2 GQiFĜoQHi2`` KQ/2H U1tKTH2 9XRXRRV Bb b 7QHHQrbX _2+HH i?i i?Bb KQ/2H +QM+2`Mb  TQTmHiBQM Q7 rQK2M 7i2`  ;Bp2M iBK2 Ubv- iBK2 0VX  rQKM #Q`M i iBK2 a > 0 ;Bp2b #B`i? /m`BM; ?2` HB72iBK2 iQ ;B`Hb ++Q`/BM; iQ  SQBbbQM T`Q+2bb rBi? BMi2MbBiv 7mM+iBQM μ(t − a)X h?2 `M/QK HB72iBK2b Q7 rQK2M `2 BM/2T2M/2Mi M/ ?p2  +QKKQM /Bbi`B#miBQM QX h?2 72KBMBM2 T`Q;2Mv Q7 HH rQK2M #Q`M #27Q`2 i?2 Q`B;BM Q7 iBK2 7Q`Kb QM i?2 TQbBiBp2 ?H7@HBM2  SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ0 (t)X G2iiBM; σn #2 i?2 HB72iBK2 Q7 i?2 rQKM #Q`M i iBK2 Tn > 0 Un ≥ 1V- i?2 TQBMi T`Q+2bb Q7 #B`i?b QM R+ ?b 7Q` BMi2MbBiv  μ(t − Tn )1{σn >t−Tn } . λ(t) = λ0 (t) + n≥1

1tKTH2 8XRXkd, ai`2bb _2H2b2 JQ/2HX h?2 Q`B;BMH TT2HHiBQM Q7 i?Bb KQ/2H Bb dzb2H7@+Q``2+iBM; TQBMi T`Q+2bbǴXj Aib biQ+?biB+ Ft @BMi2MbBiv Bb N (0,t)

λ(t) := eX(0)+ct−

i=1

Zi

,

r?2`2 X(0) Bb  ;Bp2M `M/QK p`B#H2- c > 0- M/ {Zn }n≥1 Bb M BB/ b2[m2M+2 Q7 MQM@M2;iBp2 `2H `M/QK p`B#H2bX >2`2 (N,Z)

Ft := σ(X(0)) ∨ Ft

,

r?2`2 (N, Z) Bb i?2 K`F2/ TQBMi T`Q+2bb rBi? #b2 N M/ K`F b2[m2M+2 {Zn }n≥1 X h?Bb KQ/2H M/ Bib `2}M2K2Mib `2 mb2/ iQ /2b+`B#2 b2BbKB+ +iBpBivX 1tKTH2 8XRXk3, h`MbBiBQM hBK2b Q7  *QMiBMmQmb@iBK2 ?K+- hF2  , Z)  kX h?2 ?K+ Q7 1tKTH2 8XRXRj +M #2 pB2r2/ b  K`F2/ TQBMi T`Q+2bb (N     r?2`2 N +QmMib HH i?2 i`MbBiBQMb M/ Zn := X(Tn )X G2i Nij #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 i`MbBiBQMb 7`QK bii2 i iQ jX _2+HH i?2 MQiiBQM Zi (t) = 1{X(t)=i} X  /KBib i?2 FtN @BMi2MbBiv h?2 #b2 TQBMi T`Q+2bb N    ij (t) =  = qij Zi (t) = qi Zi (t) = qX(t) . λ λ(t) i,j∈E ; i=j

i∈E j , j=i

i∈E

j (Ab?K M/ q2bi+Qii- RNdN)X  T`QQ7 Q7 2tBbi2M+2 Q7  biiBQM`v p2`bBQM Bb ;Bp2M BM (Gbikyyy)X

Rde *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu i (t) #2 i?2 F X @BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb N i +QmMiBM; i?2 i`MbBiBQMb G2i λ t

  #`BM;BM; i?2 ?K+ BMiQ bii2 iX b Ni = k∈E ; k=i Nki 

i (t) = λ

Zk (t) qki = qX(t),i .

k∈E ; k=i

h?2 7QHHQrBM; +Q`QHH`v Dmbi `2T?`b2b h?2Q`2K 8XRXkyX h?2Q`2K 8XRXkN G2i (N, Z) #2  bBKTH2 K`F2/ TQBMi T`Q+2bb QM R rBi? K`Fb BM K M/ /KBiiBM; i?2 Ft @biQ+?biB+ F2`M2H λ(t, dz)X h?2M    

 H(Tn , Zn ) = E H(t, z)λ(t, dz) E R

n∈Z

K

7Q` HH MQM@M2;iBp2 P(F· ) ⊗ K@K2bm`#H2 7mM+iBQMb H : R × Ω × K → RX

1tKTH2 8XRXjy, J2M "mbv S2`BQ/ BM i?2 M/GI/1/∞ Zm2m2X G2i i?2 b2[m2M+2 {(Tn , σn )}n∈N #2 M M/GI BMTmi ~QrX h?Bb K2Mb i?i {Tn }n∈N\{0} Bb i?2 2p2Mi@iBK2b b2[m2M+2 Q7 M ?TT A rBi? BMi2MbBiv λ- T0 := 0- M/ {σn }n∈N Bb M BB/ b2[m2M+2 Q7 MQM@M2;iBp2 `M/QK p`B#H2b rBi? +QKKQM +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM G- BM/2T2M/2Mi Q7 AX AM [m2m2BM; i?2Q`v- A +QmMib i?2 ``BpH iBK2b BM  dzbvbi2KǴ U+HH2/ i?2 [m2m2V M/ σn Bb i?2 b2`pB+2 `2[mB`2/ #v i?2 +mbiQK2` ``BpBM; i iBK2 Tn X G2i W (t) #2 i?2 rQ`FHQ/ i iBK2 t- i?i Bb- i?2 iQiH KQmMi Q7 b2`pB+2 `2KBMBM; iQ #2 /QM2 i iBK2 t- bbmKBM; i?i b2`pB+2 Bb T`QpB/2/ i mMBi `i2 b HQM; i?i i?2`2 Bb bQK2 rQ`F iQ #2 /QM2X JQ`2 2tTHB+BiHv- 7Q` t ≥ 0 t  W (t) = σ0 + σn 1(0,t] (Tn ) − 1{W (s)>0} ds . 0

n≥1

G2i R1 #2 i?2 }`bi bi`B+iHv TQbBiBp2 iBK2 i r?B+? i?2 bvbi2K Bb 2KTiv U∞ B7 i?2 bvbi2K M2p2` 2KTiB2bVX W (t)

σ1

σ2

σ4 σ3

σ0 0

t T1

T2

T3

T4

R1

*H2`Hv R1 Bb M FtN @biQTTBM; iBK2- r?2`2 N Bb i?2 TQBMi T`Q+2bb QM R+ × R+ rBi? TQBMi b2[m2M+2 {(Tn , σn )}n∈N X 6Q` HH M > 0 ∞  R1 ∧ M ≤ σ0 + σk 1(0,R1 ∧M ] (Tk ) = σ0 + σ1(0,R1 ∧M ] (t)N (dt × dσ) . U8XNV k≥1

0

E

8XRX h>1 aJPPh>AL: 6P_JlGa h?2 KTTBM;

Rdd

(t, ω, σ) → H(t, ω, σ) := σ1(0,R1 (ω)∧M ] (t),

Bb BM P(F·N ) ⊗ B(R+ ) UMQiBM; i?i Bi Bb i?2 FtN @BMi2MbBiv F2`M2H Q7 N Bb λG(dz)-

H27i@+QMiBMmQmb BM i?2 t@`;mK2MiVX aBM+2 r2 Q#iBM 7`QK U8XNV M/ h?2Q`2K 8XRXkN  ∞ 

σ1(0,R1 ∧M ] (t)λG(dσ) dt E[R1 ∧ M ] ≤ E[σ0 ] + E R+

0

= E[σ0 ] + λE[σ0 ]E[R1 ∧ M ].

U8XRyV

AM T`iB+mH`- E[R1 ∧ M ](1 − λE[σ0 ]) ≤ E[σ0 ]- M/ i?2`27Q`2- B7 λE[σ0 ] < 1E[R1 ∧ M ] ≤

E[σ0 ] 1 − λE[σ0 ]

E[σ0 ] < ∞X _2T`Q/m+BM; i?2 +H+mHiBQM 7Q` HH M > 0- M/ i?2`27Q`2 E [R1 ] ≤ 1−λE[σ 0] rBi? R1 `2TH+BM; R1 ∧ M - r2 ?p2- bBM+2 R1 Bb HKQbi bm`2Hv }MBi2- i?2 2[mHBiv

E[R1 ] =

E[σ0 ] . 1 − λE[σ0 ]

AM bmKK`v, λE[σ0 ] < 1 Bb  bm{+B2Mi +QM/BiBQM 7Q` E[R1 ] iQ #2 }MBi2 BM  M/GI/1/∞ [m2m2X Ai Bb HbQ  M2+2bb`v +QM/BiBQM B7 E[σ0 ] > 0- #2+mb2 r?2M R1 Bb }MBi2 R1 = σ0 + σk 1(0,R1 ] (Tk ) k≥1

M/ i?2`27Q`2 E[R1 ] = E[σ0 ]+λE[σ0 ]E[R1 ]- i?i Bb E[R1 ](1−λE[σ0 ]) = E[σ0 ]- r?B+? BKTHB2b i?i 1 − λE[σ0 ] > 0X G2i (N, Z) #2  bBKTH2 HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb QM R+ rBi? K`Fb BM i?2 K2bm`#H2 bT+2 (K, K) M/ H2i {Ft }t≥0 #2  ?BbiQ`v Q7 (N, Z)X G2i (N, Z) /KBi i?2 Ft @BMi2MbBiv F2`M2H λ(t, dz)X AM i?2 b2[m2H- i?2 7QHHQrBM; MQiiBQM Bb mb2/  H(s, z) MZ (ds × dz) (0,t]×K   H(s, z) NZ (ds × dz) − H(s, z) λ(s, dz) ds , := (0,t]×K

(0,t]×K

T`QpB/2/ i?2 `B;?i@?M/ bB/2 Bb r2HH /2}M2/X h?2`27Q`2- 7Q`KHHvMZ (ds × dz) := NZ (ds × dz) − λ(s, dz) ds . aiQ+?biB+ AMi2;`Hb M/ J`iBM;H2b h?2Q`2K 8XRXjR G2i {H(t, z)}t≥0 #2  `2H@pHm2/ Ft @T`2/B+i#H2 biQ+?biB+ T`Q@ +2bb BM/2t2/ #v K bm+? i?i 7Q` HH t ≥ 0 

E |H(s, z)|λ(s, dz) ds < ∞ . U8XRRV (0,t]×K

Rd3 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu h?2M i?2 biQ+?biB+ T`Q+2bb  H(s, z) MZ (ds × dz)

M (t) :=

U8XRkV

(0,t]×K

Bb  r2HH@/2}M2/ +2Mi2`2/ Ft @K`iBM;H2X  |H(s, z)| NZ (ds × dz) < ∞ 7Q`  HH t ≥ 0 Uh?2Q`2K 8XRXkNVX h?2`27Q`2- 7Q` HH t ≥ 0- (0,t]×K |H(s, z)|λ(s, dz) ds < ∞  M/ (0,t]×K |H(s, z)| NZ (ds × dz) < ∞- S@XbX AM T`iB+mH`- {M (t)}t≥0 Bb S@XbX  r2HH@/2}M2/ M/ }MBi2 biQ+?biB+ T`Q+2bbX 6Q` HH a, b ∈ R+ U0 ≤ a ≤ bV M/ 7Q` HH A ∈ Fa 

E [1A (M (b) − M (a)] = E H  (t, z) MZ (ds × dz) , S`QQ7X *QM/BiBQM U8XRRV Bb 2[mBpH2Mi iQ E



(0,t]×K

R+ ×K

r?2`2 H  (t, z) := H(t, z)1A 1(a,b] (t) /2}M2b M Ft @T`2/B+i#H2 `2H@pHm2/ biQ+?biB+ T`Q+2bb BM/2t2/ #v KX "v h?2Q`2K 8XRXkN 



E H  (t, z) NZ (ds × dz) = E H  (t, z) λ(s, dz) ds R+ ×K

R+ ×K

M/ i?2`27Q`2- 7Q` HH A ∈ Fa - E [1A (M (b) − M (a)] = 0X



*Q`QHH`v 8XRXjk _2TH+BM; bbmKTiBQM U8XRRV Q7 h?2Q`2K 8XRXjR #v i?2 +QM/B@ iBQM i?i P @HKQbi bm`2Hv  |H(s, z)|λ(s, dz) ds < ∞ 7Q` HH t ≥ 0 , U8XRjV (0,t]×K

i?2 biQ+?biB+ T`Q+2bb {M (t)}t≥0 /2}M2/ #v U8XRkV Bb i?2M  HQ+H Ft @K`iBM;H2X S`QQ7X G2i 7Q` 2+? BMi2;2` n ≥ 1 i?2 Ft @biQTTBM; iBK2b  Sn := inf{t > 0 ; |H(s, z)|λ(s, dz) ds ≥ n} , (0,t]×K

rBi? i?2 mbmH +QMp2MiBQM inf ∅ = +∞X h?2M limn↑∞ Sn = ∞- M/ KQ`2Qp2` H(t, z)1{t≤Sn } biBb}2b +QM/BiBQM U8XRRVX h?2`27Q`2 #v h?2Q`2K 8XRXjR{M (t ∧ Sn )}t≥0 Bb  r2HH@/2}M2/ Ft @K`iBM;H2X  h?2Q`2K 8XRXjj G2i H #2 M Ft @T`2/B+i#H2 `2H@pHm2/ biQ+?biB+ T`Q+2bb BM/2t2/ #v Kbm+? i?i 7Q` HH t ≥ 0- S@XbX 

E |H(s, z)|2 λ(s, dz) ds < ∞ . U8XR9V (0,t]×K

8XRX h>1 aJPPh>AL: 6P_JlGa

RdN

h?2M i?2 biQ+?biB+ T`Q+2bb  H(s, z) MZ (ds × dz)

M (t) := (0,t]×K

Bb r2HH /2}M2/ M/  b[m`2@BMi2;`#H2 Ft @K`iBM;H2X JQ`2Qp2` 

|H(s, z)|2 λ(s, dz) ds . E M (t)2 = E

U8XR8V

(0,t]×K

S`QQ7X G2i Tn #2 i?2 n@i? 2p2Mi iBK2 Q7 i?2 #b2 TQBMi T`Q+2bbX h?2 T`QQ7 i?i  M (t) Bb r2HH  /2}M2/ 7QHHQrb 7`QK h?2Q`2K 8XRXjRX AM 7+i- Q#b2`pBM; i?i E (0,Tn ] λ(s) ds = E [N (0, Tn ]] = n  E



|H(s, z)|λ(s, dz) ds ≤ E (1 + |H(s, z)|2 )λ(s, dz) ds (0,t∧Tn ]×K (0,t∧Tn ]×K 

|H(s, z)|2 λ(s, dz) ds < ∞ = E [N (0, Tn ]] + E (0,t∧Tn ]×K

 2 |H(s, z)| λ(s, dz) ds < ∞ . =n+E (0,t∧Tn ]×K

h?2`27Q`2 {M (t∧Tn )}t≥0 Bb r2HH /2}M2/- M/ bQ Bb {M (t)}t≥0 bBM+2 limn↑∞ Tn = ∞X q2 MQr im`M iQ i?2 T`QQ7 Q7 U8XR8VX "v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 +H+mHmb  M (t)2 = M (t−) dM (t) + H(s, z)2 NZ (ds × dz) . (0,t]

aBM+2

(0,t]×K





M (s−)H(s, z) MZ (ds × dz)

M (t−) dM (t) =

m(t) := (0,t]

(0,t]×K

Bb  HQ+H Ft @K`iBM;H2 rBi? `2bT2+i iQ i?2 HQ+HBxBM; biQTTBM; iBK2b    Vn := inf t ≥ 0 ; |M (t−)| + |H(s, z)|λ(s, dz) ds ≥ n ∧ Tn , (0,t]×K

r2 ?p2 i?i



H(s, z)2 NZ (ds × dz) (0,t∧Vn ]×K

 2 H(s, z) λ(s, dz) ds , =E

E M (t ∧ Vn )2 = E

(0,t∧Vn ]×K

7`QK r?B+? U8XR8V 7QHHQrb- B7 r2 +M b?Qr i?i limn↑∞ E [M (t ∧ Vn )2 ] = E [M (t)2 ]X h?Bb rBHH #2 i?2 +b2 #2+mb2 Ub r2 b?HH bQQM T`Qp2V B7 M (t ∧ Vn ) +QMp2`;2b

R3y *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu BM L2C (P ) iQ bQK2 HBKBiX h?Bb HBKBi Bb M2+2bb`BHv M (t)- i?2 HKQbi bm`2 HBKBi Q7 M (t ∧ Vn )X h?2 L2C (P )@+QMp2`;2M+2 Q7 M (t ∧ Vn ) iQ #2 T`Qp2/ 7QHHQrb 7`QK i?2 *m+?v +`Bi2`BQM bBM+2- #v  +QKTmiiBQM bBKBH` iQ i?2 QM2 #Qp2- rBi? m ≥ n E (M (t ∧ Vm ) − M (t ∧ Vn ))2 

H(s, z)2 λ(s, dz) ds = E [m(t ∧ Vm ) − m(t ∧ Vn )] + E (t∧Vn ,t∧Vm ]×K

 H(s, z)2 λ(s, dz) ds , =E (t∧Vn ,t∧Vm ]×K

 [mMiBiv i?i pMBb?2b b m, n ↑ ∞X



*Q`QHH`v 8XRXj9 A7 i?2 bbmKTiBQM U8XR9V Q7 h?2Q`2K 8XRXjj Bb `2TH+2/ #v  |H(s, z)|2 λ(s, dz) ds < ∞ , P @XbX (t ≥ 0) , U8XReV (0,t]×K

i?2 biQ+?biB+ T`Q+2bb  H(s, z) MZ (ds × dz)

M (t) := (0,t]×K

Bb  b[m`2@BMi2;`#H2 HQ+H Ft @K`iBM;H2X S`QQ7X h?Bb 7QHHQrb 7`QK h?2Q`2K 8XRXjj BM i?2 bK2 KMM2` b *Q`QHH`v 8XRXjk Bb /2`Bp2/ 7`QK h?2Q`2K 8XRXjRX  .2T2M/2M+2 mTQM i?2 >BbiQ`v AM i?2 /2}MBiBQM Q7 i?2 Ft @biQ+?biB+ BMi2MbBiv {λ(t)}t∈R Q7 i?2 bBKTH2 TQBMi T`Q+2bb N - i?2 KBMBKH `2[mB`2K2Mi QM i?2 ?BbiQ`v {Ft }t∈R Bb Ft ⊇ Ftλ ∨ FtN

U8XRdV

#2+mb2 i?2 BMi2MbBiv T`Q+2bb M/ i?2 TQBMi T`Q+2bb Bib2H7 `2 Ft @/Ti2/X q?i ?TT2Mb B7 r2 KQ/B7v i?2 ?BbiQ`v\ q2 }`bi Mbr2` i?2 7QHHQrBM; [m2biBQM, hQ r?i 2ti2Mi +M r2 +?M;2 i?2 ?BbiQ`v M/ `2iBM i?2 bK2 biQ+?biB+ BMi2MbBiv\ h?2Q`2K 8XRXj8 G2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM R rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X h?2M, UBV A7 {Gt }t∈R Bb  ?BbiQ`v bm+? i?i G∞ Bb BM/2T2M/2Mi Q7 F∞ - {λ(t)}t∈R Bb HbQ M Ft ∨ Gt @BMi2MbBiv Q7 N X UBBV A7 {Ft }t∈R Bb  ?BbiQ`v Q7 N bm+? i?i Ftλ ∨ FtN ⊆ Ft ⊆ Ft - {λ(t)}t∈R Bb HbQ M Ft @BMi2MbBiv Q7 N X

8XRX h>1 aJPPh>AL: 6P_JlGa

R3R 

S`QQ7X 1t2`+Bb2 8XNX8X

q2 MQr +QMbB/2` bBimiBQMb r?2`2 i?2 biQ+?biB+ BMi2MbBiv Bb KQ/B}2/ #v  +?M;2 Q7 ?BbiQ`vX G2i N ?p2 i?2 Ft @BMi2MbBiv {λ(t)}t∈R - r?2`2 i?2 ?BbiQ`v {Ft }t∈R biBb}2b i?2 KBMBKH `2[mB`2K2Mi U8XRdVX amTTQb2 i?i i?2 M2r ?BbiQ`v {Ft }t∈R Bb bKHH2`- i?i BbFt ⊇ Ft ⊇ FtN . U8XR3V Ai Bb TQbbB#H2 i?i {λ(t)}t∈R Bb MQi Ft @/Ti2/ M/ i?2`27Q`2 +MMQi #2 i?2 Ft @ BMi2MbBivX L2p2`i?2H2bb- i?2`2 biBHH 2tBbib M Ft @BMi2MbBiv- M/ i?Bb BMi2MbBiv Bb Q#@ iBM2/ #v dzT`QD2+iBQMǴ Q7 i?2 BMBiBH biQ+?biB+ BMi2MbBiv QM i?2 bKHH2` ?BbiQ`v- BM  b2Mb2 iQ #2 /2b+`B#2/ MQrX *QMbB/2` i?2 ?BbiQ`B2b {Ft }t∈R M/ {Ft }t∈R biBb7vBM; +QM/BiBQM U8XR3V 7Q` HH t ∈ RX G2i {Y (t)}t∈R #2  MQM@M2;iBp2 HQ+HHv BMi2;`#H2 K2bm`#H2 T`Q+2bbX G2i i?2 σ@}MBi2 K2bm`2b μ1 M/ μ2 QM (R × Ω, P(F· )) #2 /2}M2/ `2bT2+iBp2Hv #v, 



μ1 (H) := E H(t)Y (t)dt M/ μ2 (H) := E H(t)dt R

R

 7Q` HH MQM@M2;iBp2 H : R × Ω i?i `2 P(F)@K2bm`#H2X LQi2 i?i μ2 Bb i?2 T`Q/m+i K2bm`2 P × QM (Ω × R, P(F· ))X *H2`Hv μ1  μ2 - M/ i?2`27Q`2 i?2`2 1 2tBbib  _/QMĜLBFQ/ɷK /2`BpiBp2 Y (t, ω) = dμ (t, ω) i?i Bb P(F· )@K2bm`#H2 dμ2   UM/ i?2`27Q`2 /2}M2b  Ft ĜT`2/B+i#H2 T`Q+2bb {Y (t)}t∈R V bm+? i?i μ1 (H) = μ2 (Y H) = E



 H(t)Y (t) dt . R

JQ`2Qp2`- i?Bb _/QMĜLBFQ/ɷK /2`BpiBp2 Bb μ2 @mMB[m2- i?i Bb iQ bv- B7 i?2`2 2tBbib MQi?2` bm+? _/QMĜLBFQ/ɷK /2`BpiBp2- bv Y - i?2M Y (t, ω) = Y (t, ω) ,

P (dω) × dt

X2X

U8XRNV

.2}MBiBQM 8XRXje h?2 #Qp2 biQ+?biB+ T`Q+2bb {Y (t)}t∈R Bb +HH2/ i?2 T`2/B+i#H2 T`QD2+iBQM Q7 {Y (t)}t∈R QM {Ft }t∈R - Q` i?2 Ft @T`2/B+i#H2 T`QD2+iBQM9 Q7 {Y (t)}t∈R X h?2Q`2K 8XRXjd G2i i?2 bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb N QM R ?p2 i?2 biQ+?biB+ Ft @BMi2MbBiv {λ(t)}t∈R X G2i {Ft }t∈R #2 MQi?2` ?BbiQ`v- biBb7vBM; +QM/B@  t @BMi2MbBiv {λ(t)}  iBQM U8XR3VX h?2M N /KBib i?2 biQ+?biB+ F t∈R - i?2 Ft @T`2/B+i#H2 T`QD2+iBQM Q7 {λ(t)}t∈R X S`QQ7X G2i {H(t)}t∈R #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bbX Ai Bb  7Q`iBQ`B M Ft @T`2/B+i#H2 T`Q+2bb- M/ i?2`27Q`2 9

h?Bb i2`KBMQHQ;v Bb MQi i?2 biM/`/ QM2 BM i?2 ;2M2`H i?2Q`v Q7 biQ+?biB+ T`Q+2bb2bX

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 dt . H(t) N (dt) = E H(t) λ(t) dt = E H(t) λ(t)

E R

R

R

 6Q`  ;Bp2M ?BbiQ`v {Ft }t∈R M/  ;Bp2M T`Q##BHBiv P - i?2`2 2tBbi KMv p2`bBQMb Q7 i?2 (P, Ft )@BMi2MbBiv (λ(t))t∈R X am+? M BMi2MbBiv +M Hrvb #22M +?Qb2M iQ #2 Ft @T`2/B+i#H2- bBM+2 B7 Bi Bb MQi- QM2 +M `2TH+2 Bi #v Bib Ft @T`2/B+i#H2 T`QD2+iBQMr?B+? 2tBbib M/ Bb 2bb2MiBHHv mMB[m2 BM  b2Mb2 iQ #2 /2b+`B#2/ #2HQrX h?2Q`2K 8XRXj3 V h?2 Ft @T`2/B+i#H2 p2`bBQM Bb P (dω) dt@mMB[m2 M/ i?2`27Q`2 P (dω) N (ω, dt) mMB[m2X AM T`iB+mH`- B7 {λ (t)}t∈R Bb MQi?2` Ft @T`2/B+i#H2 p2`@ bBQM- i?2M- P @HKQbi bm`2Hv- λ (Tn ) = λ(Tn ) 7Q` HH n ∈ ZX U#V A7 {λ(t)}t∈R Bb  T`2/B+i#H2 p2`bBQM Q7 i?2 Ft @BMi2MbBiv Q7 i?2 HQ+HHv }MBi2 bBKTH2 TQBMi T`Q+2bb N - i?2M λ(t, ω) > 0

λ(t, ω)P (dω)dt M/ N (dt, ω)P (dω)@X2X

U8XkyV

AM T`iB+mH`- 7Q` HH nλ(Tn ) > 0

P @XbX

U8XkRV

S`QQ7X UV 7QHHQrb 7`QK i?2 mMB[m2M2bb T`QT2`iv U8XRNVX 6Q` U#V- MQi2 i?i H(t) = 1{λ(t)=0} Bb M Ft @T`2/B+i#H2 T`Q+2bbX AMb2`iBM; i?Bb BM i?2 bKQQi?BM; 7Q`KmH- r2 Q#iBM  

  E 1{λ(Tn )=0} = E 1{λ(t)=0} λ(t) dt = 0, n∈Z

R



r?B+? BKTHB2b U8XkyVX

AM T`iB+mH`- B7 i?2 HQ+HHv }MBi2 bBKTH2 K`F2/ TQBMi T`Q+2bb (N, Z) ?b i?2 Ft @T`2/B+i#H2 biQ+?biB+ BMi2MbBiv F2`M2H λ(t)Φ(t, dz)- i?2M- 7Q` HH L ∈ K- QM {Tn < ∞}λ(Tn )Φ(Tn , L) > 0 . h?2 #Qp2 `2bmHib 2ti2M/ bi`B;?i7Q`r`/Hv iQ K`F2/ TQBMi T`Q+2bb2b M/ i?2B` biQ+?biB+ BMi2MbBiv F2`M2Hb b 7QHHQrbX G2i i?2 bBKTH2 HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb (N, Z) ?p2 i?2 Ft @BMi2MbBiv F2`M2H λ(t)Φ(t, dz) r?2`2 Φ(t, dz) = λ(t)μ(t, z)Q(t, dz)

U8XkkV

7Q` bQK2 FtN,Z @T`2/B+i#H2 F2`M2H Q(t, dz)X G2i {Ft }t∈R #2  ?BbiQ`v bm+? i?iFt ⊇ Ft ⊇ FtN,Z .

U8XkjV

Ai Kv ?TT2M i?i i?2 biQ+?biB+ F2`M2H λ(t)Φ(t, dz) Bb MQi Ft @/Ti2/ M/ i?2`2@ 7Q`2 +MMQi #2 i?2 Ft @BMi2MbBiv F2`M2HX L2p2`i?2H2bb- i?2`2 biBHH 2tBbib  biQ+?biB+

8XRX h>1 aJPPh>AL: 6P_JlGa

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Ft @BMi2MbBiv F2`M2H- M/ Bi Bb Q#iBM2/ #v dzT`QD2+iBQMǴ Q7 i?2 BMBiBH biQ+?biB+ BM@ i2MbBiv F2`M2H QM i?2 bKHH2` ?BbiQ`v- BM  b2Mb2 iQ #2 /2b+`B#2/ MQrX _2+HH i?2 i2`KBMQHQ;v, B7 i?2 KTTBM; Y : (t, ω, z) → Y (t, ω, z) ∈ R Bb B(R+ )⊗ F ⊗ K- QM2 bvb i?i Y (t, z) Bb  K2bm`#H2 T`Q+2bb BM/2t2/ #v KX Ai Bb bB/ iQ #2 Ft @/Ti2/ U`2bTX Ft @T`2/B+i#H2V B7 KQ`2Qp2` 7Q` HH z ∈ K- i?2 biQ+?biB+ T`Q+2bb {Y (t, z)}t∈R Bb Ft @/Ti2/ U`2bTX Ft @T`2/B+i#H2VX *QMbB/2` i?2 ?BbiQ`B2b {Ft }t∈R M/ {Ft }t∈R biBb7vBM; +QM/BiBQM U8XkjV 7Q` HH t ∈ RX G2i {Y (t, z)}t∈R #2  MQM@M2;iBp2 K2bm`#H2 T`Q+2bb BM/2t2/ #v KX G2i i?2 sigma@}MBi2 K2bm`2b μ1 M/ μ2 QM (R × Ω × K, P(F· ) ⊗ K) #2 /2}M2/ `2bT2+iBp2Hv #v,  

μ1 (H) := E

H(t, z)Y (t, z) Q(t, dz) dt R

K

 

M/



μ2 (H) := E

H(t, z) Q(t, dz) dt R

K

 ⊗ K@K2bm`#H2X LQi2 7Q` HH MQM@M2;iBp2 KTTBM;b H : R × Ω × K i?i `2 P(F) i?i μ2 Bb i?2 T`Q/m+i K2bm`2 P (dω) × Q(t, ω, dz) dt QM (Ω × R × K, P(F· ) ⊗ K)X *H2`Hv μ1  μ2 - M/ i?2`27Q`2 i?2`2 2tBbib  _/QMĜLBFQ/ɷK /2`BpiBp2 1 (t, ω, z) i?i Bb P(F· ) ⊗ K@K2bm`#H2 M/ i?2`27Q`2 /2}M2b M Y (t, ω, z) = dμ dμ2 Ft @T`2/B+i#H2 T`Q+2bb BM/2t2/ #v K- Y (t, z)- bm+? i?i  

μ1 (H) = μ2 (Y H) = E H(t, z)Y (t, z) Q(t, dz) dt . R

K

JQ`2Qp2`- i?Bb _/QMĜLBFQ/ɷK /2`BpiBp2 Bb μ2 @mMB[m2- i?i Bb iQ bv- B7 i?2`2 2tBbib MQi?2` bm+? _/QMĜLBFQ/ɷK /2`BpiBp2- bv Y - i?2M Y (t, ω, z) = Y (t, ω, z) ,

P (dω)Q(t, ω, dz)dt

X2X

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.2}MBiBQM 8XRXjN h?2 #Qp2 biQ+?biB+ T`Q+2bb Y (t, z) BM/2t2/ #v K Bb +HH2/ i?2 T`2/B+i#H2 T`QD2+iBQM Q7 Y (t, z) QM {Ft }t∈R - Q` i?2 Ft @T`2/B+i#H2 T`QD2+iBQM Q7 Y (t, z)X h?2Q`2K 8XRX9y G2i i?2 bBKTH2 HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb (N, Z) QM R ?p2 i?2 biQ+?biB+ Ft @BMi2MbBiv F2`M2H U8XkkV 7Q` bQK2 FtN,Z @T`2/B+i#H2 F2`M2H Q(t, dz)X G2i {Ft }t∈R #2 MQi?2` ?BbiQ`v- biBb7vBM; +QM/BiBQM U8XkjVX h?2M (N, Z)    t @BMi2MbBiv F2`M2H λ(t)  ?b i?2 biQ+?biB+ F h(t, z)Q(t, dz) r?2`2 {λ(t)} t∈R Bb i?2 Ft @ T`2/B+i#H2 T`QD2+iBQM Q7 {λ(t)}t∈R M/  h(t, z) Bb i?2 Ft @T`2/B+i#H2 T`QD2+iBQM Q7 h(t, z)X S`QQ7X G2i H(t, z) #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 BM/2t2/ biQ+?biB+ T`Q+2bbX Ai Bb  7Q`iBQ`B M Ft @T`2/B+i#H2 BM/2t2/ biQ+?biB+ T`Q+2bb- M/ i?2`27Q`2  

 

E H(t, z)) N (dt × dz) = E H(t, z)) λ(t)h(t, z) Q(t, dz) dt R K R K

=E H(t, z)) v(t, z) Q(t, dz) dt , R

K

R39 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu r?2`2 v(t, z) Bb i?2 Ft @T`2/B+i#H2 T`QD2+iBQM Q7 λ(t)h(t, z)X G2i MQr v(t, ω, z)  h(t, ω, z) := ,  ω) λ(t,  [mMiBiv i?i Bb P (dω) × N (ω, dt × dz)@ M/ P (dω) × λ(t, ω)Q(t, ω, dz)dt@r2HH /2}M2/ BM pB2r Q7 h?2Q`2K 8XRXj3X q2 ?p2 i?i 7Q` HH MQM@M2;iBp2 Ft @T`2/B+i#H2 BM/2t2/ T`Q+2bb2b H  

  v  (t, z)  H(t, z) N (dt × dz) = E H(t, z)λ(t) E Q(t, dz) dt  λ(t) R K R K  

=E H(t, z)λ(t) h(t, z) Q(t, dz) dt . R

K

1[mBpH2MiHv  E R

  v(t, z)  dt H(t, z) Q(t, dz) λ(t)  λ(t) K   

=E H(t, z) h(t, z) Q(t, dz) λ(t) dt . R

LQr

K



  E R

K



H(t, z) h(t, z) Q(t, dz) λ(t) dt   

=E H(t, z) h(t, z) Q(t, dz) N (dt) R K 

 =E H(t, z) h(t, z) Q(t, dz) λ(t) dt , R

K

M/ i?2`27Q`2     v(t, z)  H(t, z) E Q(t, dz) λ(t) dt  λ(t) R K   

 dt . =E H(t, z) h(t, z) Q(t, dz) λ(t) R

_2TH+BM; H(t, z) #v

K

H(t,z)  λ(t)

  E R

  v(t, z) H(t, z) Q(t, dz) dt  λ(t) K   

=E H(t, z) h(t, z) Q(t, dz) dt , R

r?B+? b?Qrb i?i

v (t,z)  λ(t)

= h(t, z)X

K



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HbQ H2i N (ω) := N2 (ω2 )- Ft := G1 ⊗FtN2 M/ G0 := G1 ⊗(∅, Ω2 )- v(t, ω, N (ω)|[0,t) := v(t, ω1 , N2 (ω2 ) |[0,t) )X PM2 p2`B}2b i?i N Bb  TQBMi T`Q+2bb QM (Ω, F) rBi? i?2

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LQi2 i?i N ({0}) = 0X .2}M2 i?2 UbBKTH2V TQBMi T`Q+2bb NZ QM R+ × K #v L

NZ (C × L) := N L (C) .2}M2 i?2 BMi2`MH ?BbiQ`v FtN,Z

{FtN,Z }t≥0

(C ∈ B(R+ ), L ∈ K) .

Q7 NZ #v

:= σ (NZ (C × L) ; C ∈ (0, t], L ∈ K) ,

M/ i?2 ?BbiQ`v {Ft }t≥0 #v Ft = σ(Z0 ) ∨ FtN,Z ,

U8Xk8V

r?2`2 Z0 Bb  `M/QK 2H2K2Mi iFBM; pHm2b BM M `#Bi``v K2bm`#H2 bT+2 (L0 , K0 ) U7Q` BMbiM+2  bT+2 Q7 7mM+iBQMbVX Ai `2T`2b2Mib  dzT`2?BbiQ`vǴ Q7 i?2 K`F2/ TQBMi T`Q+2bb- BM i?2 b2Mb2 i?i F0 = σ(Z0 ) +QMiBMb i?2 BM7Q`KiBQM H`2/v ;i?2`2/ i iBK2 0 i?i Kv BM~m2M+2 Bib 7mim`2 #2?pBQm`X *QM/BiBQMH AMi2`pH .Bbi`B#miBQMb M/ aiQ+?biB+ AMi2MbBiv E2`M2Hb amTTQb2 i?i 7Q` HH n ≥ 0- HH L ∈ K M/ HH C ∈ B(R+ ) P (Sn+1 ∈ C , Zn+1 ∈ L | FTn ) (ω) = g (n+1) (ω, x, L) dx := G(n+1) (ω, C, L) , C

r?2`2 7Q` HH L ∈ K- i?2 KTTBM; (ω, x) → g (n+1) (ω, x, L) Bb FTn ⊗ B(R+ )@ K2bm`#H2- M/ r?2`2 7Q` HH (ω, x)- i?2 KTTBM; L → g (n+1) (ω, x, L) Bb  σ@}MBi2 K2bm`2 QM (K, K)X AM T`iB+mH` P (Sn+1 ∈ C | FTn ) (ω) = g (n+1) (ω, x) dx := G(n+1) (ω, C) , C

r?2`2 g

(n+1)

(ω, x) := g

(n+1)

(ω, x, K)) M/ G(n+1) (ω, C) := G(n+1) (ω, C, K)X

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R3d

h?2Q`2K 8XkXk 6Q` L ∈ K M/ t ≥ 0- H2i  g (n+1) (t − Tn , L) λ(t, L) := 1{Tn ≤t 0- Q7 i?2 7Q`K,  λ(n) (t, ω) 1{Tn (ω)1 ALh1LaAhu E1_L1G

R3N

 1{N ((Tn ,t])=0} = 1 −

1{N ((Tn ,s))=0} N (ds) , (Tn ,t]

M/ i?2`27Q`2 E 1A 1{N ((Tn ,t])=0} 1{t≥Tn }

   1{N ((Tn ,s))=0} N (ds) 1{t≥Tn } . = E 1A 1 − (Tn ,t]

h?2 T`Q+2bb H(t) := 1A 1{N ((Tn ,s))=0} 1{s>Tn }

(t ≥ 0)

Bb Ft @T`2/B+i#H2 U#2BM; Ft @/Ti2/ M/ H27i@+QMiBMmQmbV- M/ i?2`27Q`2- #v i?2 bKQQi?BM; 7Q`KmH

  1{N ((Tn ,s))=0} N (ds) 1{t≥Tn } E 1A 1 − (Tn ,t]  

 t = E 1A 1 − 1{N ((Tn ,s])=0} λ(s) ds 1{t≥Tn } T

   nt (n) 1{N ((Tn ,s])=0} λ (s) ds 1{t≥Tn } . = E 1A 1 − Tn

h?2 Hii2` [mMiBiv 2[mHb- 7i2` +QM/BiBQMBM; rBi? `2bT2+i iQ FTn  

 t (n+1) (n) E 1A 1 − G ([s − Tn , ∞]) λ (s) ds 1{t≥Tn } . Tn

*QK#BMBM; i?2 #Qp2 2[mHBiB2b- r2 Q#iBM E 1A G(n+1) ((t − Tn , ∞]) 1{t≥Tn }  

 t = E 1A 1 − G(n+1) ([s − Tn , ∞]) λ(n) (s) ds 1{t≥Tn } . Tn

h?Bb T`Qp2b i?2 2tBbi2M+2 Q7 i?2 +QM/BiBQMH T`Q##BHBiv /2MbBiv g (n+1) M/ UVX  G2i NZ /KBi i?2 biQ+?biB+ (P, Ft )@BMi2MbBiv F2`M2H λ(t, dz)X .2MQi2 #v {λ(t)}t≥0 M Ft @T`2/B+i#H2 biQ+?biB+ BMi2MbBiv Q7 i?2 #bB+ TQBMi T`Q+2bb N := N K - M/ 7Q` 2+? L ∈ K- HH t ≥ 0- /2}M2 Φ(t, L) #v λ(t, L) := λ(t)Φ(t, L) B7 λ(t) > 0- M/ #v Φ(t, L) := 0 B7 λ(t) = 0X aBM+2 λ(Tn ) > 0 P @XbX QM {Tn < ∞}Φ(Tn , L)1{Tn  ahP*>ahA* ALh1LaAhu S`QQ7X "v h?2Q`2K X9XR- Φ(Tn , L)1{Tn BH#2`i bm#bT+2 Q7 L2R (F1N,Z , P ) +QMbBbiBM; Q7 i?2 +2Mi2`2/ `M/QK p`B#H2b i?2`2Q7X

8XjX J_hAL:G1a a ahP*>ahA* ALh1:_Ga

RNR

h?2Q`2K 8XjXR h?2 KTTBM; ϕ : H → M20 ;Bp2M #v  ϕ(C)(t) := C(t, z) MZ (dt × dz) [0,t]×K

/2}M2b M BbQK2i`v #2ir22M H M/ M20 - M/ i?2 KTTBM; ψ : H → L2R (F1N,Z , P ) ;Bp2M #v  C(t, z) MZ (dt × dz) ψ(C) = [0,1]×K

/2}M2b M BbQK2i`v #2ir22M H M/ L2R,0 (F1N,Z , P )X Mv +2Mi2`2/ b[m`2@BMi2;`#H2 FtN,Z @K`iBM;H2 {M (t)}t∈[0,1] /KBib i?2 `2T`2@ b2MiiBQM  C(s, z) MZ (ds × dz) , M (t) = [0,t]×K

r?2`2 C ∈ HX q2 }`bi ;Bp2 irQ T`2HBKBM`v `2bmHibX G2i f : [0, 1] × K → R #2  K2bm`#H2 7mM+iBQM bm+? i?i  2  f (s,z) − 1 λ(t, z) dt Q(dz) < ∞ . e [0,1]×K

Uh?Bb Bb i?2 +b2 B7 f Bb MQM@TQbBiBp2 Q` #QmM/2/VX G2i 7Q` t ∈ [0, 1]     f (s,z)  Mf (t) := exp e f (s, z) NZ (ds × dz) + − 1 λ(s, z) Q(dz) . [0,t]×K

[0,t]×K

G2KK 8XjXk lM/2` i?2 #Qp2 +QM/BiBQMbUV,



  Mf (s−) ef (s,z) − 1 MZ (ds × dz) ,

Mf (t) = 1 +

()

[0,t]×K

U#V, {Mf (t)}t∈[0,1] Bb  b[m`2 BMi2;`#H2 K`iBM;H2- M/    2 U+V, E [Mf (t)2 − 1] = E [0,t]×K Mf (s)2 ef (s,z) − 1 λ(s, z) Q(dz) (t ∈ [0, 1])X S`QQ7X UV, Ai bm{+2b iQ Q#b2`p2 i?i i M 2p2Mi@iBK2 t Q7 i?2 #b2 TQBMi T`Q+2bb rBi? +Q``2bTQM/BM; K`F z ∈ K  Mf (t) − Mf (t−) = Mf (t−) ef (t,z) − 1 , M/ i?i i  iBK2 t #2ir22M irQ 2p2Mi iBK2b   f (t,z)  dMf (t) = Mf (t) e − 1 λ(t, z) Q(dz) . dt K U#V, Ai bm{+2b iQ b?Qr- BM pB2r Q7 h?2Q`2K 8XRXjj- i?i

RNk *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu 

1

 2 Mf (s)2 ef (s,z) − 1 λ(s, z) Q(dz) ds < ∞ .



E 0

U8XkdV

K

h?Bb Bb i`m2 r?2M Mf (t) Bb `2TH+2/ #v Mf (t ∧ Sn ) r?2`2 Sn := inf{t ≥ 0 ; Mf (t−) ≥ n} . AM T`iB+mH` E (Mf (t ∧ Sn ) − 1)2 = E

 t  0

 2 Mf (s)2 1{s≤Sn } ef (s,z) − 1 λ(s, z) Q(dz) ds . K

U8Xk3V LQr-

E (Mf (t ∧ Sn ) − 1)2 = E Mf (t ∧ Sn )2 − 1

M/ KQ`2Qp2`

E Mf (t ∧ Sn )2 ≥ E Mf (t)2 1{t≤Sn } .

h?2`27Q`2  t 

 2 Mf (s)2 1{s≤Sn } ef (s,z) − 1 λ(s, z) Q(dz) E Mf (t)2 1{t≤Sn } ≤ 1 + E 0 K   t  f (s,z) 2 2 e E Mf (s) 1{s≤Sn } − 1 λ(s, z) Q(dz) ds . =1+ 0

K

"v :`QMrHHǶb H2KK- 7Q` HH t ≥ 0 t    f (s,z) 2 2 E Mf (t) 1{t≤Sn } ≤ exp e − 1 λ(s, z) Q(dz) ds < ∞ . K

0

aBM+2 limn↑∞ Sn = ∞- #v KQMQiQM2 +QMp2`;2M+2 E Mf (t)2 ≤ exp



1



ef (s,z) − 1

0

M/ i?2`27Q`2  E

1 0





2

 λ(s, z) Q(dz)

:= C < ∞ ,

K

2  Mf (s)2 ef (s,z) − 1 λ(s, z) Q(dz) K  2  f (s,z) ≤C − 1 λ(t, z) dt Q(dz) , e [0,1]×K

 }MBi2 [mMiBiv #v ?vTQi?2bBbX h?2`27Q`2 U8XkdV Bb T`Qp2/X U+V, ai`i 7`QK U8Xk3V M/ H2i n ↑ ∞ iQ Q#iBM  t 

 f (s,z) 2 2 2 E (Mf (t)) = 1 + E Mf (s) e − 1 λ(s, z) Q(dz) ds . 0

K

h?Bb Bb  /Bz2`2MiBH 2[miBQM BM E [Mf (t)2 ] r?Qb2 bQHmiBQM ;Bp2b U+VX



8XjX J_hAL:G1a a ahP*>ahA* ALh1:_Ga

RNj

G2KK 8XjXj X h?2 +QHH2+iBQM Q7 `M/QK p`B#H2b H := {eX ; X ∈ U} r?2`2 U Bb i?2 +QHH2+iBQM Q7 `M/QK p`B#H2b k 

aj NZ (Cj × Lj ) (k ∈ N, Cj ∈ B([0, 1]), Lj ∈ K, a1 , . . . , ak ∈ R)

j=1 (N,Z)

Bb iQiH BM i?2 >BH#2`i bT+2 L2R (F1

, P )X

"X h?2 +QHH2+iBQM Q7 `M/QK p`B#H2b i?i `2 HBM2` +QK#BMiBQMb Q7 2H2K2Mib Q7 , , -   K0 := Mf (1) := exp [0,1]×K f (s, z) NZ (ds × dz) + [0,t]×K ef (s,z) − 1 λ(s, z) Q(dz) ,

r?2`2 f : [0, 1] × K → R Bb  K2bm`#H2 7mM+iBQM bm+? i?i   f (s,z) 2 e − 1 λ(t, z) dt Q(dz) < ∞ , [0,1]×K (N,Z)

Bb /2Mb2 BM i?2 >BH#2`i bT+2 L2R (F1

, P )X

S`QQ7X S`i " Bb M BKK2/Bi2 +QMb2[m2M+2 Q7 S`i X 6Q` i?2 T`QQ7 Q7 - Q#b2`p2 i?i H ⊂ L2R (F 1N,Z , P ) M/ i?i 1 ∈ HX q2 ?p2 iQ T`Qp2 i?i B7 Y ∈ L2R (F1N,Z , P ) Bb bm+? i?i E Y eX = 0 7Q` HH X ∈ U - i?2M P (Y = 0) = 1X b 1 ∈ H- E [Y ] = 0X JmHiBTHvBM; Y #v  +QMbiMi B7 M2+2bb`v- r2 Kv bmTTQb2 i?i E [|Y |] = 2 M/ BM T`iB+mH`- bBM+2 E[Y ] = 0- E [Y + ] = 1 M/ E [Y − ] = 1X h?2`27Q`2 Q+ := Y + P − X M/ Q− := Y +P X`2 T`Q##BHBiv − X K2bm`2bX aBM+2 E Y e X = 0 7Q` HH X ∈ U - r2 ?p2 i?i E Y e = E Y e Q`- 2[mBpH2MiHv- EQ+ e = EQ− eX X h?2`27Q`2 i?2 GTH+2 i`Mb7Q`Kb Q7 i?2 p2+iQ`b Q7 i?2 ivT2 (NZ (C1 × L1 ), . . . , NZ (Ck × Lk )) `2 i?2 bK2 mM/2` Q + M/ Q− X h?Bb BKTHB2b i?i Q+ M/ Q− ;`22 BM T`iB+mH` QM F1W X h?2`27Q`2 E 1{Y + >Y − } (Y + − Y − ) = 0 M/ E 1{Y + Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu h?2 T`QQ7 Bb QKBii2/X8 h?2 ?vTQi?2bBb Q7 `B;?i@+QMiBMmBiv BM h?2Q`2K 8XjX9 Bb bmT2`~mQmb B7 i?2 7QH@ HQrBM; bQ@+HH2/ mbmH +QM/BiBQMb `2 biBb}2/, UBV (Ω, F, P ) Bb +QKTH2i2UBBV F0 +QMiBMb HH i?2 P @M2;HB;B#H2 b2ib- M/ UBBBV {Ft }t≥0 Bb `B;?i@+QMiBMmQmbX h?Bb 7QHHQrb 7`QK i?2 7QHHQrBM; 7mM/K2MiH `2bmHi Q7 K`iBM;H2 i?2Q`v, h?2Q`2K 8XjX8 G2i (Ω, F, P ) #2 bQK2 T`Q##BHBiv bT+2 M/ H2i {Ft }t≥0 #2 bQK2 ?BbiQ`v /2}M2/ QM BiX A7 i?2 mbmH +QM/BiBQMb `2 biBb}2/- Mv HQ+H Ft @K`iBM;H2 {M (t)}t≥0 /KBib  `B;?i@+QMiBMmQmb KQ/B}+iBQM rBi? H27i@?M/ HBKBibX AM h?2Q`2K 8XjX9- B7 (Ω, F, P ) Bb +QKTH2i2 M/ B7 F0 +QMiBMb HH i?2 P @M2;HB;B#H2 b2ib- i?2M i?2 mbmH +QM/BiBQMb `2 biBb}2/ bBM+2 i?2 BMi2`MH ?BbiQ`v {FtN,Z }t≥0 Bb `B;?i@+QMiBMmQmbX 1tKTH2 8XjXe, JmHiBp`Bi2 SQBMi S`Q+2bb2b- hF2 kX h?2 #Qp2 `2bmHi +M #2 bT2+BHBx2/ iQ i?2 bBimiBQM /2b+`B#2/ BM 1tKTH2 8XRXk8- iQ r?B+? r2 `272` 7Q` i?2 MQiiBQMbX h?2 ?BbiQ`v i?2`2Q7 Bb MQr bbmK2/ iQ #2 Q7 i?2 7Q`K Ft = G∨FtN X 6Q` 2+? i ∈ E- H2i {λi (t)}t≥0 #2 i?2 F t @BMi2MbBiv Q7 Ni X h?2M Mv `B;?i@+QMiBMmQmb HQ+H Ft @K`iBM;H2 {M (t)}t≥0 /KBib i?2 `2T`2b2MiiBQM   Hi (s)Mi (ds) M (t) = M (0) + i∈E

(0,t]

K

r?2`2 7Q` 2+? i ∈ E- Mi (ds) = Ni (ds) − λi (s)ds M/ Hi ∈ P(F· ) Bb bm+? i?i 7Q` HH t ≥ 0   t

|Hi (s)|λi (s) ds < ∞ . 0

K

8X9 hBK2 a+HBM; _2+HH i?2 7QHHQrBM; irQ r2HH@FMQrM `2bmHib +QM+2`MBM; SQBbbQM T`Q+2bb2b,  #2 M ?TT QM i?2 TQbBiBp2 ?H7@HBM2 rBi? BMi2MbBiv 1 U dzbiM/`/Ǵ U_RV G2i N ?TTV- M/ H2i t → λ(t) #2  U/2i2`KBMBbiB+V HQ+HHv BMi2;`#H2 7mM+iBQMX h?2 TQBMi   ((0, t λ(s) ds]) Bb  SQBbbQM T`Q+2bb rBi? T`Q+2bb N /2}M2/ #v N ((0, t]) := N 0 BMi2MbBiv λ(t)X U_kV *QMp2`b2Hv- B7 N Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t)- /2}MBM; τ (t)  τ (t)  /2}M2/ #v N  ((0, t]) = N ((0, τ (t)]) Bb  #v 0 λ(s) ds = t- i?2 TQBMi T`Q+2bb N biM/`/ ?TTX 8 6Q` i?2 T`QQ7 Q7  KQ`2 ;2M2`H 7Q`K Q7 i?2 K`iBM;H2 `2T`2b2MiiBQM i?2Q`2K- b22- 7Q` mMK`F2/ TQBMi T`Q+2bb2b, (*?Qm M/ J2v2`- RNd9)- M/ 7Q` K`F2/ TQBMi T`Q+2bb2b- (C+Q/RNd8) M/ ("Q2H- o`Bv M/ qQM;- RNd8)X

8X9X hAJ1 a*GAL:

RNd

h?2b2 `2bmHib ?p2 Mim`H 2ti2MbBQMb iQ TQBMi T`Q+2bb2b rBi?  biQ+?biB+ BM@ i2MbBivX h?2 2ti2MbBQM Q7 _k Bb Q7 BMi2`2bi BM BM7Q`KiBQM i?2Q`v M/ rBHH #2 ;Bp2M BM a2+iBQM RyXjX h?2 2ti2MbBQM Q7 _R #2HQM;b iQ i?2 7QHFHQ`2 Q7 TQBMi T`Q+2bb i?2Q`v- M/  T`iB+mH`Hv BMi2`2biBM; TTHB+iBQM `2 rb ;Bp2M BM a2+iBQM kXe UEm`ixǶb h?2Q`2K kXeXkVX q2 MQr ;Bp2  ;2M2`H +QMbi`m+iBQM Q7  TQBMi T`Q+2bb rBi?  ;Bp2M biQ+?biB+ BMi2MbBiv 7`QK  biM/`/ ?TTX JQ`2 T`2+Bb2Hv- H2i (Ω, F, P ) #2  T`Q##BHBiv bT+2 QM r?B+? HBp2  Q7 BMi2MbBiv 1 M/ UV M ?TT N X U#V  bm#@σ@}2H/ G ⊂ F BM/2T2M/2Mi Q7 N G2i #2 ;Bp2M  MQM@M2;iBp2 K2bm`#H2 KTTBM; ϕ : (R+ × Ω × Mp (R), B(R+ ) ⊗ G ⊗ Mp (R)) → (R, B(R)) . .2}M2 `2+m`bBp2Hv i?2 b2[m2M+2 {Tn }n≥1 Q7  TQBMi T`Q+2bb N QM (0, ∞) #v  t   ϕ(s, ω, ∅) ds = 1} T1 (ω) := inf{t > 0 ; N 0

r?2`2 ∅ /2MQi2b i?2 2KTiv TQBMi T`Q+2bb UM/ rBi? i?2 mbmH +QMp2MiBQM i?i i?2 BM}KmK Q7 M 2KTiv b2i Bb ∞V- M/ 7Q` n ≥ 1 t   Tn+1 (ω) := inf{t > Tn (ω) ; N ϕ(s, ω, N(0,Tn ] ) ds = 1} Tn (ω)

r?2`2 7Q` HH c > 0- N(0,c] Bb i?2 TQBMi T`Q+2bb i?i +QBM+B/2b rBi? N QM (0, c] M/ Bb 2KTiv QM (c, ∞)X G2i T∞ := limn↑∞ Tn X h?2 TQBMi T`Q+2bb N Bb /2}M2/ #v i?2 #Qp2 +QMbi`m+iBQM mT iQ T∞ UMQi BM+Hm/2/VX Ai ?b- #v /2}MBiBQM- MQ TQBMib QM [T∞ , ∞)X h?2Q`2K 8X9XR .2}M2 7Q` 2+? t ∈ (0, T∞ (ω)) λ(t, ω) := ϕ(t, ω, N(0,t] (ω)) M/ 7Q` t ≥ T∞ (ω)- λ(t, ω) := 0X AM T`iB+mH`   t  (0, λ(s) ds] . N ((0, t]) = N 0

h?2M {λ(t)}t≥0 Bb i?2 biQ+?biB+ Ft @BMi2MbBiv Q7 N X c  N λ(s) ds Bb M F S`QQ7X 6Q` HH c > 0- τ(c) := 0 t ∨ G@biQTTBM; iBK2 bBM+2 7Q` HH c t > 0- i?2 2p2Mi { τ (c) ≤ t} = 0 λ(s) ds ≤ t +M #2 K2bm`#Hv 2tT`2bb2/ BM (0,t] M/ GX h?2`27Q`2 Ft := F N ∨ G ⊆ F N ∨ G := Ft X 6Q` HH 0 ≤ a ≤ bi2`Kb Q7 N t τ(t) HH A ∈ Fa    (( E [1A N ((a, b])] = E 1A N τ (a), τ(b)]) . (†)

RN3 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu HbQ- A ∈ Fa - M/ i?2`27Q`2 i?2 H27i@+QMiBMmQmb biQ+?biB+ T`Q+2bb (t, ω) → 1A (ω)1(τ (ω)(a),τ (ω,b)] (t) Bb Ft @T`2/B+i#H2X h?2`27Q`2 #v i?2 bKQQi?BM; 7Q`KmH- i?2 `B;?i@?M/ bB/2 Q7 U†V Bb 2[mH iQ 

 b

E 1A 1(τ (a),τ (b)] (t) dt = E [1A ( τ (b) − τ(a))] = E 1A λ(s) ds . R

a



8X8 *QMiBMmQmb *?M;2 Q7 S`Q##BHBiv q2 MQr +QMbB/2` +?M;2b Q7 BMi2MbBiv 2MiBH2/ #v M #bQHmi2Hv +QMiBMmQmb +?M;2 Q7 T`Q##BHBiv K2bm`2X G2i (N, Z) #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? K`Fb BM K M/ bbQ+Bi2/ HB7i2/ T`Q+2bb NZ QM R+ × KX G2i {Ft }t≥0 #2  ?BbiQ`v Q7 NZ M/ bmTTQb2 i?i NZ /KBib i?2 (P, Ft )@HQ+H +?`+i2`BbiB+b (λ(t), Φ(t, dz))X G2i {μ(t)}t≥0 #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb M/ H2i {h(t, z)}t≥0,z∈K #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 K@BM/2t2/ biQ+?biB+ T`Q+2bb bm+? i?i 7Q` HH t≥0   t

λ(s)μ(s) ds < ∞

P

U8XjRV

=1

0



M/



P

h(t, z)Φ(t, dz) dz = 1

U8XjkV

= 1.

K

.2}M2 7Q` 2+? t ≥ 0 ⎛ ⎞  L(t) := L(0) ⎝ μ(Tn )h(Tn , Zn )⎠ × · · · Tn ∈(0,t]

  exp −



 (μ(s)h(s, z) − 1)λ(s)Φ(s, dz) ds (0,t]

, U8XjjV

K

r?2`2 L(0) Bb  MQM@M2;iBp2 F0 @K2bm`#H2 `M/QK p`B#H2- E[L(0)] = 1X h?2Q`2K 8X8XR Ue V lM/2` i?2 #Qp2 +QM/BiBQMbURV {L(t)}t≥0 Bb  MQM@M2;iBp2 (P, Ft )@HQ+H K`iBM;H2X A7 KQ`2Qp2` E[L(t) = 1 7Q` HH t ≥ 0- Bi Bb  MQM@M2;iBp2 (P, Ft )@K`iBM;H2X e  bBKBH` `2bmHi TT2`2/ BM i?2 2M;BM22`BM; HBi2`im`2 BM i?2 mMK`F2/ +b2- 7Q` BMbiM+2 BM (_m#BM- RNdk)X ("`ûKm/- RNdk) +QMiBMb i?2 }`bi dz:B`bMQp@ivT2Ǵ `2bmHi Q7 i?Bb FBM/ rBi?  T`QQ7 #b2/ QM i?2 K`iBM;H2 +H+mHmbX 6Q` i?2 2ti2MbBQM iQ K`F2/ TQBMi T`Q+2bb- b22 (C+Q/RNd8) M/ ("Q2H- o`Bv M/ qQM;- RNd8)X

8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu

RNN

UkV A7 E[L(T )] = 1 7Q` bQK2 T > 0- M/ B7 r2 /2}M2 i?2 T`Q##BHBiv Q #v i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 T`Q+2bb dQ = L(T ) , dP

U8Xj9V

i?2 K`F2/ TQBMi NZ /KBib i?2 (Q, Ft )@HQ+H +?`+i2`BbiB+b (μ(t)λ(t), h(t, z)Φ(t, dz)) QM [0, T ]X

S`QQ7X URV "v i?2 2tTQM2MiBH `mH2 Q7 aiB2HiD2bĜG2#2b;m2 +H+mHmb Uam#b2+iBQM RXe- 1t@ KTH2 RXeXNV

 (μ(s)h(s, z) − 1)L(s−)MZ (ds × dz) ,

L(t) = L(0) + (0,t]

K

r?2`2 MZ (ds × dz) := NZ (ds × dz) − λ(s)Φ(s, dz) dsX G2i 7Q` n ≥ 1   t Sn = inf t ; L(t−) + μ(s)λ(s) ds ≥ n .

U8Xj8V

0

h?2M- #v h?2Q`2K 8XRXkN- {L(t ∧ Sn )}t≥0 Bb  (P, Ft ) K`iBM;H2- M/ bBM+2 mM/2` +QM/BiBQMb U8XjRV M/ U8XjkV- P (limn↑∞ Sn = ∞) = 1- {L(t)}t≥0 Bb  (P, Ft )@HQ+H K`iBM;H2X "2BM; MQM@M2;iBp2- Bi Bb HbQ  (P, Ft )@bmT2`K`iBM;H2X "mi  bmT2`@ K`iBM;H2 rBi? +QMbiMi K2M Bb  K`iBM;H2 U1t2`+Bb2 8XNXRdVX UkV q2 ?p2 iQ T`Qp2 i?i 7Q` Mv MQM@M2;iBp2 Ft @T`2/B+i#H2 K@BM/2t2/ biQ+?biB+ T`Q+2bb {H(t, z)}t≥0,z∈K M/ HH t ∈ [0, T ] 



EQ (0,t]

H(s, z)NZ (ds × dz) K

  H(s, z)μ(s)λ(s)h(s, z)Φ(s, dz) ds . = EQ (0,t]

K

h?Bb Bb /QM2 i?`Qm;? i?2 7QHHQrBM; b2[m2M+2 Q7 2[mHBiB2b UrBi? TT`QT`Bi2 DmbiB@ }+iBQMb i i?2 2M/V

kyy *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu 



EQ (0,t]

H(s, z)NZ (ds × dz) K

  H(s, z)NZ (ds × dz) = E L(t) (0,t] K

  L(s)H(s, z)NZ (ds × dz) =E (0,t] K

  L(s−)H(s, z)μ(s)h(s, z)NZ (ds × dz) =E (0,t] K

  L(s−)H(s, z)μ(s)h(s, z)λ(s)Φ(s, dz) ds =E (0,t] K

  L(s)H(s, z)μ(s)h(s, z)λ(s)Φ(s, dz) ds =E (0,t] K

  H(s, z)μ(s)h(s, z)λ(s)Φ(s, dz) ds = E L(t) (0,t] K

  H(s, z)μ(s)λ(s)h(s, z)Φ(s, dz) ds . = EQ (0,t]

K

h?2 }`bi 2[mHBiv 7QHHQrb 7`QK U8Xj9V M/ i?2 7+i i?i 7Q`  MQMM2;iBp2 Ft @ K2bm`#H2 `M/QK p`B#H2 V (t)- EQ [V (t)] = EP [L(t)V (t)]X h?2 b2+QM/ 2[mHBiv 7QHHQrb 7`QK h?2Q`2K XjXd- i?2 i?B`/ QM2 7`QK i?2 Q#b2`piBQM i?i i  TQBMi Tn < ∞ Q7 N - L(Tn ) = L(Tn −)μ(Tn )h(Tn , Zn )X h?2 i?B`/ 2[mHBiv mb2b i?2 bKQQi?@ BM; i?2Q`2K- i?2 }7i? Bb #v h?2Q`2K XjXd M/ i?2 Hbi QM2 mb2b U8Xj9VX  _2K`F 8X8Xk h?2 KBM +QM/BiBQM iQ p2`B7v r?2M mbBM; h?2Q`2K 8X8XR Bb EP [L(T )] = 1X  ;2M2`H K2i?Q/ iQ /Q i?Bb +QMbBbib BM }M/BM; bQK2 γ > 1 bm+? i?i 7Q` i?2 b2[m2M+2 Q7 biQTTBM; iBK2b {Sn }n≥1 /2}M2/ #v U8Xj8Vsup EP [L(T ∧ Sn )γ ] < ∞ . n≥1

h?Bb BKTHB2b i?i i?2 b2[m2M+2 {L(T ∧ Sn )}n≥1 Bb mMB7Q`KHv BMi2;`#H2- M/ i?2`2@ 7Q`2

lim E [L(T ∧ Sn )] = E lim L(T ∧ Sn ) . n↑∞

n↑∞

"mi- #v S`i URV Q7 h?2Q`2K 8X8XR- E [L(T ∧ Sn )] = 1 M/ Sn ↑ ∞X h?2`27Q`2 E [L(T )] = 1X

1tKTH2 8X8Xj, GBF2HB?QQ/ _iBQ 7Q`  aBKTH2 SQBMi S`Q+2bb QM  6BMBi2 AMi2`pHX G2i N #2 mM/2` T`Q##BHBiv P M Ft @SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X G2i {λ(t)}t≥0 #2  MQM@M2;iBp2 #QmM/2/ Ft @T`2/B+i#H2 T`Q+2bbX .2}M2 7Q` HH t≥0    t   λ(Tn ) exp (λ(s) − 1) ds , L(t) = L(0) n≥1

0

8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu

kyR

r?2`2 L(0) Bb  MQM@M2;iBp2 b[m`2 BMi2;`#H2 `M/QK p`B#H2 bm+? i?i E [L(0)2 ] < ∞X "v i?2 2tTQM2MiBH 7Q`KmH Q7 aiB2HiD2bĜG2#2b;m2 +H+mHmb L(t) = L(0) + L(s−)(λ(s) − 1) (N (ds) − ds) . (0,t]

"v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 +H+mHmb   L(s−) dL(s) + L(s−)ΔL(s) L(t)2 = L(0)2 + 2 

(0,t]



(0,t]

= L(0)2 + 2

s≤t



L(s−)2 (λ(s) − 1) N (ds)

L(s−) dL(s) +

= L(0)2 + 2



(0,t]

L(s−)2 (λ(s) − 1) (N (ds) − ds)

L(s−) dL(s) + (0,t]

(0,t]

U8XjeV

 L(s)2 (λ(s) − 1) ds)

+

U8XjdV

(0,t]

UMQiBM; i?i 7Q` G2#2b;m2@HKQbi HH t- L(t) = L(t−)VX .2}M2 7Q` 2+? n ≥ 1    t λ(s) ds ≥ n ∧ Tn , Sn := inf t ; L(t−) + 0

M Ft @biQTTBM; iBK2 bm+? i?i limn↑∞ Sn = ∞X AM T`iB+mH`  L(s−) dL(s) = L(s−)2 (λ(s) − 1) (N (ds) − ds) (0,t]

(0,t]

Bb M Ft @HQ+H K`iBM;H2 UrBi? HQ+HBxBM; b2[m2M+2 {Sn }n≥1 V Q7 K2M 0X _2TH+BM; BM U8XjdV t #v t ∧ Tn M/ iFBM; 2tT2+iiBQMb E L(t ∧ Sn )2 = E L(0)2 + L(s ∧ Sn )2 (λ(s) − 1) ds) . (0,t∧Sn ]

AM T`iB+mH`- BM pB2r Q7 i?2 #QmM/2/M2bb bbmKTiBQM QM i?2 λ(t)

 2 2 2 L(s ∧ Sn ) (λ(s) + 1) ds E L(t ∧ Sn ) ≤ E L(0) + E (0,t]  E L(s ∧ Sn )2 (C2 + 1) ds = C1 + (0,t]

7Q` bQK2 }MBi2 TQbBiBp2 C1 M/ C2 X h?Bb BKTHB2b- #v :`QMrHHǶb H2KK- i?i  t  E L(t ∧ Sn )2 ≤ C1 exp (C2 + 1) ds . 0

AM T`iB+mH`- 7Q` Mv T < ∞- supn≥1 E [L(T ∧ Sn )2 ] < ∞X

kyk *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu

1tKTH2 8X8X9, .2`BpiBp2 Q7 i?2 1tT2+iiBQM Q7  TQBMi T`Q+2bb 6mM+iBQMHX h?2 b2iiBM; Bb +HQb2 iQ i?i Q7 1tKTH2b jXRX8 M/ jX8Xj- rBi? i?2 bK2 Q#D2+iBp2X h?2 T`Q##BHBiv Pα Bb Q#iBM2/ 7`QK i?2 T`Q##BHBiv P1 KFBM; Q7 N  SQBbbQM T`Q+2bb QM [0, 1] rBi? T`2/B+i#H2 FtN @BMi2MbBiv {λ(t, α)} UbbmK2/ #QmM/2/ 7Q` bBKTHB+Biv BM i?Bb 2tKTH2V pB i?2 _/QMĜLBFQ/ɷK /2`BpiBp2    1 dPα = exp λ(t, α) N (dt) − (λ(t, α) − 1) dt . dP1 0 (0,1] bbmKBM; / ?Q+ ?vTQi?2b2b Q7 /Bz2`2MiB#BHBiv M/ BMi2;`#BHBiv- 7Q`KH +QKTm@ iiBQMb ;Bp2 7Q`  #QmM/2/ 7mM+iBQMH f (N )

 dEα [f (N )] ∂ = Eα f (N ) λ(t, α)(N (dt) − dt) . dα (0,1] ∂α

1tKTH2 8X8X8, GBF2HB?QQ/ _iBQb 7Q` *QMiBMmQmb@iBK2 ?K+bX G2i {X(t)}t≥0 #2- mM/2` P -  `2;mH` +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J`FQp +?BM rBi? bii2 bT+2 E M/ bi#H2 M/ +QMb2`piBp2 BM}MBi2bBKH ;2M2`iQ` {qij }i,j∈E X G2i α ij Ui = j ∈ EV #2 MQM@M2;iBp2 MmK#2`b bm+? i?i 7Q` HH i ∈ Eqi := j=i∈E, αij qij < ∞X 6Q` HH t ≥ 0- H2i Ft := FtX M/ H2i  L(t) := L(0)



 N (t) αij i,j

 exp

t



 (αij − 1)qij 1{X(s)=i} ds

.

0 i,j;i=j

i,j;i=j

amTTQb2 i?i EP [L(0)] = 1- i?i EP [L(0)2 ] < ∞ M/ i?i j=i αij qij < ∞ Ui ∈ EVX h?2M EP [L(T )] = 1 M/ mM/2` T`Q##BHBiv Q /2}M2/ #v dQ = L(T )dP i?2 T`Q+2bb {X(t)}t≥0 Bb QM i?2 BMi2`pH (0, T ]  `2;mH` bi#H2 M/ +QMb2`piBp2 +QMiBMmQmb@iBK2 ?QKQ;2M2Qmb J`FQp +?BM rBi? bii2 bT+2 E M/ BM}MBi2bBKH T`K2i2`b q˜ij := αij qij Uj = iVX S`QQ7X i  /Bb+QMiBMmBiv iBK2 t Q7 i?2 +?BM L(t−)(αij − 1)ΔNi,j (t) , L(t) − L(t−) = i,j ; j=i

r?2`2b 7Q` t bi`B+iHv #2ir22M irQ DmKTb Q7 i?2 +?BM dL(t) = L(t) (αij − 1)qij 1{X(t)=i} . dt i,j ; j=i h?2`27Q`2

 L(s−)

L(t) = L(0) + (0,t]

 i=j

(αij − 1)(Nij (ds) − qij 1{X(s)=i} ds) .

8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu

kyj

"v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 +H+mHmb  L(s−) dL(s) + L(s−)ΔL(s) L(t)2 = L(0)2 + 2 

(0,t]

2

= L(0) + 2

s≤t

L(s−) dL(s) (0,t]





L(s−)2

+ (0,t]

 +

i=j

(αij − 1)(Nij (ds) − qij 1{X(s)=i} ds)



(αij − 1)qij 1{X(s)=i} ds .

(0,t] i=j

lbBM; i?2 biQTTBM; iBK2b Q7 ivT2 U8Xj8V- r2 ?p2 i?i   L(s)2 (αij − 1)qij 1{X(s)=i} ds . L(t)2 = L(0)2 + HQ+H K`iBM;H2 + (0,t] i=j

h?2M

E L(t ∧ Sn )

2





= E L(0)

2







+E

 L(s ∧ Sn ) (αij − 1)qij 1{X(s)=i} ds 2

(0,t∧Sn ] i=j

M/ BM T`iB+mH` E L(t ∧ Sn )2 ≤ E L(0)2 +



 E L(s ∧ Sn )2 |αij − 1|qij ds . (0,t]

i=j

h?2`27Q`2- #v :`QMrHHǶb H2KK

E L(t ∧ Sn )

2







≤ E L(0) exp 2





q˜i +

i

M/ }MHHv



 qi t

i

sup E L(T ∧ Sn )2 < ∞ . n≥1



h?2 #Qp2 irQ 2tKTH2b +M #2 ;2M2`HBx2/ b 7QHHQrbX 1tKTH2 8X8Xe, GBF2HB?QQ/ _iBQb 7Q` J`F2/ SQBMi S`Q+2bb2bX *QM@ bB/2` i?2 ;2M2`H bBimiBQM Q7 h?2Q`2K 8X8XRX G2i (N, Z) #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? K`Fb BM K M/ bbQ+Bi2/ HB7i2/ T`Q+2bb NZ QM R+ × KX G2i {Ft }t≥0 #2  ?BbiQ`v Q7 (N, Z) M/ bmTTQb2 i?i (N, Z) /KBib i?2 (P, Ft )@HQ+H +?`+i2`BbiB+b (λ(t), Φ(t, dz))X G2i {μ(t)}t≥0 #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb M/ H2i {h(t, z)}t≥0,z∈K #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 K@BM/2t2/ biQ+?biB+ T`Q+2bb bm+? i?i 7Q` HH t ≥ 0

ky9 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu 



t

λ(s)μ(s) ds < ∞

P

U8Xj3V

=1

0





M/

h(t, z)Φ(t, dz) dz = 1

P

U8XjNV

= 1.

K

6Q` 2+? t ≥ 0- /2}M2 L(t) b BM U8XjjV- rBi? L(0)  MQM@M2;iBp2 F0 @K2bm`#H2 `M/QK p`B#H2 bm+? i?i E[L(0)] = 1X q2 bmTTQb2 BM //BiBQM i?i L(0) Bb b[m`2 BMi2;`#H2 M/ i?i (μ(t) + 1)h(t, z)λ(t) ≤ K(t) , T r?2`2 K : R+ → R+ Bb  /2i2`KBMBbiB+ 7mM+iBQM bm+? i?i 0 K(s) ds < ∞X h?2M E [L(T )] = 1X h?2 T`QQ7 7QHHQrb i?2 bK2 HBM2b b i?2 T`QQ7 BM 1tKTH2 8X8Xj M/ Bb H27i b M 2t2`+Bb2X

a2T`iBQM Q7 .2i2+iBQM M/ 6BHi2`BM; h?2 b2iiBM; Bb i?i Q7 i?2 +m``2Mi b2+iBQM- #mi BM //BiBQM- r2 bmTTQb2 i?i mM/2` T`Q##BHBiv Q- (N, Z) Bb  biM/`/ SQBbbQM T`Q+2bb rBi? BMi2MbBiv 1 M/ BB/ K`Fb Q7 +QKKQM /Bbi`B#miBQM q- i?i Bb- i?2 +Q``2bTQM/BM; TQBMi T`Q+2bb N Z QM R+ × K Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 × qX h?2 +QM/BiBQMb T`2pBHBM; BM h?2Q`2K 8X8XR `2 `2iBM2/ rBi? i?2 //BiBQMH bbmKTiBQM i?i EQ L(T )2 < ∞ r?B+? ;m`Mi22b i?i L(T ) Bb +imHHv  _/QMĜLBFQ/ɷK /2`BpiBp2 U_2K`F 8X8XkVX h?2Q`2K 8X8Xd lM/2` i?2 #Qp2 +QM/BiBQMb- 7Q` HH t ∈ [0, T ]⎞ ⎛  μ(Tn )h(Tn , Zn )⎠ × · · · L(t) := ⎝ Tn ∈(0,t]

  exp −



 (μ(s)h(s, z) − 1)λ(s)q(dz) ds (0,t]

,

U8X9yV

K

   := EQ L(t) | FtN,Z X Bb  p2`bBQM Q7 L(t)  S`QQ7X aBM+2 {L(t)}t∈[0,T ] Bb  b[m`2@BMi2;`#H2 (Q, Ft )@K`iBM;H2- {L(t)} t∈[0,T ] Bb N,Z  b[m`2@BMi2;`#H2 (Q, Ft )@K`iBM;H2X "2BM;  b[m`2@BMi2;`#H2 K`iBM;H2- Bi Bb mMB7Q`KHv BMi2;`#H2- r?B+? HHQrb mb iQ mb2 h?2Q`2K XjXd b i?2 M22/ `Bb2bX "v h?2Q`2K 8XjXR QM i?2 bi`m+im`2 Q7 SQBbbQMBM b[m`2@BMi2;`#H2 (Q, FtN,Z )@ K`iBM;H2b       EQ L(t) | FtN,Z = E L(T ) | FtN,Z = 1 + K(s, z)M (ds × dz) (0,t]

K

8X8X *PLhALlPla *>L:1 P6 S_P""AGAhu

ky8

Ur?2`2 M (ds × dz) := M (ds × dz) − Q(dz) dsV 7Q` bQK2 FtN,Z @T`2/B+i#H2 BM/2t2/ T`Q+2bb K(t, z) bm+? i?i

  EQ |K(s, z)|2 q(dz) ds < ∞ . (0,T ]

K

hQ T`Qp2 i?2 i?2Q`2K- Bi bm{+2b iQ b?Qr i?i 7Q` Mv b[m`2@BMi2;`#H2 FtN,Z @ K`iBM;H2 {m(t)}t∈[0,1]   () EQ [L(t)m(t)] = EQ L(t)m(t) . "mi Uh?2Q`2K 8XjXRV





H(s, z)M (ds × dz) ,

m(t) = m(0) + (0,t]

K

7Q` bQK2 FtN,Z @T`2/B+i#H2 BM/2t2/ T`Q+2bb H(t, z) bm+? i?i

  EQ |H(s, z)|2 q(dz) ds < ∞ . (0,T ]

K

6`QK UV r2 i?2M ;2i    

  EQ L(s−)(μ(s)h(s, z) − 1) M (ds × dz) H(s, z)M (ds × dz) (0,t] K (0,t] K    

  K(s, z) M (ds × dz) H(s, z)M (ds × dz) . EQ (0,t]

K

(0,t]

K

"v BbQK2i`v M/ BM pB2r Q7 h?2Q`2K XjXd    

  EQ L(s−)(μ(s)h(s, z) − 1) M (ds × dz) H(s, z)M (ds × dz) (0,t] K (0,t] K

  L(s−)(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ (0,t] K  

L(s)(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ (0,t] K

  (μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ L(t) (0,t] K

  (μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EP (0,t] K  

(μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EP (0,t] K

   μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ L(t) (0,t] K  

 μ(s)h(s, z) − 1)H(s, z) ds q(dz) = EQ L(s) (0,t] K  

 L(s−) μ(s)h(s, z) − 1)H(s, z) ds q(dz) . = EQ (0,t]

K

kye *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu amKK`BxBM;, 



EQ (0,t]

 L(s−)(μ(s)h(s, z) − 1)H(s, z) ds q(dz) K  

K(s, z)H(s, z) ds q(dz) = EQ (0,t]

K

7Q` HH #QmM/2/ FtN,Z @T`2/B+i#H2 BM/2t2/ T`Q+2bb2b H(t, z)X h?2`27Q`2  L(s−)( μ(s)h(s, z) − 1) = K(s, z) M/  =1+ L(t)

  K

 L(s−)( μ(s)h(s, z) − 1) M (ds × dz) , (0,t]

7`QK r?B+? i?2 `2bmHi 7QHHQrbX



8Xe 1ti2MbBQM Q7 i?2 h?2Q`v Q7 aiQ+?biB+ AMi2MbBiv t h?Bb 2ti2MbBQM +QMbBbib BM `2TH+BM; i?2 BMi2;`i2/ BMi2MbBiv T`Q+2bb { 0 λ(s) ds}t≥0 Ud V #v  +Q`HQH MQM@/2+`2bBM; T`2/B+i#H2 T`Q+2bb {A(t)}t≥0 bm+? i?i {N (0, t] − A(t)}t≥0 Bb  HQ+H K`iBM;H2X h?i bm+?  T`Q+2bb /Q2b 2tBbi mM/2` [mBi2 ;2M2`H +QM/BiBQMb Bb  /B`2+i +QMb2[m2M+2 Q7  `2bmHi 7`QK i?2 ;2M2`H i?2Q`v Q7 biQ+?biB+ T`Q+2bb2b Uh?2Q`2K 8XeXj #2HQrVX AM Q`/2` iQ bii2 i?Bb `2bmHi BM 7mHH `B;Q`-  /2}MBiBQM Bb BM Q`/2`X .2}MBiBQM 8XeXR  ?BbiQ`v {Ft }t≥0 /2}M2/ QM i?2 T`Q##BHBiv bT+2 (Ω, F, P ) Bb bB/ iQ biBb7v i?2 mbmH +QM/BiBQMb B7 Bi Bb `B;?i@+QMiBMmQmb M/ F0 +QMiBMb HH i?2 P @M2;HB;B#H2 b2ibX h?2 +QM/BiBQM +QM+2`MBM; i?2 M2;HB;B#H2 b2ib ?b MQ BKT+i BM TTHB+iBQMb bBM+2 bm+? 2p2Mib `2 MQi Q#b2`p#H2- r?2`2b i?2 `B;?i@+QMiBMmBiv bbmKTiBQM 7Q` i?2 ?BbiQ`v Bb ;`Mi2/ BM i?2 +b2 Q7 BMi2`2bi BM i?Bb #QQF- b i?2 M2ti 2tKTH2 `2KBM/b mbX 1tKTH2 8XeXk, h?2 >BbiQ`v Q7  J`F2/ SQBMi T`Q+2bbX A7 (N, Z) Bb  K`F2/ TQBMi T`Q+2bb M/ F0 Bb M `#Bi``v σ@}2H/ +QMiBMBM; HH i?2 P @M2;HB;B#H2 (N,Z) b2ib- i?2 ?BbiQ`v {F0 ∨ Ft }t≥0 biBb}2b i?2 mbmH +QM/BiBQMb Uh?2Q`2K XkXR8VX h?2 M2ti i?2Q`2K Bb QM2 Q7 i?2 7mM/K2MiH `2bmHib Q7 i?2 ;2M2`H i?2Q`v Q7 biQ+?biB+ T`Q+2bb2bX d

AM i?Bb b2+iBQM- i?2 iBK2 BM/2t Bb i?2 b2i Q7 MQM@M2;iBp2 MmK#2`bX

8XeX 1sh1LaAPL P6 h>1 h>1P_u P6 ahP*>ahA* ALh1LaAhu

kyd

h?2Q`2K 8XeXj G2i {N (t)}t≥0 #2  `B;?i@+QMiBMmQmb MQM@/2+`2bBM; T`Q+2bb /Ti2/ iQ i?2 ?BbiQ`v {Ft }t≥0 biBb7vBM; i?2 mbmH +QM/BiBQMbX h?2M- i?2`2 2tBbib  mMB[m2 `B;?i@+QMiBMmQmb MQM@/2+`2bBM; Ft @T`2/B+i#H2 T`Q+2bb {A(t)}t≥0 bm+? i?i {N (t) − A(t)}t≥0 Bb  HQ+H (P, Ft )@K`iBM;H2- +HH2/ i?2 Ft @+QKT2MbiQ` Q7 {N (t)}t≥0 X 6`QK MQr QM i?2 dzmbmH +QM/BiBQMbǴ `2 Hrvb bbmK2/X h?2Q`2K 8XeXj TTHB2b BM T`iB+mH` iQ i?2 +QmMiBM; T`Q+2bb {N (t)}t≥0 Q7  TQBMi T`Q+2bbX q?2M i?Bb +QKT2MbiQ` ?b i?2 7Q`K  A(t) =

t

λ(s) ds , 0

r?2`2 {λ(t)}t≥0 Bb  MQM@M2;iBp2 Ft @T`Q;`2bbBp2 biQ+?biB+ T`Q+2bb- r2 `2i`B2p2 i?2 /2}MBiBQM Q7 i?2 Ft @BMi2MbBiv Q7  TQBMi T`Q+2bbX G2i (N, Z) #2  K`F2/ TQBMi T`Q+2bb QM R+ rBi? K`Fb BM i?2 K2bm`#H2 bT+2 (K, K)- M/ H2i NL #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 TQBMib rBi?  K`F BM L ∈ KX "v h?2Q`2K 8XeXj- i?2`2 2tBbib 7Q` HH L ∈ K  mMB[m2 `B;?i@+QMiBMmQmb MQM@/2+`2bBM; Ft @T`2/B+i#H2 T`Q+2bb {AL (t)}t≥0 bm+? i?i {NL (t) − AL (t)}t≥0 Bb  HQ+H (P, Ft )@K`iBM;H2X AM 7+i- i?2 T`Q+2bb2b AL UL ∈ KV +M #2 +?Qb2M3 BM bm+?  rv i?i i?2`2 2tBbib  `M/QK K2bm`2 A QM (R+ × K, B(R+ ) ⊗ K) bm+? i?i 7Q` 2+? L ∈ K- {NZ ((0, t] × L) − A((0, t] × L)}t≥0 Bb  HQ+H Ft @K`iBM;H2X h?Bb Bb bmKK`Bx2/ BM i?2 7QHHQrBM; 7Q`K, h?2Q`2K 8XeX9 G2i (N, Z) #2  K`F2/ TQBMi T`Q+2bb QM R+ rBi? K`Fb BM i?2 K2bm`#H2 bT+2 (K, K)- M/ H2i NL #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 TQBMib rBi?  K`F BM L ∈ KX h?2M i?2`2 2tBbib  `M/QK K2bm`2 A QM (R+ × K, B(R+ ) ⊗ K) bm+? i?i 7Q` 2+? L ∈ K- {A((0, t] × L)}t≥0 Bb Ft @T`2/B+i#H2 M/ {NZ ((0, t] × L) − A((0, t] × L)}t≥0 Bb  HQ+H Ft @K`iBM;H2X .2}MBiBQM 8XeX8 h?2 `M/QK K2bm`2 A BM i?2 bii2K2Mi Q7 h?2Q`2K 8XeX9 Bb +HH2/ i?2 BMi2;`i2/ Ft @BMi2MbBiv F2`M2H Q7 i?2 K`F2/ TQBMi T`Q+2bb (N, Z)X h?2 M2ti `2bmHi Bb i?2 Mim`H 2ti2MbBQM Q7 h?2Q`2K 8XRXjR M/ Bb T`Qp2/ BM 2t+iHv i?2 bK2 rvX h?2Q`2K 8XeXe G2i (N, Z) #2  K`F2/ TQBMi T`Q+2bb QM R+ rBi? K`Fb BM i?2 K2bm`#H2 bT+2 (K, K) M/ /KBiiBM; i?2 BMi2;`i2/ Ft @BMi2MbBiv F2`M2H A M/ H2i H #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb BM/2t2/ #v KX h?2M 

 H(s, z) NZ (ds × dz) = E



E R+ 3

(C+Q/- RNd8)X

K

H(s, z) A(ds × dz) .

 R+

K

ky3 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu amTTQb2 i?i 7Q` HH n ≥ 0- HH L ∈ K M/ HH C ∈ B(R+ )P (Sn+1 ∈ C , Zn+1 ∈ L | FTn ) (ω) := G(n+1) (ω, C, L) , r?2`2 7Q` HH C ∈ B(R+ ) M/ HH L ∈ K- i?2 KTTBM; ω → G(n+1) (ω, C, L) Bb FTn @K2bm`#H2- M/ r?2`2 7Q` HH ω ∈ Ω M/ HH L ∈ K- i?2 KTTBM; C → G(n+1) (ω, C, L) Bb  bm#@T`Q##BHBiv K2bm`2 QM (R+ , B(R+ ))X G2i G(n+1) (ω, C) := G(n+1) (ω, C, K)X h?2Q`2K 8XeXd h?2 biQ+?biB+ Ft @F2`M2H A Q7 i?2 K`F2/ TQBMi T`Q+2bb (N, Z) Bb ;Bp2M #v  G(n+1) (dx, L) A((0, t], L) = A((0, Tn ], L) + (n+1) ([x − Tn , +∞]) (0,t−Tn ] G (t ∈ (Tn , Tn+1 ]L ∈ K) . h?2 T`QQ7 7QHHQrb i?i Q7 h?2Q`2K 8XkXj- KmiiBb KmiM/BbX q2 b?HH MQi bT2M/ KQ`2 iBK2 QM i?Bb Mim`H 2ti2MbBQM Q7 i?2 i?2Q`v Q7 biQ+?b@ iB+ BMi2MbBivXN

8Xd :`B;2HBQMBbǶ _2T`2b2MiiBQM  MQM@?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? U/2i2`KBMBbiB+V BMi2MbBiv 7mM+iBQM λ(t) +M #2 Q#iBM2/ #v T`QD2+iBM; QMiQ i?2 iBK2 tBb i?2 TQBMib Q7  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R2 Q7 BMi2MbBiv 1 r?B+? HB2 #2ir22M i?2 +m`p2 y = λ(t) M/ i?2 iBK2 tBb U1t2`+Bb2 jX3XRyVX AM 7+i- Mv TQBMi T`Q+2bb rBi? biQ+?biB+ Ft @BMi2MbBiv {λ(t)}t∈R MQi QMHv +M #2 Q#iBM2/ BM i?Bb rv Ui?2 /B`2+i BK#2//BM; i?2Q`2KbV #mi +M Hrvb #22M i?Qm;?i Q7 b ?pBM; #22M Q#iBM2/ BM i?Bb rv Ui?2 BMp2`b2 BK#2//BM; i?2Q`2KbVXRy h?2 2t+i 7Q`KmHiBQM M/ i?2 Ki?2KiB+H /2iBHb rBHH #2 ;Bp2M BM i?2 M2ti b2+iBQMX 6Q` i?Bb i?2 7QHHQrBM; T`2HBKBM`B2b Q7 BMi`BMbB+ BMi2`2bi `2 M22/2/X M 1ti2MbBQM Q7 qiM#2Ƕb h?2Q`2K h?2 Q`B;BMH p2`bBQM Q7 qiM#2Ƕb i?2Q`2K +QM+2`Mb ?QKQ;2M2Qmb SQBbbQM T`Q@ +2bb2b QM i?2 HBM2X Ai rBHH #2 2ti2M/2/- rBi?  T`QQ7 bBKBH` iQ i?i Q7 h?2Q`2K kX9XR- iQ SQBbbQM T`Q+2bb2b QM T`Q/m+i bT+2b Q7 i?2 ivT2 R × KX h?Bb M2r p2`bBQM rBHH THv  +2Mi`H `QH2 BM i?2 T`QQ7 Q7 i?2 BK#2//BM; i?2Q`2Kb BM i?Bb b2+iBQMX G2i N #2  TQBMi T`Q+2bb QM R × KX _2+HH i?2 MQiiBQM St N Ut ∈ RV 7Q` i?2 TQBMi T`Q+2bb Q#iBM2/ #v b?B7iBM; N #v t iQ i?2 H27i UH;2#`B+HHvVX G2i St N + /2MQi2 i?2 TQBMi T`Q+2bb Q#iBM2/ #v `2bi`B+iBM; i?2 b?B7i2/ T`Q+2bb St N iQ R+ ×KX GQQb2Hv bT2FBM;- St N + Bb i?2 7mim`2 Q7 N 7i2` iBK2 t- M/ KQ`2 7Q`KHHvSt N + ([a, b] × L) := N (R+ ∩ [a + t, b + t] × L) N Ry

6Q`  +QKTH2i2 ++QmMi- b22 (Gbi M/ "`M/i- RNN8)X (:`B;2HBQMBb- RNdR)X

([a, b] ⊂ R, L ∈ K) .

8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL

kyN

PM2 +M bBKBH`Hv /2}M2 St N − - i?2 `2bi`B+iBQM Q7 St N iQ R− × KX h?2 7QHHQrBM; p2`bBQM Q7 qiM#2Ƕb i?2Q`2K +QM+2`Mb BM T`iB+mH` *Qt T`Q+2bb2b QM R2 X h?2Q`2K 8XdXR G2i (K, K) #2 M `#Bi``v K2bm`#H2 bT+2X G2i G #2 bQK2 σ@ }2H/ Q7 FX G2i λ(t, dz) #2  HQ+HHv BMi2;`#H2 F2`M2H 7`QK R × Ω iQ (K, K) bm+? i?i 7Q` HH t ∈ R M/ HH L ∈ K- λ(t, L) Bb G@K2bm`#H2X G2i N #2  TQBMi T`Q+2bb QM R × K i?i /KBib i?2 Ft @BMi2MbBiv F2`M2H λ(t, ·)- r?2`2 Ft = Ht ∨ G M/ {Ht }t∈R Bb  ?BbiQ`v Q7 N X h?2M N Bb- +QM/BiBQMHHv QM G-  SQBbbQM T`Q+2bb rBi? i?2 BMi2MbBiv K2bm`2 ν(dt × dz) = λ(t, dz) × dtX 6m`i?2`KQ`2- +QM/BiBQMHHv QM G- 7Q` HH t ∈ R- i?2 7mim`2 St N + Q7 N 7i2` iBK2 t Bb BM/2T2M/2Mi Q7 Ht X S`QQ7X Ai bm{+2b iQ b?Qr i?i 7Q` HH a ∈ R- 7Q` Mv }MBi2 7KBHv Q7 /BbDQBMi K2bm`#H2 b2ib C1 , . . . , Cm ⊂ (a, +∞) × K- HH t1 , . . . , tm ∈ R+    m  m     E exp − tj N (Cj ) |Fa = exp ν(Cj )(e−tj − 1) . U8X9RV j=1

j=1

PM2 Kv bbmK2 i?i C1 , . . . , Cm ⊂ (a, b] × Lk 7Q` bQK2 Lk b BM .2}MBiBQM 8XRXR3X Pi?2`rBb2- `2TH+2 i?2 Cj Ƕb #v Cj ∩ ((a, b] × Lk ) M/ H2i b M/ k ;Q iQ BM}MBivX .2MQi2 i?2 #Qp2 Lk #v LX 6Q` HH j U1 ≤ j ≤ mV M/ HH t ∈ R- H2i Cj (t) := Cj ∩ {(−∞, t] × K} M/ Cjt := {z ∈ K; (t, z) ∈ Cj }. .2}M2 7Q` t ≥ a

 Z(t) := exp −

m 

 .

U8X9kV

− 1)1Cjs (z) N (ds × dz) .

U8X9jV

tj N (Cj (t))

j=1

AM T`iB+mH`-

 Z(b) = exp −

m 

 tj N (Cj )

.

j=1

HbQ- bBM+2 Z(a) = 1



 Z(s−)

Z(t) = 1 + (a,t]

K

m 

 (e

−tj

j=1

6Q` i?2 T`QQ7 Q7 i?Bb 2[mHBiv- Q#b2`p2 i?i Mv i`D2+iQ`v t → Z(t) Bb TB2+2@ rBb2 +QMbiMi rBi? /Bb+QMiBMmBiv iBK2b i?i `2 TQBMib Q7 i?2 bBKTH2 TQBMi T`Q+2bb NL (·) := N (· × L)X h?2`27Q`2  Z(t) = Z(a) + (Z(s) − Z(s−)) , s∈(a,t]

kRy *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu r?2`2 Z(s) = Z(s−) QMHv B7 i?2`2 Bb  TQBMi (s, z) Q7 N i?i #2HQM;b iQ Ui KQbiV QM2 Q7 i?2 Cj ǶbX A7 (s, z) ∈ Cj M/ Bb BM N - Z(s) = Z(s−)e−tj X LQr bvBM; i?i (s, z) ∈ Cj Bb 2[mBpH2Mi iQ bvBM; i?i z ∈ Cjs - M/ i?2`27Q`2- B7 (s, z) Bb  TQBMi Q7 N Z(s) − Z(s−) =

m 

Z(s−)(e−tj − 1)1Cjs (z),

j=1

r?B+? i?2M ;Bp2b U8X9jV bBM+2 Z(a) = 1X G2i MQr A ∈ Fa X q2 ?p2 

  1A Z(t) = 1A +

1A Z(s−) (a,t]×K

m 

 (e

−tj

− 1)1Cjs (z) N (ds × dz) .

j=1

h?2 biQ+?biB+ T`Q+2bb BM/2t2/ #v K  H(t, ω, z) := 1A (ω)1(a,t] (t)Z(t− , ω) ×

m 

 (e−tj − 1)1Cjt (z)

j=1

Bb P(F· ) ⊗ K K2bm`#H2 M/ Q7 +QMbiMi bB;M UM2;iBp2VX h?2`27Q`2   m    −tj 1A Z(s−) (e − 1)1Cjs (z) N (ds × dz) E[1A Z(t)] = P (A) + E (a,t]

K

  t = P (A) + E



1A Z(s) 

a

K

(e



−tj

j=1

 t

= P (A) + E 1A

Z(s) a

j=1 m 

 m 

K



− 1)1Cjs (z) λ(s, dz) ds 



(e−tj − 1)1Cjs (z) λ(s, dz) ds .

j=1

aBM+2 A Bb `#Bi``v BM Fa     m  t  −tj Z(s) (e − 1)1Cjs (z) λ(s, dz) ds|Fa E[Z(t)|Fa ] = 1 + E a

K

j=1



 t = 1+

E[Z(s)|Fa ] a

K

m 

 (e

−tj

− 1)1Cjs (z) λ(s, dz) ds .

j=1

h?2`27Q`2  E[Z(t)|Fa ] = exp

m 

(e

−tj



 t  − 1)

j=1

G2iiBM; t = b ;Bp2b i?2 MMQmM+2/ `2bmHiX

a

K

1Cjs (z)λ(s, dz)ds

. 

8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL

kRR

:`B;2HBQMBbǶ AK#2//BM; h?2Q`2Kb G2i (K, K) #2 bQK2 K2bm`#H2 bT+2 M/ H2i {Ft }t∈R #2  ?BbiQ`vX  bHB;?i 2ti2MbBQM Q7 i?2 /2}MBiBQM Q7  SQBbbQM T`Q+2bb rBHH #2 M22/2/, .2}MBiBQM 8XdXk h?2 TQBMi T`Q+2bb N QM R × K Bb +HH2/ M Ft @SQBbbQM T`Q+2bb B7 i?2 7QHHQrBM; +QM/BiBQMb `2 biBb}2/, UBV {Ft }t∈R Bb  ?BbiQ`v Q7 N c UBBV N Bb  SQBbbQM T`Q+2bbc M/ UBBBV 7Q` Mv t ≥ 0- St N + M/ Ft `2 BM/2T2M/2MiX h?2 M2ti `2bmHi Bb i?2 /B`2+i BK#2//BM; i?2Q`2K, h?2Q`2K 8XdXj G2i (K, K) #2 bQK2 K2bm`#H2 bT+2 M/ H2i Q #2 bQK2 T`Q#@ #BHBiv K2bm`2 QM BiX G2i N #2 M Ft @SQBbbQM T`Q+2bb QM R × K × R+ rBi? BMi2MbBiv K2bm`2 dt × Q(dz) × dsX G2i f : Ω × R × K → R #2  MQM@M2;iBp2 7mM+iBQM i?i Bb P(F· ) ⊗ K@K2bm`#H2 M/ bm+? i?i i?2 F2`M2H λ(t, dz) := f (t, z)Q(dz) Bb HQ+HHv BMi2;`#H2X h?2 K`F2/ TQBMi T`Q+2bb (N, Z) rBi? K`Fb BM K /2}M2/ #v N (dt × dz) := N ((dt × dz × [0, f (t, z)]) /KBib i?2 Ft @biQ+?biB+ BMi2MbBiv F2`M2H λ(t, dz)X S`QQ7X h?2 T`QQ7 Bb /QM2 7Q` i?2 mMK`F2/ +b2X h?2 T`QQ7 BM i?2 K`F2/ +b2 7QHHQrb 2t+iHv i?2 bK2 HBM2b M/ Bb H27i 7Q` i?2 `2/2`X G2i N #2  ?QKQ;2M2Qmb Ft @SQBbbQM T`Q+2bb QM R × R+ rBi? p2`;2 BMi2M@ bBiv RX G2i {λ(t)}t∈R #2  MQM@M2;iBp2 HQ+HHv BMi2;`#H2 Ft @T`2/B+i#H2 biQ+?biB+ T`Q+2bbX .2}M2 i?2 TQBMi T`Q+2bb N QM R #v i?2 7Q`KmH   N (C) = 1(0,λ(t)] (z)N (dt × dz) U8X99V C

R+

7Q` HH C ∈ BX h?2M- N ?b i?2 Ft @BMi2MbBiv {λ(t)}t∈R X RX q2 }`bi b?Qr i?i U8X99V /2}M2b  HQ+HHv }MBi2 TQBMi T`Q+2bb- i?i Bb N ((0, b]) < ∞ XbX 7Q` HH b ∈ RX .2}M2 7Q` HH n ≥ 1    t λ(s)ds ≥ n τn = inf t ≥ 0 ; 0

U4∞ B7 {. . . } = ∅VX "v i?2 HQ+H BMi2;`#BHBiv bbmKTiBQM- limn↑∞ τn = ∞- XbX HbQ τn Bb  Ft @biQTTBM; iBK2X "v i?2 bKQQi?BM; 7Q`KmH Q7 h?2Q`2K 8XRXky-

kRk *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu 





E [N ((0, τn ∧ b])] = E

1(0,τn ] (s)1(0,λ(s)] (z) N (ds × dz) (0,b]×R

  1(0,τn ] (s)1(0,λ(s)] (z) ds dz =E 

(0,b]×R τn ∧b



λ(s) ds < ∞ .

=E 0

h?2`27Q`2- XbX- 7Q` HH n ≥ 1N (0, τn ∧ b] < ∞ . kX h?2 bBKTHB+Biv Bb H27i b M 2t2`+Bb2 7Q` i?2 `2/2`X jX AM Q`/2` iQ T`Qp2 i?i N ?b i?2 Ft @BMi2MbBiv {λ(t)}t∈R Bi bm{+2b iQ b?Qr i?i 7Q` HH H ∈ P(F· )- H ≥ 0



 H(t)N (dt) = E H(t)λ(t)dt . E R

R

"mi i?2 H27i@?M/ bB/2 Q7 i?Bb 2[mHBiv `2/b  

E H(t)1(0,λ(t)] (z)N (dt × dz) . R

R+

aBM+2 N Bb bbmK2/ Ft @SQBbbQM- Bi /KBib i?2 Ft @BMi2MbBiv F2`M2H λ(t, dz) = dzX h?mb- #v h?2Q`2K 8XRXky- i?Bb Bb HbQ 2[mH iQ  



E H(t)1(0,λ(t)] (z)dtdz = E H(t)λ(t)dt . R

R+

R

 1tKTH2 8XdX9, *QmTHBM; Q7 .2Hv2/ M/ lM/2Hv2/ _2M2rH S`Q+2bb2bX G2i N #2  `2M2rH T`Q+2bb rBi?  +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F Q7 i?2 BMi2`@`2M2rH iBK2b /KBiiBM;  T`Q##BHBiv /2MbBiv f rBi? mM#QmM/2/ bmTTQ`i M/ /2+`2bBM; 7BHm`2 `i2 U/7`V r(t) :=

f (t) . 1 − F (t)

ULQi2 i?i i?2 /7` T`QT2`iv BKTHB2b mM#QmM/2/M2bb Q7 i?2 bmTTQ`i Q7 F c 1t2`+Bb2 8XNXkjXVX h?2 +QMbi`m+iBQMRR mb2b  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Π QM R+ × R+ Q7 BMi2MbBiv 1X 6B`bi 7Q` i?2 mM/2Hv2/ +b2 UT0 ≡ 0V i?2 TQBMib T1 - T2 - . . . Q7 i?2 mM/2Hv2/ `2M2rH T`Q+2bb `2 Q#iBM2/ `2+m`bBp2Hv b 7QHHQrb, 7Q` n ≥ 1, Tn = inf{t > Tn−1 ; i?2`2 Bb  TQBMi Q7 Π Q7 i?2 7Q`K (t, z), z ≤ r(t − Tn−1 )} , r?2`2 T0 = 0X h?2 /2Hv2/ TQBMi T`Q+2bb 7i2` Bib }`bi TQBMi T0 - T1 - T2 - . . . Bb +QMbi`m+i2/ BM  bBKBH` rv- mbBM; i?2 bK2 SQBbbQM T`Q+2bb Π, 7Q` n ≥ 1, RR

(GBM/pHH- RNNk)- p kR@kk UTX RNjzVX

8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL

kRj

 Tn = inf{t > Tn−1 ; i?2`2 Bb  TQBMi Q7 Π Q7 i?2 7Q`K (t, z), z ≤ r(t − Tn−1 )} ,

r?2`2 T0 ≥ 0 Bb i?2 BMBiBH /2HvX G2i V0 #2 i?2 Hbi TQBMi Q7 i?2 mM/2Hv2/ T`Q+2bb #27Q`2 T0 M/ H2i V1 i?2 }`bi TQBMi Q7 i?2 mM/2Hv2/ T`Q+2bb 7i2` V0 UM/ i?2`27Q`2 7i2` T0 VX V1 rBHH HbQ #2  TQBMi Q7 i?2 /2Hv2/ TQBMi T`Q+2bb bBM+2 7Q` t ≥ T0 r(t − V0 ) ≤ r(t − T0 ) M/ 7`QK i?2M QM i?2 irQ `2M2rH T`Q+2bb2b +QBM+B/2X h?2 7+i i?i i?2`2 2tBbib M 2t+i +QmTHBM; Bb FMQrM Ur2 `2 BM i?2 bT`2/@Qmi +b2V#mi ?2`2 r2 +M BM T`BM+BTH2 +QKTmi2 i?2 /Bbi`B#miBQM Q7 i?2 +QmTHBM; iBK2X qBi?Qmi +QKTmiiBQM- r2 BKK2/Bi2Hv `2K`F i?i V1 Bb biQ+?biB+HHv bKHH2` i?i T0 + X r?2`2 X Bb BM/2T2M/2Mi Q7 T0 M/ Q7 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F X 1tKTH2 8XdX8, h?BMMBM;  _2M2rH SQBMi S`Q+2bbX G2i N0 #2 M mM/2@ Hv2/ `2M2rH b2[m2M+2 QM (0, +∞), 7Q` n ≥ 1- X0,n = S1 + · · · + Sn r?2`2 i?2 b2[m2M+2 Q7 MQM@M2;iBp2 `M/QK p`B#H2b {Sn }n≥1 Bb BB/- rBi?  +QKKQM /Bbi`B@ #miBQM /KBiiBM;  /2MbBiv f M/  +Q``2bTQM/BM; 7BHm`2 `i2 r(t) := 1− ft (t) 0 f (s) ds mMB7Q`KHv #QmM/2/ #v M < ∞X h?2 biQ+?biB+ BMi2MbBiv Q7 bm+? TQBMi T`Q+2bb Bb λ(t) = r(t − θt ) r?2`2 θt Bb i?2 TQbBiBQM Q7 i?2 Hbi TQBMi Q7 N0 i?i Bb < tX A7 QM2 ?b i /BbTQbBiBQM  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM i?2 bi`BT (0, +∞) × [0, M ] rBi? BMi2MbBiv 1i?2 biM/`/ `2+m`bBp2 T`Q+2/m`2 iQ ;2M2`i2 N0 - #b2/ QM i?2 `2T`2b2MiiBQM UVBb i?2 7QHHQrBM;X :Bp2M Xn - Xn+1 Bb i?2 bKHH2bi t > Xn bm+? i?i N ?b  TQBMi #2HQr i?2 +m`p2 y = r(t − Xn )- Ut ≥ Xn VX Pm` ;QH Bb iQ bKTH2 i?2 #Qp2 `2M2rH T`Q+2bb i?BMM2/ rBi? i?2 i?BMMBM; ∞ T`Q##BHBiv 7mM+iBQM p(t) bm+? i?i 0 p(t) dt < ∞X 6Q` i?Bb- i?2 i?BMMBM; rBHH #2 /QM2 #27Q`2 i?2 +QMbi`m+iBQM Q7 i?2 #bB+ `2M2rH T`Q+2bbX h?Bb Bb ?QrX 6B`bi +QMbi`m+i  }MBi2 U/m2 iQ i?2 BMi2;`#BHBiv +QM/BiBQM QM i?2 i?BMMBM; T`Q##BHBiv 7mM+iBQMV SQBbbQM TQBMi T`Q+2bb QM i?2 TQbBiBp2 HBM2 Q7 BMi2MbBiv M p(t)- r?Qb2 TQBMib `2 t1 , . . . , tk BM i?Bb Q`/2`X // iQ i?2b2 TQBMib i?Qb2 Q7  SQBbbQM T`Q+2bb Q7 BMi2MbBiv M (1 − p(t)) iQ Q#iBM  b2[m2M+2 t1 , t2 , . . . Ui?2`2 Bb M BM}MBiv Q7 i?2K #mi QMHv i?Qb2 #27Q`2 tk rBHH #2 mb2/VX J2`;2 i?Bb b2[m2M+2 rBi? i?2 b2[m2M+2 t1 , . . . , tk X h?Bb K2`;BM; T`Q/m+2b  bKTH2 Q7  SQBbbQM T`Q+2bb Q7 BMi2MbBiv M QM i?2 iBK2 tBb, T1 , T2 , . . .X A7 Tn ∈ {t1 , . . . , tk }- b2i Xn = 1- Qi?2`rBb2 b2i Xn = 0X :Bp2M T1 , T2 , . . .- i?2 b2[m2M+2 {Xn }n≥1 Bb BM/2T2M/2Mi M/ i?2 T`Q##BHBiv i?i Xn = 1 Bb p(Tn )X LQr H2i {Wn }n≥1 #2 M BB/ b2[m2M+2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1)X h?2 b2@ [m2M+2 {Tn , Wn }n≥1 Q7 TQBMib Q7 R2 7Q`K  SQBbbQM T`Q+2bb N Q7 BMi2MbBiv 1 QM i?2 bi`BT R+ × (0, M )X *QMbi`m+i i?2 `2M2rH TQBMi T`Q+2bb, N0 (dt) := N (dt × (0, r(t − θt )) , r?2`2 θt Bb i?2 Hbi TQBMi Q7 N0 i?i Bb < t Q` 0 B7 N0 ((0, t)) = 0X  TQBMi Q7 N0 Bb bQK2 Tn X E22T Bi B7 M/ QMHv B7 i?2 +Q``2bTQM/BM; Xn Bb 1X h?2 bm`pBpBM; TQBMib `2 i?2 TQBMib Q7  `2M2rH T`Q+2bb rBi? 7BHm`2 `i2 r(t) i?BMM2/ rBi? i?2 T`Q##BHBiv 7mM+iBQM p(t)X

kR9 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu h?2 M2ti `2bmHi Bb i?2 BMp2`b2 BK#2//BM; i?2Q`2K, h?2Q`2K 8XdXe G2i (N, Z) #2  HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb rBi? K`Fb BM i?2 K2bm`#H2 bT+2 (K, K) M/ Ft @biQ+?biB+ BMi2MbBiv F2`M2H Q7 i?2 7Q`K λ(t, dz) = f (t, z)Q(dz) , r?2`2 f : Ω × R × K → R Bb  MQM@M2;iBp2 7mM+iBQM i?i Bb P(F· ) ⊗ K@K2bm`#H2 M/ Q Bb  T`Q##BHBiv K2bm`2 QM (K, K)X h?2M- i?2 T`Q##BHBiv bT+2 Kv #2 2MH`;2/ iQ ++QKKQ/i2 M Ft @SQBbbQM T`Q+2bb N QM R × K × R+ rBi? BMi2MbBiv K2bm`2 dt × Q(dz) × ds bm+? i?i N (dt × dz) := N (dt × dz × [0, f (t, z)]) . S`QQ7X h?2 `2bmHi rBHH #2 T`Qp2/ 7Q` i?2 mMK`F2/ +b2- i?2 ;2M2`H +b2 7QHHQrBM; 2t+iHv i?2 bK2 HBM2bX h?2 i?2Q`2K BM i?Bb bBKTHB}2/ 7Q`K Bb b 7QHHQrbX G2i N #2  bBKTH2 TQBMi T`Q+2bb QM R rBi? i?2 Ft @T`2/B+i#H2 BMi2MbBiv {λ(t)}t∈R X h?2M- i?2`2 2tBbib  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R × R+ rBi? i?2 p2`;2 BMi2MbBiv Rbm+? i?i U8X99V ?QH/bX JQ`2Qp2`- i?Bb T`Q+2bb N Bb M Ft ∨ FtN ĜSQBbbQM T`Q+2bbX b bm+?- 7Q` HH a ∈ R- Sa N + Bb BM/2T2M/2Mi Q7 Fa X G2i {Un }n∈Z #2 M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]- M/ H2i N 1 #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R × R+ - Q7 BMi2MbBiv Rbm+? i?i {Un }n∈Z - N 1 M/ F∞ `2 BM/2T2M/2MiX .2}M2 N #v    N (A) = 1(λ(t),∞) (σ)N 1 (dt × dσ) + 1A ((Tn , Un λ(Tn ))) A

n∈Z

7Q` HH A ∈ B ⊗ B(R+ )X A7 H Bb  MQM@M2;iBp2 7mM+iBQM 7`QK R × Ω × R+ iQ R  H(t, σ)N (dt × dσ) = R

  R

R+

R

H(t, σ)1(λ(t),∞) (σ)N 1 (dt × dσ) +



H(Tn , Un λ(Tn )) .

n∈Z

.2MQi2 #v N U i?2 K`F2/ TQBMi T`Q+2bb Q#iBM2/ #v ii+?BM; i?2 K`F Un iQ TQBMi Tn Q7 N X G2i U Gt = Ft ∨ FtN 1 ∨ FtN M/ bmTTQb2 i?i H Bb P(Gt ) ⊗ B(R+ )@K2bm`#H2 M/ MQM@M2;iBp2X ×

×

× U0 λ(T0 )

U2 λ(T2 )

U−1 λ(T−1 ) U1 λ(T1 ) T−1

T0

0

T1

T2

t

8XdX :_A:1GAPLAaǶ _1S_1a1LhhAPL

kR8

aBM+2 N 1 ?b i?2 FtN 1 @BMi2MbBiv F2`M2H λ1 (t, dσ) = dσ- i?2 Hii2` Bb HbQ i?2 Gt Ĝ U N1 M/ F∞ `2 BM/2T2M/2Mi Q7 F∞ VX "v i?2 BMi2MbBiv F2`M2H Q7 N 1 U`2+HH i?i F∞ bKQQi?BM; i?2Q`2K   E R

R+

  1(λ(t),∞) (σ)H(t, σ)N 1 (dt × dσ) = E R

R+

1(λ(t),∞) (σ)H(t, σ)dt × dσ

7Q` Mv MQM@M2;iBp2 H ∈ P(F· ) ⊗ B(R+ )X LQr- 7Q` Mv MQM@M2;iBp2 H ∈ P(F· ) ⊗ B(R+ ) E



 H(Tn , Un λ(Tn ))

 

H(t, uλ(t))N U (dt × du)

= E R

n∈Z

 

[0,1] 1

= E R

H(t, uλ(t)) dt du .

0

UAM 7+i- N U /KBib i?2 Ft ∨ FtN @BMi2MbBiv F2`M2H λ(t)1[0,1] (u) duX h?Bb Bb HbQ  Gt @biQ+?biB+ F2`M2H 7Q` N U bBM+2 N 1 Bb BM/2T2M/2Mi Q7 N U M/ F∞ X h?2 KT (t, ω, u) → H(t, uλ(t)) Bb P(Gt ) ⊗ B(R)@K2bm`#H2 BM pB2r Q7 i?2 Ft @T`2/B+i#BHBiv Q7 {λ(t)}t∈R - M/ Q7 i?2 K2bm`#BHBiv bbmKTiBQMb QM HX h?Bb DmbiB}2b i?2 #Qp2 mb2 Q7 i?2 bKQQi?BM; i?2Q`2KXV h?Bb i2`K Bb HbQ 2[mH iQ U

  E R

R+

H(t, σ)1(0,λ(t)] (σ)dtdσ ,

#v i?2 +?M;2 Q7 p`B#H2b σ = uλ(t)X h?2`27Q`2  

H(t, σ)N (dt × dσ) R R+  



 =E H(t, σ)1(0,λ(t)] (σ)dtdσ H(t, σ)1(λ(t),∞) (σ)dtdσ + E R R R×R+   +

=E H(t, σ)dtdσ

E

R

R+

7Q` HH MQM@M2;iBp2 H ∈ P(G) ⊗ B+ X h?2`27Q`2- #v h?2Q`2K 8XdXR- N Bb  ?QKQ@ ;2M2Qmb Gt @SQBbbQM T`Q+2bb QM (R × R+ , B(R) ⊗ B(R+ )) rBi? BMi2MbBiv RX Ai Bb   7Q`iBQ`B M Ft ∨ FtN @SQBbbQM T`Q+2bbX

o`BMib Q7 i?2 AK#2//BM; h?2Q`2Kb h?2 7QHHQrBM; `2bmHib `2 Q7 i?2 bK2 FBM/ b i?2 T`2pBQmb QM2b- #mi i?2v bbmK2 #QmM/2/M2bb Q7 i?2 BMi2MbBiv F2`M2HX q2 bii2 i?2K BM i?2 mMK`F2/ +b2X h?2B` T`QQ7b `2 H27i b M 2t2`+Bb2 U1t2`+Bb2 8XNXkRVX

kRe *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu  M/ H2i {Un }n∈Z #2 M BB/  #2 M ?TT QM R rBi? BMi2MbBiv λ h?2Q`2K 8XdXd G2i N  U #2 i?2 bbQ+Bi2/ HB7i2/ b2[m2M+2 Q7 K`Fb- mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X G2i N U - M/ H2i 7Q` TQBMi T`Q+2bb QM R × [0, 1]X G2i {Gt }t∈R #2  ?BbiQ`v BM/2T2M/2Mi Q7 N HH t ∈ R Ft := Gt ∨ FtNU .  < ∞- M/ G2i {λ(t)}t∈R #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb #QmM/2/ #v λ /2}M2  TQBMi T`Q+2bb N QM (R, B(R)) #v  N (C) = 1C (Tn )10≤U ≤ λ(Tn )  (C ∈ B(R)) . U8X98V n

n∈Z

 λ

h?2M N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R X h?2Q`2K 8XdX3 G2i N #2  bBKTH2 TQBMi T`Q+2bb QM (R, B(R)) rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R UbbmK2/ Ft @T`2/B+i#H2- rBi?Qmi HQbb Q7 ;2M2`HBivVX amTTQb2 i?i i?2`2  < ∞ bm+? i?i P @XbX 2tBbib  +QMbiMi λ  λ(t, ω) ≤ λ

(t ∈ R).

 , U ) QM (R, B(R)) rBi? K`Fb h?2M- i?2`2 2tBbib  +QKTQmM/ SQBbbQM T`Q+2bb (N  BM ([0, 1]- B([0, 1]))- +?`+i2`BbiB+b (λ, Q) r?2`2 Q Bb i?2 mMB7Q`K /Bbi`B#miBQM QM [0, 1]- M/ bm+? i?i U8X98V ?QH/bX AK#2//BM; i?2Q`2Kb +M #2 mb2/ BM KMv rvbX h?2v b2`p2 7Q` BMbiM+2 b  #bBb 7Q` i?2 +QMbi`m+iBQM Q7 irQ TQBMi T`Q+2bb2b rBi?  biQ+?biB+ BMi2MbBiv QM i?2 bK2 T`Q##BHBiv bT+2- bQ i?i i?2v +M #2 +QKT`2/ i`D2+iQ`vrBb2X 6Q` BMbiM+2/2}MBM; i?2 BM2[mHBiv N1 ≥ N2 `2HiBp2 iQ TQBMi T`Q+2bb2b QM R iQ bB;MB7v i?i N1 (C) ≥ N2 (C) 7Q` HH C ∈ B(R)- Bi Bb +H2` i?i B7 i?2b2 TQBMi T`Q+2bb2b /KBi biQ+?biB+ BMi2MbBiB2b λ1 (t, N1t ) M/ λ2 (t, N2t ) Ur?2`2 N t Bb i?2 `2bi`B+iBQM Q7 N iQ (−∞, t]V rBi? `2bT2+i iQ i?2B` `2bT2+iBp2 BMi2`MH ?BbiQ`B2b- M/ B7 7Q` HH t ∈ RN1t ≥ N2t ⇒ λ1 (t, N1t ) ≥ λ2 (t, N2t ) , i?2M N1 ≥ N2 X :`B;2HBQMBbǶ +QMbi`m+iBQM Bb Q7 T`iB+mH` BKTQ`iM+2 BM i?2 T`QQ7b Q7 +QmTHBM; Q7 TQBMi T`Q+2bb2b M/ rBHH #2 TTHB2/ BM a2+iBQM RkXe iQ i?2 bim/v Q7 i?2 E2`biM TQBMi T`Q+2bbX aBKmHiBQM H;Q`Bi?Kb M Q#pBQmb TTHB+iBQM Q7 T`+iB+H BKTQ`iM+2 Q7 i?2 /B`2+i BK#2//BM; i?2Q`2Kb Bb iQ i?2 bBKmHiBQM Q7 TQBMi T`Q+2bb2b rBi?  biQ+?biB+ BMi2MbBivX q2 bi`i rBi?  bBKTH2 2tKTH2 Q7 i?2 K2i?Q/QHQ;vXRk amTTQb2 i?i r2 rBb? iQ bBKmHi2  TQBMi T`Q+2bb QM i?2 TQbBiBp2 ?H7@HBM2 rBi?  biQ+?biB+ BMi2MbBiv Q7 i?2 7Q`K λ(t, ω) = v(t, N[0,t) (ω)) , Rk

(G2rBb M/ a?2/H2`- RNde)- (P;i- RN3R)X

8X3X P_A:AL L. JPhAohAPL P6 h>1 J_hAL:G1 SS_P*> kRd r?2`2 v : R+ × Mp (R+ ) → R+ Bb K2bm`#H2 rBi? `2bT2+i iQ B(R+ ) ⊗ Mp (R+ ) M/ B(R+ ) M/ #QmM/2/- bv- #v K < ∞X 6Q` i?Bb r2 +M mb2 h?2Q`2K 8XdX3  QM R+ r?B+? i2HHb mb i?i Bi bm{+2b iQ i?BM  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N  rBi? BMi2MbBiv K- 2tKBMBM; Bib TQBMib b2[m2MiBHHv- F22TBM;  TQBMi Tn B7 M/ QMHv v(Tn ,N



)

[0,Tn ) B7 Un ≤ r?2`2 {Un }n≥1 Bb M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b mMB7Q`KHv K X /Bbi`B#mi2/ QM [0, 1] M/ BM/2T2M/2Mi Q7 N

h?Bb K2i?Q/ Bb 2bBHv /Ti2/ iQ i?2 +b2 Q7 TQBMi T`Q+2bb2b rBi? M FtN ∨ G0 @ BMi2MbBiv Q7 i?2 7Q`K λ(t, ω) = v(t, X(t−, ω), N[0,t) (ω)) ,

()

r?2`2 UBV {X(t)}t≥0 Bb bQK2 +Q`HQH biQ+?biB+ T`Q+2bb rBi? pHm2b BM bQK2 K2bm`@ X #H2 bT+2 (K, K) M/ G0 = F∞ := ∨t≥0 FtX UBBV v : R+ ×Mp (R+ ) → R+ Bb K2bm`#H2 rBi? `2bT2+i iQ B(R+ )⊗K⊗Mp (R+ ) M/ B(R+ )- M/ Bb #QmM/2/ #v KUAM Qi?2` rQ`/b- i?2 TQBMi T`Q+2bb r2 b22F iQ bBKmHi2 Bb  b2KB@*Qt TQBMi T`Q+2bbXV  - M/ UBBBV {X(t)}t≥0 Bb BM/2T2M/2Mi Q7 N UBpV r2 bmTTQb2 i?i r2 ?p2 i /BbTQbBiBQM i Mv iBK2 t i?2 pHm2 X(t)X  Bb F2Ti B7 M/ QMHv B7 Un ≤  TQBMi Tn Q7 N

v(Tn ,X(Tn −),N(0,T ) ) n

K

X

q2 +M KF2 MQi?2` bi2T iQr`/b ;2M2`HBxiBQM bbmKBM; M FtN ∨ FtX @ BMi2MbBiv Q7 i?2 7Q`K UVX "mi i?Bb iBK2 r2 Kmbi // i?2 +QM/BiBQM i?i- /2MQiBM; #v Tn i?2 n@i? TQBMi Q7 N - QM2 +M +QMbi`m+i X(t) 7`QK Tn QMr`/ #b2/ QM i?2 FMQrH2/;2 Q7 N[0,t) X PM2 2tKTH2 Bb r?2M i?2 T`Q+2bb {X(t)}t≥0 Bb Q7 i?2 7Q`K  t  t ϕ(N (t−)) dt + σ(N (t−)) dW (t) , X(t) = X(0) + 0

0

X r?2`2 {W (t)}t≥0 Bb  ;Bp2M qB2M2` T`Q+2bb BM/2T2M/2Mi Q7 N Pi?2` 2tKTH2b- +QM+2`MBM; 7Q` BMbiM+2 KmimHHv 2t+BiBM; TQBMi T`Q+2bb2b- `2 H27i iQ i?2 BK;BMiBQM Q7 i?2 `2/2`X h?2 `2HtiBQM Q7 i?2 ?vTQi?2bBb Q7 #QmM/@ 2/M2bb Q7 i?2 biQ+?biB+ BMi2MbBiv Bb MQi  T`Q#H2K bBM+2 QM2 +M Hrvb p`v i?2  r?2M M22/2/X BMi2MbBiv Q7 i?2 SQBbbQM T`Q+2bb N

8X3 P`B;BM M/ JQiBpiBQM Q7 i?2 J`iBM;H2 TT`Q+? h?Bb b2+iBQM T`2b2Mib i?2 i?2Q`v Q7 biQ+?biB+ BMi2MbBiv BM  #`Q/2` T2`bT2+iBp2 M/ ;Bp2b i?2 KQiBpiBQMb M/ i?2 B/2b i i?2 Q`B;BM Q7 i?2 K`iBM;H2 TT`Q+? iQ TQBMi T`Q+2bb2bXRj h?2 bivH2 Q7 2tTQbBiBQM Q7 i?Bb b2+iBQM rBHH #2 Tm`TQb2Hv HQQb2 BM Rj

("`ûKm/- RNdk)X

kR3 *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu Q`/2` MQi iQ Q#b+m`2 i?2 bBKTH2 B/2b rBi? i?2 T2iiv UM2p2`i?2H2bb BM/BbT2Mb#H2V /2iBHb i?i  bi`B+iHv Ki?2KiB+H i`2iK2Mi rQmH/ `2[mB`2X J`iBM;H2 *?`+i2`BxiBQMb UGûpv M/ qiM#2V Ai ?b #22M FMQrM 7Q`  HQM; iBK2 i?i 7Q`  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N Q7 BMi2MbBiv 1- {N (t) − t}t≥0 Bb  K`iBM;H2X AM 7+i- 7Q`  bBKTH2 TQBMi T`Q+2bb N iQ #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1 Bi Bb HbQ bm{+B2Mi i?i {N (t)−t}t≥0 #2  K`iBM;H2 UqiM#2Ƕb i?2Q`2Kc *Q`QHH`v kX9XkVX h?Bb rb i?2 }`bi bB;MB}+Mi i?2Q`2iB+H `2bmHi Q7 K`iBM;H2 i?2Q`v +QM+2`MBM; TQBMi T`Q+2bb2bX h?Bb +?`+i2`BxiBQM BM i2`Kb Q7 K`iBM;H2b Bb iQ i?2 SQBbbQM T`Q+2bb r?i GûpvǶb +?`+i2`BxiBQM Bb iQ i?2 qB2M2` T`Q+2bbR9 Bb iQ i?2 qB2M2` T`Q+2bbX AM i?2 K`iBM;H2 TT`Q+? iQ TQBMi T`Q+2bb2b- qiM#2Ƕb i?2Q`2K +M #2 Q#iBM2/ BM irQ rvb, UBV, b  +Q`QHH`v Q7 i?2 KQ`2 ;2M2`H `2bmHi biiBM; BM T`iB+mH` i?i i?2 /Bbi`B#miBQM Q7  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb Bb +?`+i2`Bx2/ #v Bib FtN @BMi2MbBiv r?2M Bi 2tBbibX UBBV, pB i?2 bKQQi?BM; 7Q`KmH Q7 SQBbbQM +H+mHmb- b rb /QM2 BM i?2 T`QQ7 Q7 h?2Q`2K kX9XRX LQi2 i?i i?Bb i?2Q`2K rb Q`B;BMHHv bii2/ BM i2`Kb Q7 i?2 BMi2`MH ?BbiQ`v Q7 N - BM +QMi`bi rBi? i?2 p2`bBQM ;Bp2M BM i?Bb #QQFX h?Bb Bb M BKTQ`iMi Q#b2`@ piBQM #2+mb2 r?i +?`+i2`Bx2b i?2 biQ+?biB+ BMi2MbBiv i?2Q`v Bb T`2+Bb2Hv i?2 7+i i?i Bi iF2b BMiQ ++QmMi i?2 BM7Q`KiBQM i?i Bb MQi bQH2Hv +QMiBM2/ BM i?2 TQBMi T`Q+2bb Bib2H7-  7mM/K2MiH bT2+i BM KMv TTHB+iBQMb- 2bT2+BHHv BM i?2 bvbi2Kb M/ +QKKmMB+iBQMb b+B2M+2b- r?B+? `2 +QM+2`M2/ rBi? i?2 T`Q#H2Kb Q7 2biBKiBQM- /2i2+iBQM M/ +QMi`QHX AM i?Bb `2 Q7 TTHB2/ Ki?2KiB+b- i?2`2 Bb  H`;2 +Q`Tmb Q7 FMQrH2/;2 `QmM/ i?2b2 i?2K2b- r?B+? Kv #2 +HH2/ i?2 i?2Q`v Q7 qB2M2`@/`Bp2M biQ+?biB+ bvbi2Kb- r?2`2 i?2 #bB+ Q#D2+i Bb i?2 biQ+?biB+ T`Q+2bb+HH2/ i?2 dzQ#b2`piBQMǴ

t

X(t) =

ϕ(s) ds + W (t) ,

(†)

0

r?2`2 {W (t)}t≥0 Bb  qB2M2` T`Q+2bb UBM T`iB+mH`  T`Q+2bb rBi? Q`i?Q;QMH BM+`2K2MibV- +HH2/ i?2 dzBMi2;`i2/ MQBb2Ǵ- M/ {ϕ(t)}t≥0 +``B2b i?2 BM7Q`KiBQM Q7 BMi2`2bi Ui?2 dzbB;MHǴVX AM i?2 2M;BM22`BM; HBi2`im`2- UV Bb i?2 BMi2;`i2/ 7Q`K Q7 x(t) = ϕ(t) + b(t) , (††) r?2`2 {b(t)}t≥0 Bb  dzr?Bi2 MQBb2Ǵ U7Q`KHHv i?2 /2`BpiBp2 Q7 i?2 qB2M2` T`Q+2bb#mi b Bb r2HH FMQrM- bm+?  /2`BpiBp2 /Q2b MQi 2tBbi- M/ i?Bb Bb r?v i?2 BMi2;`i2/ 7Q`K Bb mb2/XV R9 aiiBM; i?i  +QMiBMmQmb `2H@pHm2/ T`Q+2bb {W (t)}t≥0 bm+? i?i {W (t)}t≥0 M/ {W (t)2 − t}t≥0 `2 K`iBM;H2b Bb  biM/`/ qB2M2` T`Q+2bbX

8X3X P_A:AL L. JPhAohAPL P6 h>1 J_hAL:G1 SS_P*> kRN h?2 K`iBM;H2 TT`Q+? iQ TQBMi T`Q+2bb2b rb Q`B;BMHHv KQiBpi2/ #v i?2 +QMbi`m+iBQM Q7  i?2Q`v Q7 bvbi2Kb /`Bp2M #v TQBMi T`Q+2bb2b MHQ;Qmb iQ i?2 i?2Q`v Q7 qB2M2`@/`Bp2M biQ+?biB+ bvbi2KbX Ai im`Mb Qmi i?i i?2 `M;2 Q7 i?2 7Q`KH MHQ;v TT`2Mi BM i?2 +QKT`BbQM Q7 i?2 irQ 7mM/K2MiH i?2Q`2Kb Q7 qiM#2 M/ Gûpv BM 7+i 2ti2M/b 7` #2vQM/ i?2b2 irQ `2bmHibX AM i?2 i?2Q`v Q7 TQBMi T`Q+2bb /`Bp2M biQ+?biB+ bvbi2Kb- i?2 #bB+ Q#D2+i Bb  TQBMi T`Q+2bb N i?i +M HbQ #2 `2T`2b2Mi2/ BM i?2 7Q`K dzbB;MH THmb MQBb2Ǵ,  t N (t) = λ(s) ds + M (t) , () 0

r?2`2- bv- {M (t)}t≥0 Bb M Ft @K`iBM;H2 M/ i?2`27Q`2- BM T`iB+mH`-  biQ+?biB+ T`Q+2bb rBi? Q`i?Q;QMH BM+`2K2MibX 6`QK i?2 BM7Q`KiBQM i?2Q`2iB+ TQBMi Q7 pB2r- i?2 KQ/2H U††V T2`iBMb iQ KTHB@ im/2 KQ/mHiBQM i?2Q`v- i?2 mb27mH BM7Q`KiBQM #2BM; +QMiBM2/ BM i?2 KTHBim/2 Q7 i?2 bB;MH {ϕ(t)}t≥0 M/ i?Bb BM7Q`KiBQM Bb iQ #2 2ti`+i2/ 7`QK  +Q``mTi2/ p2`bBQM Q7 Bi- i?2 +Q``mTiBQM #2BM; /m2 iQ M //BiBp2 :mbbBM MQBb2X AM i?2 /B7@ 72`2MiBi2/ p2`bBQM Q7 UV- i?2 BM7Q`KiBQM pBH#H2 Bb i?2 bTBF2 i`BM  Δ(t) = δ(t − Tn ) . n≥1

h?Bb +Q``2bTQM/b BM +QKKmMB+iBQMb i?2Q`v iQ 7`2[m2M+v KQ/mHiBQM- r?2`2 i?2 `i2 Udz7`2[m2M+vǴV Q7 TQBMib Bb T`QTQ`iBQMH iQ i?2 bB;MH KTHBim/2X h?2 /Bz2`2M@ iBi2/ p2`bBQM Q7 i?2 KQ/2H UV Bb ˙ Δ(t) = λ(t) + p(t) ,

()

˙ r?2`2 Δ(t) := 1N ({t}=1 M/ p(t) := 1N ({t}=1 − λ(t)X h?Bb Bb Q7 +Qm`b2  Tm`2Hv 7Q`KH /2b+`BTiBQM- M/ Bi Bb i?2 BMi2;`i2/ p2`bBQM U†V r?B+? Bb i?2 2{+B2Mi QM2- Dmbi b i?2 BMi2;`i2/ p2`bBQM UV Bb i?2 T`+iB+H QM2 7Q` i?2 bB;MH THmb MQBb2 KQ/2HX h?2 i?2Q`v Q7 qB2M2`@/`Bp2M biQ+?biB+ bvbi2Kb Bb +QM+2`M2/ rBi? i?2 BMiB@ Ki2Hv +QMM2+i2/ T`Q#H2Kb Q7 2biBKiBQM- /2i2+iBQM M/ +QMi`QH- M/ rb /2p2H@ QT2/ BM T`iB+mH` 7Q` i?2 M22/b Q7 bT+2 2tTHQ`iBQMX 6`QK i?2 Ki?2KiB+H TQBMi Q7 pB2r- Bi `2bib QM i?`22 TBHH`b, ∞ UBV i?2 AiƬ biQ+?biB+ BMi2;`H 0 ϕ(s) dW (s) /2}M2/ 7Q` dzbmBi#H2Ǵ BMi2;`M/bt UBBV i?2 Q#b2`piBQM i?i i?2 T`Q+2bb { 0 ϕ(s) dW (s)}t≥0 Bb  K`iBM;H2- M/ UBBBV AiƬǶb /Bz2`2MiBiBQM `mH2,  t  1 t  F (W (t)) = F (W (0)) + F  (W (s)) ds + F (W (s)) dW (s) , 2 0 0 r?2`2 F Bb  dzbmBi#H2Ǵ irB+2 +QMiBMmQmbHv /Bz2`2MiB#H2 7mM+iBQMX

kky *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu aQK2r?i MHQ;QmbHv- i?2 biQ+?biB+ BMi2MbBiv T`/B;K 7Q` TQBMi T`Q+2bb /`Bp2M bvbi2Kb `2bib QM, ∞ UV i?2 biQ+?biB+ BMi2;`H 0 ϕ(s) N (ds) 7Q` bmBi#H2 BMi2;`M/b∞ U#V i?2 Q#b2`piBQM i?i { 0 ϕ(s) (N (ds) − λ(s)ds)}t≥0 Bb  K`iBM;H2 B7 {ϕ(t)}t≥0 Bb M Ft @T`2/B+i#H2 T`Q+2bb- i?2M- M/ U+V i?2 aiB2HiD2bĜG2#2b;m2 +H+mHmbX q2 b?HH MQr HBbi  b2`B2b Q7 2tKTH2b BHHmbi`iBM; i?2 MHQ;v #2ir22M i?2 bvbi2Kb /`Bp2M #v TQBMi T`Q+2bb2b M/ i?Qb2 /`Bp2M #v  qB2M2` T`Q+2bbX 1ti2MbBQMb Q7 i?2 /2}MBiBQMb Q7 #Qi? i?2 qB2M2` T`Q+2bb M/ i?2 SQBbbQM T`Q+2bb rBHH #2 M22/2/X .2}MBiBQM 8X3XR G2i {Ft }t≥0 #2 bQK2 ?BbiQ`v QM i?2 T`Q##BHBiv bT+2 (Ω, F, Q)M/ H2i {W (t)}t≥0 #2  +QMiBMmQmb Ft @/Ti2/ biQ+?biB+ T`Q+2bb bm+? i?i 7Q` HH 0 ≤ a ≤ b < ∞- W (b)−W (a) Bb  +2Mi2`2/ :mbbBM `M/QK p`B#H2 rBi? p`BM+2 b − a BM/2T2M/2Mi Q7 Fa X h?2M {W (t)}t≥0 Bb +HH2/  (Q, Ft )@qB2M2` T`Q+2bbX .2}MBiBQM 8X3Xk G2i {Ft }t≥0 #2 bQK2 ?BbiQ`v Q7  bBKTH2 TQBMi T`Q+2bb N QM i?2 T`Q##BHBiv bT+2 (Ω, F, Q) M/ bmTTQb2 i?i 7Q` HH 0 ≤ a ≤ b < ∞- N (a, b] Bb  SQBbbQM p`B#H2 rBi? K2M b − a BM/2T2M/2Mi Q7 Fa X h?2M N Bb +HH2/ M (Q, Ft )@SQBbbQM T`Q+2bbX :B`bMQp@ivT2 _2bmHib q2 #2;BM rBi? i?2 Q`B;BMH :B`bMQp i?2Q`2KX G2i {Ft }t≥0 M/ {X(t)}t≥0 #2 b BM .2}MBiBQM 8X3XR M/ H2i {ϕ(t)}t≥0 #2  #QmM/2/ U7Q` bBKTHB+BivV Ft @/Ti2/ biQ+?biB+ T`Q+2bbX h?2 T`Q+2bb  t   1 t L(t) := exp ϕ(s) dX(s) − ϕ(s)2 ds 2 0 0 Bb  bQHmiBQM Q7 i?2 2[miBQM 

t

L(s)ϕ(s) dX(s)

L(t) = 1 + 0

M/ M Ft @K`iBM;H2X JQ`2Qp2`- 7Q` Mv T > 0- i?2 T`Q##BHBiv P QM (Ω, F) /2}M2/ #v dP = L(T ) dQ Bb bm+? i?i QM i?2 BMi2`pH [0, T ] t ϕ(s) ds + W (t) X(t) =

(t ∈ [0, T ])

0

r?2`2 {W (t)}t≥0 Bb  (P, Ft )@qB2M2` T`Q+2bbX q2 MQr `2+HH i?2 MHQ;Qmb `2bmHi 7Q` SQBbbQM@/`Bp2M TQBMi T`Q+2bb2bX

8X3X P_A:AL L. JPhAohAPL P6 h>1 J_hAL:G1 SS_P*> kkR G2i N #2  bBKTH2 TQBMi T`Q+2bb /Ti2/ iQ bQK2 ?BbiQ`v {Ft }t≥0 M/ H2i Q #2  T`Q##BHBiv- #Qi? QM i?2 bK2 K2bm`#H2 bT+2 (Ω, F)X amTTQb2 i?i N Bb  (Q, Ft ) SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X G2i {λ(t)}t≥0 #2  TQbBiBp2 M/ #QmM/2/ U7Q` bBKTHB+BivV Ft @/Ti2/ biQ+?biB+ T`Q+2bbX h?2 T`Q+2bb   t  t log(λ(s) N (ds) − (λ(s) − 1) ds L(t) := exp 0

Bb  bQHmiBQM Q7 i?2 2[miBQM

0



t

L(s−)(λ(s) − 1) (N (ds) − ds)

L(t) = 1 + 0

M/ M Ft @K`iBM;H2X JQ`2Qp2`- 7Q` Mv T > 0- i?2 T`Q##BHBiv P QM (Ω, F) /2}M2/ #v dP = L(T ) dQ Bb bm+? i?i QM i?2 BMi2`pH [0, T ]- N /KBib i?2 (P, Ft ) BMi2MbBiv {λ(t)}t∈[0,T ] X AM bvK#QHb t

λ(s) ds + M (t) (t ∈ [0, T ]) ,

N (t) = 0

r?2`2 {M (t)}t∈[0,T ] Bb  (P, Ft )@K`iBM;H2X AMMQpiBQMb AM i?2 bB;MH THmb MQBb2 KQ/2H- 7Q`  dzbmBi#H2Ǵ p2`bBQM Q7 i?2 2biBKi2/ bB;MH T`Q+2bb ϕ(t) := E ϕ(t) | FtX (t ≥ 0) , i?2 Q#b2`piBQM /KBib i?2 `2T`2b2MiiBQM  t 2(t) , ϕ(s) ds + W X(t) = 0  2(t)}t≥0 Bb  (P, FtW )@qB2M2` T`Q+2bbX h?2 bBKBH` `2bmHi 7Q` TQBMi T`Q@ r?2`2 {W +2bb2b Bb i?i  t 2(t) , N (t) = λ(s) ds + M 0

2(t)}t≥0 Bb M F N @K`iBM;H2- +HH2/ i?2 BMMQpiBQMb T`Q+2bb- #2+mb2 r?2`2 {M t 2(t) Bb 2[mH iQ dzr?i vQm ;2iǴ UdN (t)V BM i?2 BM}MBi2bBKH BMi2`pH dt KBMmb dM dzr?i vQm 2tT2+i2/Ǵ UE dN (t) | FtN VX *QM+2`MBM; i?2 TQBMi T`Q+2bb N -  `i?2` i`BpBH Q#b2`piBQM Bb i?i 7`QK i?2 2(t)}t≥0 QM2 +M BKK2/Bi2Hv Q#iBM i?2 TQBMi T`Q+2bb Bib2H7X AM MQBb2 T`QD2+iBQM {M bvK#QHb,  FtN ≡ FtM .  bBKBH` UHi?Qm;? i?Bb iBK2 MQi Q#pBQmbV `2bmHi 7Q` i?2 bB;MH THmb r?Bi2 MQBb2 KQ/2H i2HHb mb i?i- mM/2` dzbmBi#H2Ǵ +QM/BiBQMb 

FtX ≡ FtW . h?Bb Bb +HH2/ i?2 BMMQpiBQMb 2[mBpH2M+2X

kkk *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu J`iBM;H2b b aiQ+?biB+ AMi2;`Hb G2i {W (t)}t≥0 #2 M FtW @qB2M2` T`Q+2bbX h?2M Mv +2Mi2`2/ b[m`2@BMi2;`#H2 FtW @K`iBM;H2 Bb Q7 i?2 7Q`K  t m(t) = H(s) dW (s) , 0

t r?2`2 {H(t)}t≥0 Bb  T`2/B+i#H2 T`Q+2bb bm+? i?i 0 H(s)2 ds < ∞ Ut ≥ 0VX aBKBH`Hv- B7 N Bb  bBKTH2 TQBMi T`Q+2bb rBi? i?2 FtN @ BMi2MbBiv {λ(t)}t≥0 - i?2M Mv +2Mi2`2/ b[m`2@BMi2;`#H2 FtN @K`iBM;H2 Bb Q7 i?2 7Q`K  t 2(s) , m(t) = H(s) dM 0

r?2`2 {H(t)}t≥0 Bb  T`2/B+i#H2 T`Q+2bb bm+? i?i

t 0

H(s)2 ds < ∞ Ut ≥ 0VX

JQ`2 2tKTH2b rBHH #2 ;Bp2M b i?2 QTTQ`imMBiv `Bb2b- BM T`iB+mH` BM *?Ti2` Ry QM i?2 BM7Q`KiBQM@i?2Q`2iB+H bT2+ib Q7 TQBMi T`Q+2bb2b Ua2+iBQMb RyXR M/ RyXk QM i?2 }Hi2`BM; T`Q#H2K- M/ 1tKTH2 RyXjX8VX

8XN 1t2`+Bb2b 1t2`+Bb2 8XNXRX h?2 6mM/K2MiH J`iBM;H2 t a?Qr i?i .2}MBiBQM 8XRXR BKTHB2b i?i 7Q` HH a ∈ R- Ma (t) := N (a, t] − a λ(s) ds Ut ≥ aV Bb M Ft @HQ+H K`iBM;H2X 1t2`+Bb2 8XNXkX *QMM2+iBM; iQ i?2 AMimBiBp2 .27BMBiBQM G2i i?2 bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb N QM R ?p2 i?2 Ft @BMi2MbBiv {λ(t)}t∈R - M/ bmTTQb2 i?i t → λ(t, ω) Bb 7Q` HH ω ∈ Ω  `B;?i@+QMiBMmQmb 7mM+iBQMM/ i?i (t, ω) → λ(t, ω) Bb mMB7Q`KHv #QmM/2/X a?Qr i?i lim h↓0

1 E[N (t, t + h]|Ft ] = λ(t) , h

P @XbX

1t2`+Bb2 8XNXjX *Qt S`Q+2bb2b G2i N #2  /Qm#Hv biQ+?biB+ SQBbbQM T`Q+2bb U*Qt T`Q+2bbV rBi? `2bT2+i iQ i?2 σ@}2H/ G- rBi? i?2 HQ+HHv BMi2;`#H2 biQ+?biB+ BMi2MbBiv {λ(t)}t∈R X h?Bb K2Mb λ U.2}MBiBQM RXRXRyV i?i G ⊇ F∞ := σ (λ(s), s ∈ R) M/ i?i r?2M2p2` C1 , . . . , CK `2 #QmM/2/ /BbDQBMi K2bm`#H2 bm#b2ib Q7 R  K   K     iuj E exp i uj N (Cj ) | G = exp (e − 1) λ(s)ds j=1

j=1

Cj

7Q` HH u1 , . . . , uK ∈ RX a?Qr i?i N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R - r?2`2 Ft = G ∨ FtN X

8XNX 1s1_*Aa1a

kkj

1t2`+Bb2 8XNX9X Ft @T`Q;`2bbBp2M2bb G2i {X(t)}t∈R+ #2  HQ+HHv BMi2;`#H2 Ft @T`Q;`2bbBp2 T`Q+2bbX S`Qp2 i?i i?2 t biQ+?biB+ T`Q+2bb { 0 X(s) ds}t∈R+ Bb K2bm`#H2 M/ Ft @/Ti2/X 1t2`+Bb2 8XNX8X Pi?2` >BbiQ`B2b G2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM R rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X S`Qp2 i?2 7QHHQrBM;, UBV A7 {Gt }t∈R Bb  ?BbiQ`v bm+? i?i G∞ Bb BM/2T2M/2Mi Q7 F∞ - i?2M {λ(t)}t∈R Bb M Ft ∨ Gt @BMi2MbBiv Q7 N X UBBV A7 {Ft }t∈R Bb  ?BbiQ`v Q7 N bm+? i?i Ftλ ∨ FtN ⊆ Ft ⊆ Ft - i?2M {λ(t)}t∈R Bb M Ft @BMi2MbBiv Q7 N X 1t2`+Bb2 8XNXeX #Qmi S`2/B+i#BHBiv UV a?Qr i?i  /2i2`KBMBbiB+ K2bm`#H2 T`Q+2bb Bb Ft @T`2/B+i#H2- M/ i?i M Ft @ T`2/B+i#H2 T`Q+2bb Bb Ft @T`Q;`2bbBp2Hv K2bm`#H2- M/ BM T`iB+mH` K2bm`#H2 M/ Ft @/Ti2/X U#V G2i S M/ τ #2 irQ Ft @biQTTBM; iBK2b bm+? i?i S ≤ τ - M/ H2i ϕ : R+ ×R → R #2  K2bm`#H2 7mM+iBQMX h?2M X(t, ω) = ϕ(S(ω), t)1{S(ω)Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu 1t2`+Bb2 8XNXRyX  JmHiBp`Bi2 qiM#2ǰb h?2Q`2K G2i Nj Uj ≥ 1V #2 HQ+HHv }MBi2 bBKTH2 TQBMi T`Q+2bb2b rBi?Qmi +QKKQM TQBMib M/ H2i Ft #2 Q7 i?2 7Q`K U8X9eV rBi? FtN =

∞ 3

Nj

Ft

j=1

UN := (N1 , N2 , . . .)VX 6Q` HH j ≥ 1- H2i {λj (t)}t≥0 #2  G@K2bm`#H2 HQ+HHv BMi2@ ;`#H2 biQ+?biB+ T`Q+2bb M/ bmTTQb2 i?i 7Q` HH (a, b] ⊆ R

b

E[Nj (a, b] | Fa ] =

λj (s) ds . a

a?Qr i?i Nj Bb  G@+QM/BiBQMH SQBbbQM T`Q+2bb rBi? G@+QM/BiBQMH bbQ+Bi2/  K2bm`2 C → E[Nj (C)|G] = C λj (t)dt M/ i?i i?2 Nj Ƕb `2 BM/2T2M/2Mi +QM/B@ iBQMHHv rBi? `2bT2+i iQ GX 1t2`+Bb2 8XNXRRX ai`QM; J`FQp S`QT2`iv Q7 ?TTb AM i?2 b2iiBM; Q7 1t2`+Bb2 8XNXRy rBi? TQbbB#Hv K = ∞- H2i T #2  }MBi2 Ft @biQTTBM; iBK2 M/ H2i 7Q` HH t ≥ 0 Ft := FTN+t . .2}M2 i?2 TQBMi T`Q+2bb2b Nj U1 ≤ j ≤ KV #v Nj (a, b] = Nj (a + T, b + T ] . 

AM T`iB+mH` Ft = G  ∨ FtN - r?2`2 G  = G ∨ FTN X S`Qp2 i?i 7Q` HH j- Nj /KBib i?2 Ft @BMi2MbBiv {λj (t + T )}t≥0 X *QKK2Mi, M BMi2`2biBM; bT2+BH +b2 Bb r?2M 7Q` HH j U1 ≤ j ≤ KV- λj (t) ≡ λj 7Q` HH t- i?i Bb- i?2 Q`B;BMH TQBMi T`Q+2bb2b Nj `2 SQBbbQM T`Q+2bb2b rBi? p2`;2 BMi2MbBiB2b λj - KmimHHv BM/2T2M/2Mi M/ BM/2T2M/2Mi Q7 GX AM i?Bb +b2- i?2 /2Hv2/ TQBMi T`Q+2bb2b ST Nj U/2}M2/ #v ST Nj (C) = Nj (C + τ )V `2 BM/2T2M/2Mi SQBbbQM Nj T`Q+2bb2b rBi? BMi2MbBiB2b λj - M/ i?2v `2 BM/2T2M/2Mi Q7 G M/ Q7 FTN = ∨K j=1 FT X h?2 Hii2` BM/2T2M/2M+2 T`QT2`iv Bb i?2 bi`QM; J`FQp T`QT2`iv Q7 KmHiBp`Bi2 SQBbbQM T`Q+2bb2bX Ai rb T`Qp2/ #v Qi?2` K2Mb BM h?2Q`2K kXRXdX t 1t2`+Bb2 8XNXRkX M (t)2 − 0 λ(s) ds G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+rBi? i?2 UHQ+HHv BMi2;`#H2V Ft @ t BMi2MbBiv {λ(t)}t≥0 X G2i 7Q` t ≥ 0- M (t) := N (t) − 0 λ(s) dsX S`Qp2 i?i {M (t)2 − t λ(s) ds}t≥0 Bb M Ft @HQ+H K`iBM;H2X 0 1t2`+Bb2 8XNXRjX "Q`2Hė*Mi2HHB 7Q` SQBMi S`Q+2bb2b S`Qp2 i?i B7 N Bb  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? i?2 Ft @BMi2MbBiv {λ(t)}t≥0  ∞

N (∞) < ∞ ⇐⇒ 0

λ(s) ds < ∞

P @XbX

8XNX 1s1_*Aa1a

kk8

1t2`+Bb2 8XNXR9X SQBbbQM rBi? AM/2T2M/2Mi BB/ J`Fb a?Qr i?i B7 N Bb  SQBbbQM T`Q+2bb QM R × K rBi? BMi2MbBiv K2bm`2 ν(dt × dz) = λ(t)dt × ν(dz) , r?2`2 ν Bb  σ@}MBi2 K2bm`2 QM (K, K) M/ t → λ(t) Bb  /2i2`KBMBbiB+ MQM@ M2;iBp2 HQ+HHv BMi2;`#H2 7mM+iBQM- i?2M N Bb  HQ+HHv }MBi2 bBKTH2 TQBMi T`Q+2bb rBi? i?2 biQ+?biB+ FtN @BMi2MbBiv F2`M2H λ(t, dz) = λ(t)ν(dz). 1t2`+Bb2 8XNXR8X BB/ J`Fb AM/2T2M/2Mi Q7 i?2 "bB+ SQBMi S`Q+2bb G2i (N, Z) #2  bBKTH2 M/ HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb rBi? K`Fb BM KrBi? bbQ+Bi2/ UHB7i2/V TQBMi T`Q+2bb NZ QM R × KX amTTQb2 i?i N ?b i?2 Ft @BMi2MbBiv {λ(t)}t∈R M/ i?i i?2 K`F b2[m2M+2 {Zn }n∈Z Bb BB/- rBi? +QKKQM /Bbi`B#miBQM P (Z1 ∈ C) = Q(C)- M/ BM/2T2M/2Mi Q7 F∞ X a?Qr i?i NZ ?b i?2 (FtNZ ∨ Ft )@BMi2MbBiv F2`M2H λ(t, dz) = λ(t)Q(dz). 1t2`+Bb2 8XNXReX GBF2HB?QQ/ _iBQ S`Qp2 i?2 bb2`iBQM Q7 1tKTH2 8X8XeX 1t2`+Bb2 8XNXRdX *QMbiMi J2M amT2`K`iBM;H2 S`Qp2 i?i  bmT2`K`iBM;H2 rBi? +QMbiMi K2M Bb M2+2bb`BHv  K`iBM;H2X 1t2`+Bb2 8XNXR3X J`iBM;H2b b aiQ+?biB+ AMi2;`Hb G2i {M (t)}t≥0 #2  `B;?i@+QMiBMmQmb HQ+H Ft @K`iBM;H2X amTTQb2 BM //BiBQM i?i Bi Bb b[m`2@BMi2;`#H2- i?i Bb, supt≥0 E [|M (t)|2 ] < ∞X S`Qp2 i?i   H(s, z)M Z (ds × dz) , M (t) = M (0) + (0,t]

K

r?2`2 M Z (ds × dz) := NZ (ds × dz) − λ(s)Φ(s, dz) ds M/ H ∈ P(F· ) ⊗ K Bb bm+? i?i

 t  |H(s, z)|2 λ(s) Φ(s, dz) ds < ∞ E 0

K

7Q` HH t ≥ 0X 1t2`+Bb2 8XNXRNX h?2 bBimiBQM Bb i?i Q7 h?2Q`2K XkXR8X a?Qr i?i 7Q` 2+? n ≥ 0- Tn Bb M Ft @biQTTBM; iBK2 M/ i?i FTn ⊇ σ (T0 , Z0 , T1 , Z1 , . . . , Tn , Zn ) , M/ i?i 7Q` 2+? t ≥ 0-

kke *>Sh1_ 8X SPALh S_P*1aa1a qAh>  ahP*>ahA* ALh1LaAhu Ft = FtY := σ(Y (s) ; s ∈ [0, t]) , r?2`2- 7Q` HH t ≥ 0- HH n ∈ N- Y (t) = Zn QM [Tn , Tn+1 ) M/ Y (t) = Δ 7Q` t ≥ T∞ r?2`2 Δ Bb M `#Bi``v dz/mKKvǴ 2H2K2MiX 1t2`+Bb2 8XNXkyX 1tTHB+BiHv /2b+`B#2 i?2 `2bmHi Q7 h?2Q`2K X9Xj 7Q` S = T5 M/ 7Q` S = inf{Tn > 0 ; Tn − Tn−1 ≥ 3}X 1t2`+Bb2 8XNXkRX MQi?2` hvT2 Q7 1K#2//BM; S`Qp2 i?2 7QHHQrBM; `2bmHib,  M/ H2i  #2 M ?TT QM R rBi? BMi2MbBiv λh?2Q`2K R, /B`2+i BK#2//BM;X G2i N  U #2 {Un }n∈Z #2 M BB/ b2[m2M+2 Q7 K`Fb- mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X G2i N i?2 bbQ+Bi2/ HB7i2/ TQBMi T`Q+2bb QM R×[0, 1]X G2i {Gt }t∈R #2  ?BbiQ`v BM/2T2M/2Mi U - M/ /2}M2 7Q` HH t ∈ RQ7 N 

Ft = Gt ∨ FtNU .  < ∞- M/ G2i {λ(t)}t∈R #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bb #QmM/2/ #v λ /2}M2  TQBMi T`Q+2bb N QM (R, B) #v  N (C) = 1C (Tn )10≤U ≤ λ(Tn )  (C ∈ B) . () n∈Z

n

 λ

h?2M N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R X h?2Q`2K k, BMp2`b2 BK#2//BM;X G2i N #2  bBKTH2 TQBMi T`Q+2bb QM (R, B(R)) rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R UbbmK2/ Ft @T`2/B+i#H2- rBi?Qmi HQbb Q7 ;2M2`HBivVX  < ∞ bm+? i?i amTTQb2 i?i i?2`2 2tBbib  +QMbiMi λ , λ(t, ω) ≤ λ

∀t ∈ R ,

P @XbX

 , U ) QM (R, B(R)) rBi? K`Fb h?2M- i?2`2 2tBbib  +QKTQmM/ SQBbbQM T`Q+2bb (N  BM ([0, 1]- B([0, 1]))- +?`+i2`BbiB+b (λ, Q) r?2`2 Q Bb i?2 mMB7Q`K /Bbi`B#miBQM QM [0, 1]- M/ bm+? i?i UV ?QH/bX 1t2`+Bb2 8XNXkkX ∞ G2i h : R → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM bm+? i?i 0 th(t) dt < ∞X G2i N #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM R+ rBi? i?2 FtN @BMi2MbBiv λ(t) = (−∞,t) h(t − s) N (ds) M/ p2`;2 BMi2MbBiv λX S`Qp2 i?i B7 λ < ∞- i?2M P (N (R+ ) = 0) > 0X .2/m+2 7`QK i?Bb i?i i?2 QMHv bBKTH2 biiBQM`v 2`;Q/B+ TQBMi T`Q+2bb N QM R+ rBi? i?2 FtN @BMi2MbBiv λ(t) = (−∞,t) h(t − s) N (ds) Bb i?2 2KTiv TQBMi T`Q+2bbX 1t2`+Bb2 8XNXkjX /7` AMi2``2M2rH .Bbi`B#miBQM a?Qr i?i i?2 /7` T`QT2`iv Q7  +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F BKTHB2b mM@ #QmM/2/M2bb Q7 i?2 bmTTQ`i Q7 F X Ua22 1tKTH2 8XdX9XV

8XNX 1s1_*Aa1a

kkd

1t2`+Bb2 8XNXk9X aiQ+?biB+ F2`M2H Q7 i?2 biiBQM`v ;`B/ QM R+ *QMbB/2` i?2 7QHHQrBM; TQBMi T`Q+2bb QM R+ , Tn = (n − 1) + U

(n ≥ 1) ,

r?2`2 U Bb mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X *QKTmi2 Bib FtN @+QKT2MbiQ`X

*?Ti2` e 1tpBbB#H2 AMi2MbBiv Q7 6BMBi2 SQBMi S`Q+2bb2b h?Bb +?Ti2` BMi`Q/m+2b i?2 MQiBQM Q7 2tpBbB#H2 BMi2MbBivXR h?Bb Bb /QM2 BM i?2 7`K2rQ`F Q7 }MBi2 TQBMi T`Q+2bb2b MQi QMHv #2+mb2 i?2 i?2Q`v Bb 2bB2` i?M BM i?2 ;2M2`H +b2- #mi HbQ #2+mb2 i?Bb `2bi`B+iBQM 7mH}HHb i?2 M22/b Q7 KMv TTHB+iBQMb- BM 2+QHQ;v M/ 2Hb2r?2`2X Hi?Qm;? i?2 i?2Q`v Q7 2tpBbB#H2 BMi2MbBiv 7Q` bTiBH TQBMi T`Q+2bb2b Kv #2 i }`bi bB;?i +QMbB/2`2/ M 2ti2MbBQM Q7 i?2 i?2Q`v Q7 biQ+?biB+ BMi2MbBiv QM i?2 HBM2 U/m2 iQ  72r 7Q`KH MHQ;B2b i?i rBHH #2 2t?B#Bi2/ Hi2`V- Bi #2HQM;b BM 7+i iQ i?2 SHK i?2Q`v BM bT+2 Q7 *?Ti2` 3X >Qr2p2`-  KQ`2 2H2K2Mi`v TT`Q+? Bb /QTi2/ BM i?Bb +?Ti2`X

eXR

h?2 CMQbbv .2MbBiv

6BMBi2 TQBMi T`Q+2bb2b rBHH #2 /2b+`B#2/ BM  KMM2` bHB;?iHv /Bz2`2Mi 7`QK i?2 QM2 /QTi2/ T`2pBQmbHv 7Q` ;2M2`H TQBMi T`Q+2bb2bX G2i W #2  K2bm`#H2 bm#b2i Q7 Rm UBM TTHB+iBQMb- Bi Bb mbmHHv  +QKT+i b2iV M/ H2i W #2 i?2 "Q`2H σ@}2H/ QM BiX G2i Mpf (W )- +HH2/ i?2 +QM};m`iBQM bT+2 QM W - #2 i?2 +QHH2+iBQM Q7 }MBi2 b2ib x Q7 TQBMib Q7 W - rBi? TQbbB#H2 `2T2iB@ iBQMbX JQ`2 T`2+Bb2Hv-  +QM};m`iBQM x ∈ Mpf (W ) Q7 +`/BMHBiv n Bb M mMQ`/2`2/ b2[m2M+2 Q7 TQBMib Q7 W x = {x1 , x2 , . . . , xn } , r?2`2 KmHiBTH2 TQBMib Ui?i Bb- xi = xj 7Q` i = jV `2 HHQr2/X G2i Mfp (W ) #2 i?2 σ@}2H/ QM Mpf (W ) ;2M2`i2/ #v i?2 KTTBM;b x → |x ∩ C| 7Q` HH K2bm`#H2 b2ib Ur?2`2 |A| /2MQi2b i?2 +`/BMHBiv Q7 i?2 b2i AV C ∈ WX  }MBi2 TQBMi T`Q+2bb X QM W Bb  `M/QK 2H2K2Mi X : (Ω, F) → (Mpf (W ), Mfp (W )) . hQ +QMM2+i rBi? i?2 T`2pBQmb MQiiBQM N 7Q`  TQBMi T`Q+2bb- Dmbi H2i 7Q` Mv C ∈ WN (C) := |X ∩ C| . R

(STM;2HQm- RNdk)X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_6

kkN

kjy *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a PM2 rv Q7 /2}MBM;  }MBi2 TQBMi T`Q+2bb QM W Bb i?2 7QHHQrBM;, RX a2H2+i i?2 MmK#2` Q7 TQBMib ++Q`/BM; iQ  +QmMi /Bbi`B#miBQM {pn }n≥0 QM N, P (N (W ) = n) = pn

(n ∈ N) .

kX :Bp2M i?2 MmK#2` Q7 TQBMib n ≥ 1- TH+2 i?2K QM W #v /`rBM; (X1 , . . . , Xn ) ∈ W n ++Q`/BM; iQ  P ((X1 , . . . , Xn ) ∈ C | N (W ) = n) =

jn (x1 , . . . , xn ) λ(dx1 ) · · · λ(dxn ) , C

r?2`2 λ Bb bQK2 }MBi2 K2bm`2 QM W M/ jn (x1 , . . . , xn ) Bb  bvKK2i`B+ 7mM+iBQM+HH2/ i?2 /2MbBiv Q7 Q`/2` nX UAi ?b iQ #2 bvKK2i`B+ bBM+2 (X1 , . . . , Xn ) Bb M Q`/2`2/ p2+iQ`- r?2`2b i?2 +QM};m`iBQM {X1 , . . . , Xn } Bb mMQ`/2`2/XV P7 +Qm`b2i?2 /2MbBiB2b Kmbi biBb7v i?2 B/2MiBiB2b  jn (x1 , . . . , xn ) λ(dx1 ) · · · λ(dxn ) = 1

(n ∈ N) .

Wn

h?2 `2bmHiBM; TQBMi T`Q+2bb Bb bBKTH2 B7 λ Bb MQM@iQKB+X AM i?Bb b2+iBQM- i?Bb rBHH #2 bbmK2/X G2i {πn }n≥0 #2  T`Q##BHBiv /Bbi`B#miBQM QM N M/ H2i λ #2  MQM@iQKB+ T`Q##BHBiv K2bm`2 QM (W, W)X G2i Q #2 i?2 /Bbi`B#miBQM Q7  }MBi2 TQBMi T`Q+2bb QM W Q7 i?2 7Q`K ∞  π n λn , n=0

r?2`2 λn := λ⊗n Bb i?2 T`Q##BHBiv K2bm`2 QM Mpf (W ) +Q``2bTQM/BM; iQ 2t+iHv n TQBMib TH+2/ BM/2T2M/2MiHv ++Q`/BM; iQ i?2 /Bbi`B#miBQM λ- M/ r?2`2 λ0 Bb i?2 T`Q##BHBiv K2bm`2 QM Mpf (W ) rBi? HH Bib Kbb QM i?2 2KTiv TQBMi K2bm`2X h?mb- mM/2` Q-  `M/QK +QM};m`iBQM Bb ;2M2`i2/ b 7QHHQrbX qBi? T`Q##BHBiv π0 Bi Bb i?2 2KTiv +QM};m`iBQM- M/ rBi? T`Q##BHBiv πn Un ≥ 1V- Bi +QMbBbib Q7 n TQBMib /`rM BM/2T2M/2MiHv QM W ++Q`/BM; iQ i?2 T`Q##BHBiv /Bbi`B#miBQM λX h?2 2tT2+iiBQM rBi? `2bT2+i iQ Q Bb /2MQi2/ #v EQ X h?2 7mM+iBQM x →   jn (x1 , . . . , xn ) := n!pn jn (x1 , . . . , xn ) Bb +HH2/ i?2 CMQbbv k /2MbBiv Q7 Q`/2` nX Aib BMi2`T`2iiBQM Bb i?i  jn (x1 , . . . , xn )λ(dx1 ) · · · λ(dxn ) Bb i?2 T`Q##BHBiv Q7 }M/BM; 2t+iHv n TQBMib- 1 BM dx1 - Ę- 1 BM dxn - r?2`2 i?2 BM}MBi2bBKH b2ib dx1 , . . . , dxn `2 KmimHHv /BbDQBMiX k

(CMQbbv- RN93)X

eXRX h>1 CLPaau .1LaAhu

kjR

.2}MBiBQM BM h2`Kb Q7  _/QMĜLBFQ/ɷK .2`BpiBp2 G2i : Mpf (W ) → [0, ∞] #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM bm+? i?i EQ [ (X)] = 1 .

UeXRV

h?Bb +QM/BiBQM HHQrb mb iQ /2}M2  T`Q##BHBiv P QM Mpf (W ) #v Bib _/QMĜ LBFQ/ɷK /2`BpiBp2 rBi? `2bT2+i iQ Qc dP (x) = (x) . dQ

UeXkV

S`Q##BHBiv Q Bb +HH2/ i?2 `272`2M+2 K2bm`2X h?2Q`2K eXRXR h?2 _/QMĜLBFQ/ɷK /2`BpiBp2 (x) Bb `2Hi2/ iQ i?2 +QmMi /Bb@ i`B#miBQM {pn }n∈N M/ i?2 /2MbBiB2b {jn }n≥1 b 7QHHQrb, p0 = π0 0 (∅) M/ 7Q` n ≥ 1

n ({x1 , . . . , xn }) λ(dx1 ) . . . λ(dxn ) pn = πn Wn

Ui?2 MQiiBQM n ({x1 , . . . , xn }) biM/b 7Q` (x) r?2M x = {x1 , . . . , xn }V M/ jn (x1 , . . . , xn ) = 

n ({x1 , . . . , xn }) .

({x , 1 . . . , xn }) λ(dx1 ) . . . λ(dxn ) Wn n

S`QQ7X h?2 +b2 n = 0 Bb +H2`X 6Q` n ≥ 1pn = P (N (W ) = n) = EQ [1{N (W )=n} (X)] = EQ [1{N (W )=n} n ({X1 , . . . , Xn })] = Q(N (W ) = n) × EQ [ (X) | N (W ) = n] = πn EQ [ n ({X1 , . . . , Xn }) | N (W ) = n]  = πn

n ({x1 , . . . , xn }) λ(dx1 ) · · · λ(dxn ) . Wn

G2i MQr ϕ : Mpf (W ) → [0, ∞] #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX 6Q` n ≥ 1E[ϕ(X) | N (W ) = n] =

E[1{N (W )=n} ϕ(X)] P (N (W ) = n)

M/ E[ϕ(X)1{N (W )=n} ] = EQ [1{N (W )=n} ϕ(X) (X)] = Q(N (W ) = n) EQ [ϕ(X) (X) | N (W ) = n]  ϕ({x1 , . . . , xn }) n ({x1 , . . . , xn }) λ(dx1 ) · · · λ(dxn ). = πn Wn

h?2`27Q`2

 ϕ({x1 , . . . , xn }) jn (x1 , . . . , xn ) λ(dx1 ) . . . λ(dxn ) ,

E[ϕ(X) | N (W ) = n] = Wn

r?2`2 jn Bb b /2}M2/ BM h?2Q`2K eXRXRX



kjk *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a _2K`F eXRXk LQi2 i?i +QM/BiBQM UeXRV `2/b π0 (∅) +

∞  n=1

i?i Bb-

∞ n=0



n ({x1 , . . . , xn }) λ(dx1 ) . . . λ(dxn ) = 1 ,

πn Wn

pn = 1X

_2K`F eXRXj AM T`BM+BTH2- i?2 KQ/2HH2` ?b BM KBM/ UV,  bT2+B}+ +QmMi /Bbi`B@ #miBQM {pn }n≥0 M/- T2`?Tb KQ`2 BKTQ`iMiHv- U#V, i?2 dzb?T2Ǵ Q7 i?2 /2MbBiB2b jn +Q``2bTQM/BM; iQ i?2 K2bm`2 λX "v i?2 b?T2 Q7 jn - r2 K2M  7mM+iBQM jn 2[mH iQ  +QMbiMi UTQbbB#Hv /2T2M/BM; QM nV iBK2b jn X h?2 b?T2 Q7 i?2 /2M@ bBiB2b `2T`2b2Mib i?2 ivT2 Q7 BMi2`+iBQM #2ir22M TQBMib /QTi2/ #v i?2 KQ/2HH2`X b i?2 2tT`2bbBQM Q7 i?2 /2MbBiv BM h?2Q`2K eXRXR b?Qrb- i?2 b?T2 Q7 jn Bb i?2 bK2 b i?i Q7 n X h?2`27Q`2- KmHiBTHvBM; n - 7Q` HH n- #v  TQbBiBp2 +QMbiMi Kn /Q2b MQi +?M;2 i?2 /2MbBiB2b jn X >Qr2p2`- Bi +?M;2b i?2 +QmMi /Bbi`B#miBQM 7`QK pn iQ pn = Kn pn X h?Bb 7`22/QK +M MQr #2 mb2/ BM i?2 2tT`2bbBQM Q7 pn BM h?2Q`2K eXRXR iQ Q#iBM i?2 /2bB`2/ +QmMiBM; /Bbi`B#miBQMX h?2`2 Bb ?Qr2p2`  /B{+mHiv BM +``vBM; i?Bb T`Q;`K iQ Bib 2M/ bBM+2 i?2 +QKTmiiBQM Q7 i?2 BMi2;`H 

({x1 , . . . , xn })λ(dx1 ) . . . λ(dxn ) Bb BM ;2M2`H /B{+mHiX Wn n _2K`F eXRX9 LQi2 HbQ i?i r2 +MMQi mb2 M `#Bi``v KmHiBTHB+iBp2 7+iQ` Kn bBM+2 r2 ?p2 iQ `2bT2+i i?2 +QM/BiBQM n≥0 pn = 1- i?i Bb Kn p n = 1 . n≥0

AM i?2 TTHB2/ HBi2`im`2- QM2 Q7i2M KF2b i?2 +?QB+2 Q7  SQBbbQM `272`2M+2 K2bm`2X 1tKTH2 eXRX8, SQBbbQM _272`2M+2 J2bm`2- hF2 RX hF2 Q = P ν iQ #2 i?2 /Bbi`B#miBQM Q7  SQBbbQM T`Q+2bb QM W rBi? i?2 }MBi2 MQM@iQKB+ BMi2MbBiv K2bm`2 ν Ui?2 +Q``2bTQM/BM; TQBMi T`Q+2bb Bb }MBi2 bBM+2 i?2 p2`;2 MmK#2` Q7 TQBMib ν(W ) Bb }MBi2V M/ H2i : Mpf (W ) → [0, ∞] #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM bm+? i?i Eν [ (X)] = 1 . h?Bb +QM/BiBQM HHQrb mb iQ /2}M2  T`Q##BHBiv P QM Mpf (W ) #v i?2 _/QMĜ LBFQ/ɷK /2`BpiBp2 dP (x) = (x) . dP ν h?Bb Bb  T`iB+mH` +b2 Q7 i?2 ;2M2`H +QMbi`m+iBQM #Qp2- rBi? πn = M/ λ(dx) = ν(dx)/ν(W )X ++Q`/BM; iQ h?2Q`2K eXRXR e−ν(W ) pn =

n ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) n! Wn M/ jn (x1 , . . . , xn ) =  Wn

n ({x1 , . . . , xn }) .

n ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn )

e−ν(W ) ν(W )n n!

eXRX h>1 CLPaau .1LaAhu

kjj

AM T`iB+mH`- 7Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM ϕ : Mpf (W ) → RE [ϕ(X)] =

 ∞  e−ν(W ) n=0

n!

ϕ(x1 , . . . , xn ) n ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) . UeXjV Wn

1tKTH2 eXRXe, >`/@+Q`2 SQBMi S`Q+2bb- hF2 RXj h?2 `272`2M+2 K2bm`2 Bb b BM 1tKTH2 eXRX8X AM i?2 ?`/@+Q`2 KQ/2H i `M;2 R > 0 n ({x1 , . . . , xn }) = αβ n Π1≤i2R} , r?2`2 β > 0 M/ α > 0 `2 +?Qb2M bm+? i?i (x) BMi2;`i2b iQ R rBi? `2bT2+i iQ P ν X AM i?2 +b2 i?i β = 1- i?2 T`Q##BHBiv P Bb P ν +QM/BiBQM2/ #v i?2 2p2Mi i?i i?2`2 Bb MQ TB` Q7 TQBMib i KmimH /BbiM+2 ≤ RX *HHBM; i?Bb 2p2Mi F - r2 i?2M ?p2 P (·) = P ν (· | F ). PM2 UBM2{+B2MiV rv iQ bKTH2 P Bb iQ KF2 BM/2T2M/2Mi bKTHBM;b Q7 i?2 /Bbi`B@ #miBQM P ν mMiBH  bKTH2 BM F Bb Q#iBM2/X h?2 BMi2;`H BMpQHp2/ BM i?2 2tT`2bbBQM 7Q` pn /Q2b MQi ?p2  +HQb2/ 7Q`KX amTTQb2 r2 +?M;2 α M/ β iQ α = a α M/ β  = b β `2bT2+iBp2HvX h?2 /2MbBiB2b Q7 Mv Q`/2` `2 MQi Hi2`2/ #mi i?2 T`Q##BHBiv Q7 ?pBM; n TQBMib BM i?2 +QM};m`iBQM #2+QK2b pn = abn pn X h?2 +QM/BiBQM E ν [ (x)] = 1 Bb i?2M   pn = a bn p n = 1 . n≥0

n≥0

6BMBi2 J`F2/ SQBMi S`Q+2bb2b q2 MQr BMi`Q/m+2 K`FbX G2i G #2  #QmM/2/ K2bm`#H2 bm#b2i Q7 Rm M/ H2i G #2 i?2 σ@}2H/ Q7 "Q`2H bm#b2ib Q7 GX  }MBi2 K`F2/ TQBMi T`Q+2bb QM G rBi? K`Fb BM i?2 K2bm`#H2 bT+2 (K, K) Bb  `M/QK 2H2K2Mi X ∈ Mpf (G × K) Q7 i?2 7Q`K {(Y1 , Z1 ), . . . , (Yn , Zn )} r?2M |X| = n- r?2`2 i?2 Zn Ƕb iF2 i?2B` pHm2b BM KX h?Bb +b2 i?2`27Q`2 /Q2b MQi `2[mB`2 M /TiiBQM Q7 i?2 i?2Q`v i i?Bb bi;2bBM+2 X Bb  }MBi2 +QM};m`iBQM QM E := G × KX 1tKTH2 eXRXd, h?2 SQBbbQM _272`2M+2 J2bm`2- hF2 kX AM 1tKTH2 eXRX8- i?2 K2bm`2 ν +M #2 +?Qb2M Q7 i?2 T`Q/m+i 7Q`K ν(dx) = ν(dy, dz) = σ(dy)QZ (dz) , r?2`2 QZ Bb  T`Q##BHBiv /Bbi`B#miBQM QM (K, K) M/ σ Bb  }MBi2 MQM@iQKB+ K2bm`2 QM G U+QMb2[m2MiHv- ν Bb MQM@iQKB+VX h?Bb BKTHB2b BM T`iB+mH` i?i j

(ai`mbb- RNd8)X

kj9 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a mM/2` i?2 `272`2M+2 T`Q##BHBiv Q- Y Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 σ M/ i?i i?2 K`Fb `2 BB/ rBi? +QKKQM /Bbi`B#miBQM QZ M/ BM/2T2M/2Mi Q7 i?2 #bB+ TQBMi T`Q+2bb YX 1tKTH2 eXRX3, h?2 S2M2i`#H2 aT?2`2b JQ/2H- hF2 RX AM i?Bb 2tKTH2G Bb  K2bm`#H2 bm#b2i Q7 Rm Q7 }MBi2 pQHmK2 U7Q` BMbiM+2  }MBi2 m@+m#2Vσ = λ m - K = {1, 2}- QZ ({1}) = α1 M/ QZ ({2}) = α2 X AM T`iB+mH`- Y Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ M/ i?2 TQBMi T`Q+2bb2b Y1 M/ Y2 QM E i?i +QmMi i?2 TQBMib Q7 Y rBi? K`Fb 1 M/ 2 `2bT2+iBp2Hv `2 BM/2T2M/2Mi ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b rBi? BMi2MbBiv `2bT2+iBp2 BMi2MbBiB2b λ1 := λα1 M/ λ2 := λα2 X AM i?2 T2M2i`#H2 bT?2`2b KQ/2H i `M;2 R > 0 (x) = αβ |y| 1{d(y1 ,y2 )>R} , r?2`2 d(y1 , y2 ) Bb i?2 bKHH2bi /BbiM+2 #2ir22M  TQBMi Q7 y1 M/  TQBMi Q7 y2 r?2`2 β > 0 M/ α > 0 `2 +?Qb2M bm+? i?i (x) BMi2;`i2b iQ R rBi? `2bT2+i iQ P ν X AM i?2 +b2 i?i β = 1- i?2 T`Q##BHBiv P Bb P ν +QM/BiBQM2/ #v i?2 2p2Mi i?i i?2`2 Bb MQ TB` Q7 TQBMib- QM2 BM Y1 i?2 Qi?2` BM Y2 - i KmimH /BbiM+2 ≤ RX *HHBM; i?Bb 2p2Mi F - r2 ?p2 P (·) = P ν (· | F ). ;BM- QM2 UBM2{+B2MiV rv iQ bKTH2 P Bb iQ KF2 BM/2T2M/2Mi bKTHBM;b Q7 i?2 /Bbi`B#miBQM P ν mMiBH  bKTH2 BM F Bb Q#iBM2/X

h?2 1tpBbB#H2 *QM/BiBQMH AMi2MbBiv h?2 #bB+ /2}MBiBQM Bb i?2 7QHHQrBM;9 .2}MBiBQM eXRXN G2i X #2  }MBi2 TQBMi T`Q+2bb QM i?2 K2bm`#H2 b2i W rBi?  /Bbi`B#miBQM /2}M2/ #v  T`Q##BHBiv /2MbBiv (x) rBi? `2bT2+i iQ i?2 SQBbbQM T`Q##BHBiv K2bm`2 Q = P ν U1tKTH2 eXRX8V b BM UeXkVX h?2 7mM+iBQM λ : W × Mpf (W ) → [0, ∞) /2}M2/ #v λ(u, x) =

(x ∪ {u})

(x)

(u ∈ / x)

UeX9V

(= 0 B7 (x) = 0) Bb +HH2/ i?2 2tpBbB#H2 BMi2MbBiv Q7 i?2 }MBi2 TQBMi T`Q+2bb XX _2K`F eXRXRy  7Q`KH `2b2K#HM+2 rBi? i?2 MQiBQM Q7 biQ+?biB+ BMi2MbBiv 7Q`  TQBMi T`Q+2bb QM i?2 HBM2 Bb i?2 7QHHQrBM;X _2im`MBM; KQK2Mi`BHv iQ i?2 MQiiBQM Q7 *?Ti2` 8- H2i N #2  bBKTH2 TQBMi T`Q+2bb QM i?2 BMi2`pH [0, T ] rBi?  T`2@ /B+i#H2 FtN @BMi2MbBiv {λ(t, N )}t∈[0,T ] - bbmK2/ #QmM/2/ BM i?Bb 2tKTH2X q2 Kv +QMbB/2` i?i i?2 mM/2`HvBM; T`Q##BHBiv P Bb Q#iBM2/ 7`QK  T`Q##BHBiv P0 QM 9

a22 ?Qr2p2` a2+iBQM 3X8X

eXRX h>1 CLPaau .1LaAhu

kj8

i?2 bK2 T`Q##BHBiv bT+2 i?i KF2b Q7 N M ?TT rBi? BMi2MbBiv 1 pB M #bQ@ Hmi2Hv +QMiBMmQmb +?M;2 Q7 T`Q##BHBiv K2bm`2 2K#Q/B2/ #v i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 L(T, N ) r?2`2   t  L(t, N ) := Πn;Tn ∈(0,t] λ(Tn , N ) exp − (λ(s, N ) − 1) ds . 0

AM T`iB+mH`- B7 QM2 T`2i2M/b i?i i?2`2 Bb M //BiBQMH TQBMi i t- i?Bb 2tT`2bbBQM #2+QK2b L(t, N ∪ t) = λ(t, N )L(t, N ) , M/ i?2`27Q`2 λ(t, N ) =

L(t, N ∪ t) . L(t, N )

1tKTH2 eXRXRR, h?2 S2M2i`#H2 aT?2`2b JQ/2H- hF2 kX AM i?2 T2M2@ i`#H2 bT?2`2b KQ/2H i `M;2 R > 0- i?2 2tpBbB#H2 BMi2MbBiv ;Bp2M #v UeX9V BM i?Bb +b2 iF2b i?2 7QHHQrBM; 7Q`K, B7 u = (v, 1)λ(u, x) = β1{d(v,y2 )>R} , r?2`2 d(v, y2 ) Bb i?2 bKHH2bi /BbiM+2 #2ir22M v M/  TQBMi Q7 y2 - rBi?  bBKBH` `2bmHi 7Q` u = (v, 2)X S`QQ7X q2 mb2 i?2 MQiiBQM x = (y, z)- i?mb b2T`iBM; i?2 K`Fb 7`QK i?2 TQBMibX h?Bb Bb  bQK2r?i #mbBp2 MQiiBQM- #mi BMMQ+mQmb B7 r2 /QTi i?2 +QMp2MiBQM i?i- r?2M |x| = n- {x1 , . . . , xn } = {(y1 , z1 ), . . . , (yn , zn )} Ui?2 BM/2t Q7  K`F BM/B+i2b i?2 TQBMi iQ r?B+? Bi Bb ii+?2/VX h?2`27Q`2- rBi? i?Bb MQiiBQM- (x) =

(y, z)X G2i v M/ ζ #2 TQBMib Q7 G M/ K `2bT2+iBp2HvX h?2 2tpBbB#H2 BMi2MbBiv i (v, ζ) Bb- #v /2}MBiBQMλ((v, ζ), (y, z)) =

(y ∪ {v}, z ∪ {ζ}) ,

(y, z)

λ((v, 1), (y, z)) =

(y ∪ {v}, z ∪ {1}) ,

(y, z)

M/ BM T`iB+mH`() .

*HH i?2 TQBMi T`Q+2bb y1 i?2 1@T`Q+2bb Q7 y- rBi?  bBKBH` K2MBM; 7Q` i?2 2@ T`Q+2bb Q7 yX h?2`2 `2 irQ TQbbB#BHBiB2b +QM+2`MBM; i?2 /DmM+iBQM Q7  1@TQBMi i vX 1Bi?2` d(v, y2 ) > R- BM r?B+? +b2 i?2 /BbiM+2 #2ir22M i?2 M2r 1@T`Q+2bb M/ y2 /Q2b MQi z2+i i?2 +QM/BiBQM Q7 b2T`iBQM Q7 i?2b2 irQ TQBMi T`Q+2bb2b M/ i?2 MmK2`iQ` M/ /2MQKBMiQ` BM UV `2 i?2 bK2X AM i?2 Qi?2` +b2- i?Bb +QM/BiBQM Bb #`QF2M M/ i?2 MmK2`iQ` BM UV Bb MmHHX >2M+2 i?2 MMQmM+2/ `2bmHiX 

kje *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a

eXk h?2 aTiBH aKQQi?BM; 7Q`KmH h?2 7QHHQrBM; `2bmHi 72im`2b v2i MQi?2` pi` Q7 *KT#2HHǶb 7Q`KmH-  bKQQi?@ BM; 7Q`KmH Q7 i?2 bK2 ivT2 b i?2 QM2 7Q` TQBMi T`Q+2bb2b QM i?2 HBM2 Q7 i?2 T`2pBQmb +?Ti2`X h?2Q`2K eXkXR G2i Q = P ν #2 b BM 1tKTH2 eXRX8X h?2M- 7Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM g : W × Mpf (W ) → [0, ∞]  

 g(x, X\x) = E g(u, X) λ(u, X)ν(du) , UeX8V E W

x∈X

r?2`2 X\x := X\{x}X S`QQ7X 6`QK UeXjV

 g(u, X)λ(u, X)ν(du) A := E W

 ∞  e−ν(W )

=

n!

n=0

g(u, {x1 , . . . , xn }) λ(u, {x1 , . . . , xn }) × · · · W n+1

· · · ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn )ν(du)

 ∞  e−ν(W )

=

n!

n=0

g(u, {x1 , . . . , xn }) ({x1 , . . . , xn } ∪ {u}) W n+1

ν(dx1 ) . . . ν(dxn )ν(du) . qBi? i?2 +?M;2 Q7 p`B#H2b u → xi - x1 → x1 , . . . , xi−1 → xi−1 - xi → xi+1 , . . . , xn → xn+1 ,  g(u, {x1 , . . . , xn }) ({x1 , . . . , xn } ∪ {u}) ν(dx1 ) . . . ν(dxn )ν(du) = W n+1

 g(xi , {x1 , . . . , xi−1 ,xi+1 , . . . , xn+1 }) ({x1 , . . . , xi−1 , xi+1 , . . . , xn+1 } ∪ {xi }) W n+1

ν(dx1 ) . . . ν(dxi−1 ) ν(dxi+1 ) . . . ν(dxn+1 ) ν(dxi ) . h?2`27Q`2 A=

 ∞  e−ν(W ) n=0

=

∞  n=1



n! e−ν(W ) n!

W n+1



n+1 1  g(xi , {x1 , . . . , xi−1 , xi+1 , . . . , xn+1 }) × · · · n + 1 i=1

· · · ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) ν(du) n 

g(xi , {x1 , . . . , xi−1 , xi+1 , . . . , xn }) × · · ·

W n i=1

· · · ({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) .

eXkX h>1 aShAG aJPPh>AL: 6P_JlG

kjd

g(x, X\x) = 0 B7 |X| = 0   g(x, X\x) B := E

LQr- bBM+2

x∈X

x∈X

=

 ∞  e−ν(W ) n=1

n!

n 

g(xi , {x1 , . . . , xi−1 , xi+1 , . . . , xn }) × · · ·

W n i=1

({x1 , . . . , xn }) ν(dx1 ) . . . ν(dxn ) , 

M/ i?2`27Q`2 A = BX

1tKTH2 eXkXk, h?2 aKQQi?BM; 6Q`KmH 7Q` SQBbbQM S`Q+2bb2bX A7 P ≡ P ν Q`- 2[mBpH2MiHv- (x) ≡ 1 M/ +QMb2[m2MiHv λ(u, x) ≡ 1- r2 ?p2 i?2 dzbKQQi?BM; 7Q`KmHǴ   

 E g(x, X\x) = E g(u, X) ν(du) . W

x∈X

_2K`F eXkXj M BMi2`T`2iiBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv bBKBH` iQ i?2 BM}MBi2b@ BKH BMi2`T`2iiBQM Q7 i?2 biQ+?biB+ BMi2MbBiv 7Q` TQBMi T`Q+2bb2b QM i?2 HBM2 Bb pBH#H2X G2i B(u, ε) #2 i?2 +HQb2/ #HH +2Mi2`2/ i u ∈ Rm M/ rBi? `/Bmb ε > 0X X G2i F #2 M `#Bi``v b2i BM FB(u,ε) i?2 σ@}2H/ ;2M2`i2/ #v i?2 `M/QK 2H2K2Mi X ∩ B(u, ε)X LQiB+2 i?i 1F (X) = 1F (X\v) B7 v ∈ B(u, ε)X h?2M    E [1F (X)X(B(u, ε))] = E 1B(u,ε) (v)1F (X) v∈X

 =E



 1B(u,ε) (v)1F (X\v)

v∈X



1F (X)λ(v, X) ν(dv) B(u,ε)

 λ(v, X) ν(dv) . = E 1F (X) =E

B(u,ε) X FB(u,ε)

r2 ?p2 T`Qp2/ i?i 

  X X =E . λ(v, X) ν(dv) | FB(u,ε) E X(B(u, ε)) | FB(u,ε)

aBM+2 F Bb `#Bi``v BM

B(u,ε)

amTTQb2 i?i i?2 `272`2M+2 SQBbbQM T`Q+2bb Bb M ?TT rBi? mMBi BMi2MbBiv- i?i Bbν(dx) = dxX b ε Bb `#Bi``BHv bKHH- i?Bb ;Bp2b +QMbBbi2M+v iQ i?2 7QHHQrBM; HQ+H BMi2`T`2iiBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv, λ(u, X) = lim

1

u↓0 m (B(u, ε))

E [X(B(u, ε)) | X\{u}] .

UeXeV

>Qr2p2` iQ ;m`Mi22 i?Bb- bQK2 i2+?MB+H +QM/BiBQMb `2 M22/2/- 7Q` BMbiM+2 i?i u → λ(u, X) #2 HKQbi bm`2Hv #QmM/2/ M/ +QMiBMmQmb U1t2`+Bb2 eXdXjVX

kj3 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a

eXj 1tpBbB#BHBiv M/ S`2/B+i#BHBiv AM i?Bb b2+iBQM- r2 b?HH b22 ?Qr 7` i?2 7Q`KH MHQ;v #2ir22M i?2 i?2Q`B2b Q7 biQ+?biB+ BMi2MbBiv M/ Q7 2tpBbB#H2 BMi2MbBiv +M #2 Tmb?2/X .2}MBiBQM eXjXR  biQ+?biB+ T`Q+2bb Q7 i?2 7Q`K {H(x, X)}x∈W r?2`2 H(x, X) = g(x, X\x) 7Q` bQK2 K2bm`#H2 7mM+iBQM g : (W × Mp (W ), B(W ) ⊗ Mp (W ) → (R, B(R)) Bb +HH2/ M 2tpBbB#H2 T`Q+2bbX P#b2`pBM; i?i bBM+2 ν Bb  /Bzmb2 K2bm`2- i?2 `B;?i@?M/ bB/2 Q7 UeX8V +M #2 r`Bii2M b 

g(u, X\u) λ(u, X)ν(du) .

E W

h?2`27Q`2 h?2Q`2K eXkXR +M #2 `2T?`b2/ BM i?2 7QHHQrBM; 7Q`K, h?2Q`2K eXjXk G2i Q = P ν #2 b BM 1tKTH2 eXRX8X h?2M- 7Q` Mv MQM@M2;iBp2 2tpBbB#H2 T`Q+2bb {H(x, X)}x∈X   

 H(x, X) = E H(u, X) λ(u, X)ν(du) . UeXdV E x∈X

W

aBKBH`Hv r2 Kv `2TH+2 BM UeXdV λ(u, X) #v Bib dz2tpBbB#H2 p2`bBQMǴ λ(u, X\u)M/  T`QQ7 bBKBH` iQ i?i Q7 h?2Q`2K 8XRXj3 ;Bp2b, h?2Q`2K eXjXj h?2 T`2/B+i#H2 p2`bBQM λ(u, X\u) Q7 STM;2HQmǶb 2tpBbB#H2 BM@ i2MbBiv Bb bm+? i?i- P @HKQbi@bm`2Hvλ(u, X\u) > 0 7Q` HH u ∈ X . _2K`F eXjX9 G2i (t, μ) → ϕ(t, μ) #2  K2bm`#H2 MQM@M2;iBp2 KTTBM; 7`QK R × Mp (R) iQ R bm+? i?i B7 i?2 `2bi`B+iBQMb Q7 μ M/ μ iQ (−∞, t] `2 i?2 bK2i?2M ϕ(t, μ) = ϕ(t, μ )X AM T`iB+mH`- `2im`MBM; KQK2Mi`BHv iQ i?2 MQiiBQM Q7 *?Ti2` 8- i?2 T`Q+2bb {ϕ(t, N )}t∈R Bb FtN /Ti2/X G2iiBM; H(t, N ) := ϕ(t, N \t) r2 +HH {H(t, N )}t∈R M FtN @T`2/B+i#H2 T`Q+2bbX "v h?2Q`2K eXkXR- M/ H2iiBM; λ(t, N ) #2 i?2 2tpBbB#H2 BMi2MbBiv Q7 N

 T

 ϕ(t, N \t) N (dx) = E ϕ(t, N )λ(t, N ) dt . E 0

[0,T ]

aBM+2 ϕ(t, N ) M/ ϕ(t, N\t) /Bz2` QMHv QM  b2i  Q7 MmHH G2#2b;m2 K2bm`2- i?2 T `B;?i@?M/ bB/2 2[mHb E 0 ϕ(t, N \t)λ(t, N ) dt - M/ r2 Q#iBM  E [0,T ]

 H(t, N )N (dt) = E

T 0

H(t, N )λ(t, N ) dt .

eXjX 1soAaA"AGAhu L. S_1.A*h"AGAhu

kjN

h?2Q`2K eXjX8 G2i Q = P ν #2 b BM 1tKTH2 eXRX8 M/ H2i g : W × Mpf (W ) →  [0, ∞] #2  K2bm`#H2 7mM+iBQM bm+? i?i E W |g(u, X)| λ(u, X)ν(du) < ∞ bQ i?i BM T`iB+mH`   1A (x)g(x, X\x) − 1A (u)g(u, X) λ(u, X)ν(du) UeX3V MA (g) := W

x∈X

Bb r2HH /2}M2/ 7Q` HH K2bm`#H2 b2ib A ⊆ W X G2i 7Q` Mv K2bm`#H2 b2i C ⊆ W FC := σ(X ∩ C). h?2M- 7Q` HH K2bm`#H2 b2ib A ⊆ W E [MA (g) | FA ] = 0 . S`QQ7X Ai bm{+2b iQ T`Qp2 i?i 7Q` HH #QmM/2/ K2bm`#H2 7mM+iBQMb f : Mpf (W ) → [0, ∞] E MA (g)f (X ∩ A) = 0 , i?i Bb- α = β- r?2`2  α := E f (X ∩ A)



 1A (x)g(x, X\x)

x∈X

M/

 1A (u)g(u, X) λ(u, X)ν(du) . β := E f (X ∩ A) W

hFBM; BMiQ ++QmMi i?2 7+i i?i B7 x ∈ A- i?2M f ((X\x) ∩ A) = f (X ∩ A) 

r2 ?p2 i?i α=E



 1A (x)f ((X\x) ∩ A))g(x, X\x) ,

x∈X

 [mMiBiv 2[mH iQ β #v h?2Q`2K eXkXRX



_2K`F eXjXe h?2Q`2K eXjX8 b22Kb iQ #`BM; mb +HQb2 iQ i?2 MQiBQM Q7 K`iBM;H2X AM/22/- B7 A1 M/ A2 `2 K2bm`#H2 bm#b2ib Q7 W bm+? i?i A1 ⊂ A2 - r2 ?p2 i?i MA2 (g) − MA1 (g) = MA2 −A1 (g) M/ i?2`27Q`2

E MA2 −A1 (g) | FA2 −A1 = 0 .

h?Bb Bb b 7` b r2 +M ;Q- M/ Bi Bb MQi +H2` b Q7 iQ/v B7 i?Bb /2}MBiBQM Q7  K`iBM;H2 UBM/2t2/ #v b2ibV ?b Mv +?M+2 iQ #2 b T`Q/m+iBp2 b i?i Q7  K`iBM;H2 rBi?  iBK2 BM/2tX

k9y *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a h?2 aTiBH qiM#2Ƕb h?2Q`2K Hi?Qm;? i?2 2tpBbB#H2 BMi2MbBiv i?2Q`v H+Fb BM Bib bTiBH 7Q`K  HBMF rBi? K`@ iBM;H2 i?2Q`v Ur?B+? Bb i?2 bHB2Mi 72im`2- i?2 Q`B;BMH ;QH M/ i?2 KBM BMi2`2bi Q7 i?2 i?2Q`v Q7 biQ+?biB+ BMi2MbBivV- i?2`2 2tBbib  +?`+i2`BxiBQM Q7 SQBbbQM T`Q+2bb2b `2KBMBb+2Mi Q7 qiM#2Ƕb i?2Q`2KX h?2Q`2K eXjXd G2i X #2  bBKTH2 }MBi2 TQBMi T`Q+2bb QM W X amTTQb2 i?i 7Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM g : W × Mpf (W ) → [0, ∞] E









g(x, X\x) = E

x∈X

g(u, X) ν(du) ,

UeXNV

W

r?2`2 ν Bb  σ@}MBi2 /Bzmb2 K2bm`2 QM EX h?2M X Bb  SQBbbQM T`Q+2bb rBi? K2M BMi2MbBiv K2bm`2 νX  S`QQ7X 6Q` 2+? n ≥ 0- i?2 [mMiBiv E E 1A (x)1{X(A)=n+1} X(dx) rBHH #2 2pHm@ i2/ BM irQ /Bz2`2Mi rvbX 6B`bi- Q#pBQmbHv- i?Bb [mMiBiv 2[mHb (n + 1)P (X(A) = n + 1)X LQr 



E 1A (x)1{X(A)=n+1} X(dx) = E 1A (x)1{(X\x)(A)=n} X(dx) , E

E

 [mMiBiv r?B+? Bb- ++Q`/BM; iQ i?2 ?vTQi?2bBb- 2[mH iQ

 1A (x)1{X(A)=n} ν(dx) = ν(A)P (X(A) = n) . E E

h?2`27Q`2- 7Q` HH n ≥ 0ν(A)P (X(A) = n) = (n + 1)P (X(A) = n + 1) , 7`QK r?B+? Bi 7QHHQrb i?i P (X(A) = n) = P (X(A) = 0)

ν(A)n n!

(n ≥ 0)

M/- #v MQ`KHBxiBQM- P (X(A) = 0) = e−ν(A) X h?2 `2bmHi i?2M 7QHHQrb 7`QK _ûMvBǶb pQB/M+2 i?2Q`2K Uh?2Q`2K RXjXk8VX 

eX9 6BMBi2 J`FQp SQBMi S`Q+2bb2b G2i MQr W #2 2M/Qr2/ rBi?  bvKK2i`B+ M/ `2~2tBp2 `2HiBQM ∼ Ui?i Bb- 7Q` HH u- v BM q- u ∼ u M/ u ∼ v ⇔ v ∼ uVX A7 u ∼ v- u M/ v `2 +HH2/ M2B;?#Qm`bX  MQM@2KTiv +QM};m`iBQM x ∈ Mpf (W ) Bb +HH2/  +HB[m2 UrBi? `2bT2+i iQ ∼V B7 u ∼ v 7Q` HH u, v ∈ xX AM T`iB+mH`- Mv bBM;H2iQM {u} Bb  +HB[m2 bBM+2 u ∼ uX h?2 2KTiv +QM};m`iBQM ∅ Bb  +HB[m2- #v +QMp2MiBQMX

eX9X 6ALAh1 J_EPo SPALh S_P*1aa1a

k9R

.2}MBiBQM eX9XR X Bb +HH2/  J`FQp TQBMi T`Q+2bb rBi? `2bT2+i iQ ∼ B7 7Q` HH x ∈ Mpf (W ) bm+? i?i (x) > 0UV (y) > 0 7Q` HH y ∈ Mpf (W ) bm+? i?i y ⊆ x- M/ U#V 7Q` HH u ∈ W - λ(u, x) /2T2M/b QMHv QM u M/ QM {x ∈ x ; x ∼ u}- i?i Bbλ(u, x) = λ(u, x ∪ {v}) r?2M2p2` v ∼ uX S`QT2`iv UV Bb +HH2/ i?2 ?2`2/Bi`v T`QT2`ivX S`QT2`iv U#V bvb i?i λ(u, x) /2@ T2M/b QMHv QM u M/ QM i?2 M2B;?#Qm`b Q7 u BM xX h?2 7QHHQrBM; Bb  +?`+i2`BxiBQM Q7 i?2 J`FQp T`QT2`iv,8 h?2Q`2K eX9Xk G2i X #2  }MBi2 TQBMi T`Q+2bb /2}M2/ #v  T`Q##BHBiv /2MbBiv

(x) rBi? `2bT2+i iQ i?2 /Bbi`B#miBQM P ν Q7  SQBbbQM T`Q+2bb Q7 MQM@iQKB+ }MBi2 BMi2MbBiv K2bm`2 ν b BM 1tKTH2 eXRX8X 6Q` X iQ #2  J`FQp T`Q+2bb rBi? `2bT2+i iQ ∼- Bi Bb M2+2bb`v M/ bm{+B2Mi i?i 

(x) = ϕ(y) UeXRyV y⊆x

7Q` bQK2 7mM+iBQM ϕ : Mpf (W ) → [0, ∞) bm+? i?i ϕ(y) = 1 r?2M2p2` y Bb MQi  +HB[m2X S`QQ7X am{+B2M+v, amTTQb2 i?i UeXRyV ?QH/b 7Q` bQK2 ϕ : Mpf (W ) → [0, ∞) bm+? i?i ϕ(y) = 1 B7 y Bb MQi +HB[m2X q2 +?2+F UV M/ U#V Q7 .2}MBiBQM eX9XRX UV "v i?2 ?2`2/Bi`v T`QT2`iv-! (x) > 0 BKTHB2b i?i ϕ(y) > 0 7Q` HH y ⊆ xM/ BM T`iB+mH` B7 z ⊆ x- (z) = y⊆z ϕ(y) > 0X U#V q`Bi2 7Q` u ∈ /x

! ϕ(y) ϕ(y ∪ {u}) 

(x ∪ {u}) y⊆x y⊆x ! = = ϕ(y ∪ {u}) . λ(u, x) =

(x) ϕ(y) y⊆x !

y⊆x

A7 y ⊆ x M/ y ∪ {u} Bb MQi  +HB[m2- i?2M ϕ(y ∪ {u}) = 1 UM/ i?2`27Q`2 /Q2b MQi /2T2M/ QM Mvi?BM;VX A7 y ⊆ x M/ y ∪ {u} Bb  +HB[m2- i?2M ϕ(y ∪ {u}) /2T2M/b QM u M/ QM HH i?2 TQBMib x ∈ y- r?B+? HH biBb7v i?2 +QM/BiBQM x ∼ u bBM+2 y ∪ {u} Bb  +HB[m2X L2+2bbBivX q2 bmTTQb2 i?i (x) Bb  J`FQp /2MbBiv rBi? `2bT2+i iQ ∼X AM T`iB+mH`- 7Q` HH z ∈ Mpf (W )- HH v- w ∈ W bm+? i?i v ∼ w (z ∪ {v, w})

(z ∪ {w}) = ,

(z ∪ {v})

(z) #2+mb2 i?Bb Bb 2[mBpH2Mi iQ λ(w, z ∪ {v}) = λ(w, z) , 8

(_BTH2v M/ E2HHv- RNdd)X

UeXRRV

k9k *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a r?B+? 2tT`2bb2b T`QT2`iv U#V Q7 .2}MBiBQM eX9XRX .2}M2 ϕ : Mpf (W ) → [0, ∞) #v BM/m+iBQM, ϕ(∅) = (∅) , ϕ(x) = 1 B7 x Bb MQi  +HB[m2 ,

(x) ϕ(x) = ! ϕ(y)

B7 x Bb  +HB[m2 ,

UeXRkV

y:y⊂x

rBi? i?2 +QMp2MiBQM i?i 0/0 = 0X h?2 7mM+iBQM ϕ Bb r2HH /2}M2/ #2+mb2 B7 i?2 /2MQKBMiQ` BM UeXRkV Bb MmHH- i?2M (y) = 0 7Q` bQK2 y ⊂ x- M/ i?2`27Q`2- BM pB2r Q7 i?2 ?2`2/Bi`v T`QT2`iv- (x) = 0c i?Bb H2/b BM UeXRkV iQ i?2 7Q`K 0/0 = 0X q2 b?Qr i?i (x) Bb Q7 i?2 7Q`K UeXRyV #v BM/m+iBQM QM i?2 MmK#2` Q7 TQBMib Q7 xX bbmK2 i?i UeXRyV ?QH/b 7Q` HH +QM};m`iBQMb rBi? n − 1 Q` 72r2` TQBMibM/ +QMbB/2`  +QM};m`iBQM x bm+? i?i |x| = nX q2 T`Qp2 UeXRyV #v +QMbB/2`BM; i?2 /Bz2`2Mi +b2b QM2 #v QM2X UBV (x) > 0- x Bb  +HB[m2X h?2M- #v UeXRkV 

(x) = ϕ(x) ϕ(y) = ϕ(y) . y:y⊂x

y⊆x

UBBV (x) > 0- x Bb MQi  +HB[m2X aBM+2 x Bb MQi  +HB[m2- i?2`2 2tBbib z ∈ Mpf (W )u, w ∈ W bm+? i?i v ∼ w M/ x = z ∪ {u, w}X "v i?2 ?2`2/Bi`v T`QT2`iv- i?2 +QM/BiBQMb z ⊂ x- z ∪ {v} ⊂ x M/ (x) > 0 BKTHv i?i (z) > 0 M/ (z ∪ {v}) > 0X h?2`27Q`2

(z ∪ {u, w})

(z ∪ {v})

(z ∪ {v})

(z ∪ {w})

(z ∪ {v}) #v UeXRRV =

(z) ! ! ϕ(y ∪ {w}) ϕ(y)  y⊆z y⊆z ! = ϕ(y) ϕ(y)

(x) =

=



y⊆z



ϕ(y ∪ {w})

y⊆z

y⊆z∪{v}

ϕ(y) =

y∈z∪{v}



+HB[m2b

ϕ(y) , y⊆x

bBM+2  +HB[m2 y ⊆ x +MMQi i i?2 bK2 iBK2 +QMiBM #Qi? v M/ wX UBBBV (x) = 0- x Bb !MQi  +HB[m2X h?2M- b BM UBBV- x = z ∪ {v, ! w} r?2`2 v ∼ wX q2 rBHH b?Qr i?i ϕ(y) = 0 UM/ i?2`27Q`2 0 = (x) = ϕ(y)VX AM pB2r Q7 y:y⊂x

+QMi`/B+iBQM- bmTTQb2 i?i

y⊆x



ϕ(y) > 0 .

y:y⊂x

h?2M- #v i?2 BM/m+iBQM ?vTQi?2bBb QM ϕ- 7Q` HH +QM};m`iBQMb y ⊂ x- (y) > 0M/ i?2M

eX9X 6ALAh1 J_EPo SPALh S_P*1aa1a 0=

k9j

(z ∪ {w})

(z ∪ {v, w})

(z ∪ {w}) =

(z ∪ {w}) > 0 ,

(z ∪ {v})

(z)

 +QMi`/B+iBQMX ! UBpV (x) = 0- x Bb  +HB[m2X h?2M- UeXRkV TTHB2b M/ vB2H/b ϕ(x) = 0X h?2`27Q`2 ϕ(y) = 0 M/ i?Bb 2[mHb (x)X 

y⊆x

_2+Q`/ BM T`iB+mH` i?2 7QHHQrBM; 2tT`2bbBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv, λ(u, x) =



ϕ(y ∪ u) .

UeXRjV

y⊆x

1tKTH2 eX9Xj, 6BMBi2 SQBbbQM S`Q+2bb2bX h?2 `272`2M+2 K2bm`2 Bb b BM 1tKTH2 eXRX8X hF2 ({x1 , . . . , xn }) = λn exp{(1 − λ)ν(W )}- BM r?B+? +b2 X Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 λνX q2 ?p2

(x) = e(1−λ)ν(W )



λ,

x∈x

M/ i?2`27Q`2- X Bb  J`FQp TQBMi T`Q+2bb U7Q` Mv +?QB+2 Q7 ∼V- rBi? ϕ(∅) = e(1−λ)ν(W ) M/ ϕ({u}) = λ

(u ∈ W )

M/ ϕ(x) = 1 7Q` b2ib rBi? k Q` KQ`2 TQBMibX h?Bb Bb +QMbBbi2Mi rBi? i?2 B/2 i?i BM  SQBbbQM T`Q+2bb i?2`2 Bb MQ BMi2`+iBQM #2ir22M TQBMibX

1tKTH2 eX9X9, h?2 >`/@+Q`2 SQBMi S`Q+2bb- hF2 kX h?2 `272`2M+2 K2bm`2 Bb b BM 1tKTH2 eXRX8X AM i?2 ?`/@+Q`2 KQ/2H i `M;2 R > 0 Q7 1tKTH2 eXRXe- u ∼ v ⇔ 'u − v' ≤ 2Rϕ(∅) = α ,

ϕ({u}) = β ,

ϕ({u, v}) = 1u−v>2R .

1tKTH2 eX9X8, h?2 SB`rBb2 AMi2`+iBQM JQ/2HX h?2 ?`/@+Q`2 KQ/2H #Qp2 Bb  bT2+BH +b2 Q7 TB`rBb2 BMi2`+iBQM KQ/2Hb- 7Q` r?B+?

(x) = α

 x∈x

β(x)



γ(u, v) ,

u,v∈x u∼v

r?2`2 α > 0- β : W → [0, ∞) M/ γ : W × W → [0, ∞)X

k99 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a

eX8 aTiBH "B`i?@M/@.2i? SQBMi S`Q+2bb2b h?Bb KQ/2He +QM+2`Mb i?2 2pQHmiBQM BM iBK2 Q7  TQBMi T`Q+2bb QM  +QKT+i rBM/Qr W Q7 Rm i?i +QMp2`;2b BM /Bbi`B#miBQM iQ  TQBMi T`Q+2bb rBi? bT2+B}2/ 2tpBbB#H2 BMi2MbBivX *QMbB/2` i?2 7QHHQrBM; biQ+?biB+ T`Q+2bb {X(t)}t≥0 rBi? pHm2b BM Mpf (W )X Ai Bb +?`+i2`Bx2/ #v  #B`i? `i2 7mM+iBQM b : Mpf (W ) × W → [0, ∞) M/  /2i? `i2 7mM+iBQM d : Mpf (W ) × W → [0, ∞)- r?2`2 Bi Bb bbmK2/ i?i 7Q` HH x ∈ Mpf (W )  B(x) := b(x, u)ν(du) < ∞, UeXR9V W

7Q` bQK2 }MBi2 MQM@iQKB+ K2bm`2 ν QM (W, W)X h?2 [mMiBiv B(x) Bb i?2 iQiH #B`i? `i2 BM i?2 +QM};m`iBQM xX G2i  D(x) := d(x, x) UeXR8V x∈x

#2 i?2 iQiH /2i? `i2 BM i?2 +QM};m`iBQM xX h?2 2pQHmiBQM Q7 {X(t)}t≥0 Bb b 7QH@ HQrbX G2i T0 = 0- T1 - T2 , . . . #2 i?2 i`MbBiBQM iBK2b- M/ H2i X0 = X(0), . . . , Xn = X(Tn ), . . . #2 i?2 pHm2b i i?2 i`MbBiBQM iBK2bX AMBiBHBx2 rBi? X0 = x0 ∈ Mpf (W )X 6Q` ;Bp2M n ≥ 0- B7 Xn = x- i?2M Tn+1 − Tn Bb M 2tTQM2MiBH `M/QK p`B#H2 rBi? K2M 1 , D(x) + B(x) M/ BM/2T2M/2Mi Q7 T1 , . . . , Tn - X0 , . . . , Xn−1 X i iBK2 Tn+1 - i?2 i`MbBiBQM Bb •  /2i? rBi? T`Q##BHBiv rBi? T`Q##BHBiv

D(x) B(x)+D(x) d(x,x) c M/ D(x)

•  #B`i? rBi? T`Q##BHBiv

B(x) D(x)+B(x)

M/ i?2 TQBMi x iQ #2 2`b2/ Bb +?Qb2M BM x

M/ i?2 TQBMi u //2/ iQ x Bb +?Qb2M BM W

++Q`/BM; iQ i?2 T`Q##BHBiv /Bbi`B#miBQM

b(x,u)ν(du) X B(x)

P7 +Qm`b2- i?Bb +QMbi`m+iBQM Bb BKTH2K2Mi#H2 QMHv mT iQ iBK2 T∞ = limn↑∞ Tn UMQi BM+Hm/2/VX 6Q` Mv iBK2 t ≥ T∞ - H2i X(t) = ∅ Ui?2 2KTiv TQBMi T`Q+2bbVX h?2 T`Q+2bb {X(t)}t≥0 - +HH2/  bTiBH #B`i?@M/@/2i? TQBMi T`Q+2bb- Bb  DmKT J`FQp T`Q+2bb rBi? bii2 bT+2 Mpf (W )X h?2Q`2K eX8XR G2i {λn }n≥0 M/ {μn }n≥1 #2 i?2 #B`i? M/ /2i? T`K2i2`b Q7  +QMiBMmQmb@iBK2 N@pHm2/ #B`i?@M/@/2i? J`FQp +?BM {Y (t)}t≥0 X amTTQb2 i?i 7Q` HH x ∈ Mpf (W )B(x) ≤ λ|x| , D(x) ≥ μ|x| . UeXReV h?2M, UV B7 {Y (t)}t≥0 Bb `2;mH`- bQ Bb {X(t)}t≥0 - M/ U#V B7 {Y (t)}t≥0 Bb 2`;Q/B+- bQ Bb {X(t)}t≥0 X e

(S`2biQM- RNd8)X

eX8X aShAG "A_h>@mail protected]h> SPALh S_P*1aa1a

k98

U"v 2`;Q/B+Biv- Bi Bb K2Mi i?i  biiBQM`v p2`bBQM Q7 i?2 bTiBH #B`i?@M/@ /2i? TQBMi T`Q+2bb rBi? i?2 `2[mB`2/ +?`+i2`BbiB+b b M/ d 2tBbib M/ i?i Mv p2`bBQM Q7 Bi +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 biiBQM`v /Bbi`B#miBQMX AM 7+i- Bi rBHH #2 b?QrM i?i- mM/2` i?2 #Qp2 +QM/BiBQMb 7Q` i?2 #B`i?@M/@/2i? T`K2i2`b+QMp2`;2M+2 Bb BM p`BiBQMXV 1tKTH2 eX8Xk, S`QQ7 Q7 G2KK eX8XRX _272` iQ a2+iBQM eX8 7Q` i?2 7`K2rQ`F M/ MQiiBQMb Q7 i?2 T`2b2Mi 2tKTH2X h?2 b2[m2M+2 {(Tn , Zn )}n≥0 r?2`2 Zn := X(Tn ) +M #2 pB2r2/ b  TQBMi T`Q@ +2bb QM R+ rBi? K`Fb BM Mpf (W )- r?Qb2 FtX @HQ+H +?`+i2`BbiB+b (λ(t), Φ(t, dx)) `2 λ(t) = B(X(t)) + D(X(t)), Φ(t, F ) = Q(X(t−), F ) , r?2`2 B(x) Q(x, F ) := B(x) + D(x)

 b(x, u)1F (x ∪ u)ν(du) W

+

 D(x) d(x, x)1F (x\x) . B(x) + D(x) x∈x

h?2 TQBMi T`Q+2bb2b B M/ D +QmMiBM; i?2 iBK2b Q7 Q++m``2M+2 Q7 #B`i?b M/ /2i?b ?p2 FtX @BMi2MbBiv {B(X(t))}t≥0 M/ {D(X(t))}t≥0 `2bT2+iBp2HvX q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 G2KK eX8XRX UV "v i?2 BK#2//BM; i?2Q`2K Uh?2Q`2K 8XdXeV-  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R2 rBi? BMi2MbBiv 1 +M #2 +QMbi`m+i2/ BM bm+?  rv i?i 7Q` HH t ∈ [0, T∞ )  1{0≤z≤B(X(s))} N (dz × ds) B(0, t] = (0,t]

M/





D(0, t] = (0,t]

6Q` HH t ∈ (0, T∞ )-

R

R

1{0≥z≥−D(X(s))} 1{X(s−)=∅} N (dz × ds) .

|X(t)| = |X(0)| + B(0, t] − D(0, t] .

*QMbi`m+i MQr {Y˜ (t)}t≥0 b 7QHHQrbX 6B`bi b2i Y˜ (0) = |X(0)|X h?2M- 7Q` t ∈ (0, T∞ )  1{0≤z≤λY˜ (s−) } N (dz × ds) Y˜ (t) = Y˜ (0) + (0,t] R   1{0≥z≥−μY˜ (s−) } 1{Y˜ (s−)>0} N (dz × ds) . − (0,t]

R

h?2 biQ+?biB+ T`Q+2bb {Y˜ (t)}t∈(0,T∞ ) /QKBMi2b {|X(t)|}t∈(0,T∞ ) U#Qi? T`Q+2bb2b ?p2 DmKTb Q7 K;MBim/2 i KQbi 1- i?2v ?p2 i?2 bK2 BMBiBH pHm2- M/ r?2M2p2` Y˜ (t) = |X(t)|- B(X(t)) ≤ λY˜ (t) , D(X(t)) ≥ μY˜ (t) VX AM T`iB+mH`- bBM+2 {Y˜ (t)}t≥0 Bb bbmK2/ `2;mH`- bQ Bb {|X(t)|}t≥0 M/ i?2`27Q`2 {X(t)}t≥0 X

k9e *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a U#V G2i X(0) #2 i?2 BMBiBH pHm2 Q7 i?2 #B`i?@M/@/2i? TQBMi T`Q+2bb iQ #2 +QMbi`m+i2/X bbmK2 i?i i?2 T`K2i2`b λn M/ μn `2 i?Qb2 Q7 M 2`;Q/B+ #B`i?@ M/@/2i? T`Q+2bb QM i?2 TQbBiBp2 HBM2X *QMbi`m+i 7`QK  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb N QM R2 Q7 BMi2MbBiv 1 bm+? M 2`;Q/B+ #B`i?@M/@/2i? T`Q+2bb {Y˜ (t)}t≥0 #v   Y˜ (t) = Y˜ (0) + 1{0≤z≤λY˜ (s−) } N (dz × ds) (0,t] R   1{0≥z≥−μY˜ (s−) } 1{Y˜ (s−)>0} N (dz × ds) , − (0,t]

R

r?2`2 Y˜ (0) = |X(0)|X LQr bmTTQb2 i?i r2 ?p2 bm++2bb7mHHv +QMbi`m+i2/ i?2 bTiBH #B`i?@M/@/2i? T`Q+2bb mT iQ iBK2 t BM bm+?  rv i?i 7Q` HH r ≤ t|X(r)| ≤ Y˜ (r) UMQi2 i?i i?Bb Bb i?2 bBimiBQM i iBK2 0VX GQQF MQr i i?2 }`bi TQBMi (s, z) Q7 N 7i2` t BM [−D(X(t)), +B(X(t))] × (t, t + a(t)]- r?2`2 t + a(t) Bb i?2 }`bi DmKT iBK2 Q7 {Y˜ (t)}t≥0 7i2` tX A7 Bi Bb  TQBMi BM i?2 mTT2` ?H7 THM2 Uz ≥ 0V- // iQ i?2 +m``2Mi bTiBH #B`i?@M/@/2i? T`Q+2bb  TQBMi u +?Qb2M i `M/QK ++Q`/BM; ν(du) iQ i?2 T`Q##BHBiv /Bbi`B#miBQM b(X(t−),u) X A7 z < 0- `2KQp2  TQBMi 7`QK i?2 B(X(t−)) +m``2Mi TQBMi T`Q+2bb- +?QQbBM; i?2 TQBMi x ∈ X(t−) iQ #2 `2KQp2/ rBi? T`Q##BHBiv d(X(t−),x) X i i?Bb TQBMi s i?2 /QKBMiBQM T`QT2`iv Y˜ (s) ≥ |X(s)| `2KBMb biBb}2/X D(X(t−)) UAM 7+i- i?2 QMHv T`Q#H2K +QmH/ `Bb2 7`QK i?2 +b2 r?2`2 |X(t)| = Y˜ (t) M/ s = t + a(t) M/ z ≥ 0- #mi i?2M t + a(t) Bb M2+2bb`BHv M mTr`/ DmKT Q7 i?2 Y˜ T`Q+2bb- #2+mb2 i?2M D(X(s−)) ≤ λ|X(s−)| = λY˜ (s−) - M/ i?2`27Q`2 s Bb M2+2bb`BHv M mTr`/ DmKT Q7 i?2 Y˜ T`Q+2bbXV q2 +M MQr `2T`Q/m+2 i?2 #Qp2 +QMbi`m+iBQMi?Bb iBK2 rBi? s b bi`iBM; iBK2X h?2 /QKBMiBQM T`QT2`iv Dmbi T`Qp2/ rBHH 2MiBH i?i irQ bTiBH #B`i?@M/@ /2i? T`Q+2bb2b {X1 (t)}t≥0 M/ {X2 (t)}t≥0 rBi? ;Bp2M BMBiBH bii2b X1 (0) M/ X2 (0) +M #2 +QMbi`m+i2/ BM bm+?  rv i?i i?2v +QmTH2 BM }MBi2 iBK2X hQ b22 i?Bb- T2`7Q`K 7Q` 2+? Q7 i?2K i?2 #Qp2 +QMbi`m+iBQM rBi? i?2 bK2 N M/ H2i {Y˜1 (t)}t≥0 M/ {Y˜2 (t)}t≥0 #2 i?2 +Q``2bTQM/BM; Y˜ T`Q+2bbX _2+HH i?i Y˜1 (0) = |X1 (0)| M/ Y˜2 (0) = |X2 (0)|X amTTQb2 i?i |X1 (0)| ≥ |X2 (0)|- M/ i?2`2@ 7Q`2 Y˜1 (0) ≥ Y˜2 (0)X 6`QK i?2 +QMbi`m+iBQM Q7 i?2 Y˜ T`Q+2bb2b M/ i?2 7+i i?i r2 `2 mbBM; 7Q` i?Bb +QMbi`m+iBQM i?2 bK2 N - r2 b22 i?i 7Q` HH t ≥ 0- Y˜1 (t) ≥ Y˜2 (t)X AM T`iB+mH`- i?2b2 irQ T`Q+2bb2b rBHH +QmTH2 BM }MBi2 iBK2 Ub bQQM b Y˜1 (t) = 0r?B+? ?TT2Mb 7Q` }MBi2 t #v i?2 2`;Q/B+Biv bbmKTiBQM 7Q` i?2 #B`i?@M/@/2i? T`Q+2bb rBi? T`K2i2`b λn M/ μn V- M/ bQ rBHH i?2 #B`i?@M/@/2i? TQBMi T`Q@ +2bb2b X1 M/ X2 X q2 MQr T`Qp2 i?2 2tBbi2M+2 Q7  biiBQM`v #B`i?@M/@/2i? TQBMi T`Q+2bb rBi? i?2 #Qp2 #B`i?@M/@/2i? +?`+i2`BbiB+bX 6Q` i?Bb r2 }`bi +QMbi`m+i  biiBQM`v +QMiBMmQmb@iBK2 #B`i?@M/@/2i? J`FQp +?BM Y˜ QM i?2 2MiB`2 HBM2 R- M/ i?2M  SQBbbQM T`Q+2bb N QM R2 bm+? i?i   ˜ ˜ Y (t) = Y (0) + 1{0≤z≤λY˜ (s−) } N (dz × ds) (0,t] R   1{0≥z≥−μY˜ (s−) } 1{Y˜ (s−)>0} N (dz × ds) . =− (0,t]

R

eX8X aShAG "A_h>@mail protected]h> SPALh S_P*1aa1a

k9d

G2i {Vn }n∈Z #2 i?2 b2[m2M+2 Q7 iBK2b bm+? i?i Y˜ (Vn ) = 0 M/ Y˜ (Vn −) > 0 UBM 7+i- = 1VX *QMbi`m+i 7Q` 2+? n ∈ Z- X(t), t ∈ [Vn , Vn+1 ) rBi? X(Vn ) = 0- b mbmHmbBM; i?2 TQBMi T`Q+2bb N X h?Bb T`Q+2bb Bb biiBQM`v U 7Q`KH T`QQ7 Bb `2[mB`2/ BM 1t2`+Bb2 eXdXRVX AM pB2r Q7 i?2 Dmbi T`Qp2/ +QmTHBM; T`QT2`iv- i?2 biiBQM`v /Bbi`B#miBQM Bb mMB[m2 M/ Mv i`MbB2Mi bQHmiBQM +QMp2`;2b BM p`BiBQM iQ i?2 biiBQM`v /Bbi`B@ #miBQMX _2+HH i?i  M2+2bb`v M/ bm{+B2Mi +QM/BiBQM 7Q` `2;mH`Biv Q7  +QMiBMmQmb@ iBK2 #B`i?@M/@/2i? J`FQp +?BM rBi? TQbBiBp2 #B`i? M/ /2i? T`K2i2`b Bb

 1 μn μn · · · μ1 = ∞, + + ··· + λn λn λn−1 λn · · · λ 0 n≥1 r?2`2b  M2+2bb`v M/ bm{+B2Mi +QM/BiBQM 7Q` 2`;Q/B+Biv Bb  λ0 λ1 · · · λn−1 n≥1

μ1 μ2 · · · μn

< ∞.

h?2 #B`i?@M/@/2i? T`Q+2bb Bb  J`FQp T`Q+2bb r?Qb2 bii2 bT+2 Bb Mpf (W )M/ Bi +M #2 b?QrM d i?i B7 i?2 7QHHQrBM; /2iBH2/ #HM+2 2[miBQMb b(x, u) (x) = d(x ∪ {u}, u) (x ∪ {u})

UeXRdV

`2 biBb}2/ 7Q` HH x, u bm+? i?i (x ∪ {u}) > 0 Ui?2M (x) > 0 #v i?2 ?2`2/Bi`v bbmKTiBQMV- Bi +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 }MBi2 TQBMi T`Q+2bb rBi? T`Q##BHBiv /2MbBiv X h?2`2 `2 KMv +?QB+2b 7Q` i?2 #B`i?@M/@/2i? `i2b biBb7vBM; i?2 /2iBH2/ #HM+2 2[miBQMbX PM2 Q7 i?2K Bb d(x, x) ≡ 1 ,

(x ∪ {u}) = λ(u, x) . b(x, u) =

(x)

UeXR3V

1tKTH2 eX8Xj, h?2 SQBbbQM T`Q+2bbX G2i λ : E → [0, ∞) #2  K2bm`#H2 MQM@M2;iBp2 7mM+iBQM bm+? i?i  B= λ(u)ν(du) < ∞ . E

h?2 +?QB+2 d(x, x) ≡ 1- b(x, u) = λ(u) +Q``2bTQM/b iQ  bTiBH #B`i?@M/@/2i? T`Q+2bb {Z(t)} r?Qb2 biiBQM`v /Bbi`B#miBQM +Q``2bTQM/b iQ  SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 λ(u)ν(du)X >2`2 B(x) = B- D(x) = n(x)- i?2 MmK#2` Q7 TQBMib BM i?2 +QM};m`iBQM xX h?2 T`Q+2bb Y (t) = n(Z(t)) +QmMib i?2 MmK#2` Q7 TQBMib Q7 i?2 bTiBH #B`i?@ M/@/2i? T`Q+2bb i iBK2 tX AM i?Bb T`iB+mH` +b2- r2 b22 i?i B7 i  ;Bp2M iBK2 d

(S`2biQM- RNd8)X

k93 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a t- Y (t) = n- Bi bivb i?2`2 7Q` M 2tTQM2MiBH iBK2 Q7 K2M (B + n)−1 X i i?2 2M/ Q7 B i?Bb 2tTQM2MiBH iBK2- Bi KF2b  i`MbBiBQM iQ n + 1 rBi? T`Q##BHBiv B+n - iQ n − 1 n rBi? T`Q##BHBiv B+n M/ i?2 T`Q+2bb Bb `2T2i2/ bi`iBM; 7`QK i?2 M2r TQbBiBQM n ± 1- BM/2T2M/2MiHv Q7 i?2 TbiX h?2`27Q`2 {Y (t)} Bb i?2 mbmH #B`i?@M/@/2i? T`Q+2bb rBi? T`K2i2`b βn = B , δn = n. am+?  T`Q+2bb Bb  `2;mH` DmKT J`FQp T`Q+2bb- B``2/m+B#H2 M/ 2`;Q/B+X h?2 bTiBH #B`i?@M/@/2i? T`Q+2bb {Z(t)}t∈R Bb  `2;mH` DmKT J`FQp T`Q+2bb UBib `2;mH`Biv 7QHHQrb 7`QK i?i Q7 {Y (t)}t∈R VX AM pB2r Q7 i?2 /2iBH2/ #HM+2 2[miBQMb- Bi /KBib P λν Ui?2 SQBbbQM K2bm`2 rBi? BMi2MbBiv K2bm`2 λ(u)ν(du)V b BMp`BMi T`Q##BHBivX

eXe M Hi2`MiBp2 JQ/2H q2 ?p2 H`2/v K2MiBQM2/  i?2Q`2iB+H M/ T2`?Tb T`+iB+H BM+QMp2MB2M+2 BM i?2 KQ/2HBM; Q7 }MBi2 TQBMi T`Q+2bb2b #v K2Mb Q7 _/QMĜLBFQ/ɷK /2`BpiBp2bMK2Hv i?2 /B{+mHiv Q7 `2HiBM; i?2 /2MbBiB2b jn Un ≥ 1V iQ i?2 +QmMi /Bbi`B#miBQMX JQ`2 T`2+Bb2Hv- i?2 KQ/2H2` BMi2`2bi2/ QMHv BM i?2 BMi2`+iBQM #2ir22M i?2 TQBMib rBHH +QK2 mT rBi? i?2 /2MbBiB2b jn Un ≥ 1V M/ i?2M rBHH ;2M2`HHv #2 +QM7`QMi2/ rBi? i?2 TQbbB#Hv mM72bB#H2 +QKTmiiBQM Q7 i?2 +QmMi /Bbi`B#miBQMX 6Q` BMbiM+2BM i?2 ?`/@+Q`2 KQ/2H- 2p2M i?2 K2M K2bm`2 Q7 i?2 TQBMi T`Q+2bb Bb /B{+mHi iQ Q#iBMX MQi?2` ivT2 Q7 KQ/2H Bb MQr T`2b2Mi2/- r?B+? /Q2b MQi pQB/ i?2 /B{+mHiB2b K2MiBQM2/- #mi +QMM2+ib iQ  r2HH@bim/B2/ `2X h?2 ;2M2`H B/2 Q7 i?2 ivT2 Q7 KQ/2HHBM; i?i Bb MQr T`QTQb2/ Bb iQ +QMbi`m+i  SQBbbQM T`Q+2bb Q` Mv Qi?2` ivT2 Q7 bBKTH2 TQBMi T`Q+2bb r?B+? rBHH #2 i?2 bmTTQ`i Q7 i?2 TQBMi T`Q+2bb i?i QM2 rBb?2b iQ +QMbi`m+iX h?Bb K2Mb i?i  T`Q+2/m`2 /Bb+`/BM; bQK2 TQBMib Q7 i?2 bmTTQ`i TQBMi T`Q+2bb Bb ;Bp2MX am+?  T`Q+2/m`2 Bb ;Bp2M BM i2`Kb Q7 }MBi2 :B##bĜ J`FQp `M/QK }2H/b- Q7 r?B+? i?2 KBM 72im`2b Q7 i?2 2H2K2Mi`v i?2Q`v rBHH #2 `2+HH2/ BM  72r HBM2bX LQi2 ?Qr2p2` i?i i?2 T?BHQbQT?v Bb bHB;?iHv /Bz2`2Mi 7`QK i?2 QM2 BM i?2 T`2pBQmb b2+iBQMb- M/ i?2 +?QB+2 rBHH /2T2M/ QM i?2 TTHB+iBQMb +QM+2`M2/X G2i V #2  }MBi2 b2i- r?Qb2 2H2K2Mib `2 +HH2/ MQ/2b- Q` bBi2b- Q` p2`iB+2b++Q`/BM; iQ i?2 +QMi2tiX h?2`2 Bb  #BM`v bvKK2i`B+ `2HiBQM ∼ #2ir22M i?2 MQ/2b- M/ i?2 MQ/2b v1 M/ v2 bm+? i?i v1 ∼ v2 `2 +HH2/ M2B;?#Qm`bX 6Q` v ∈ V /2MQi2 #v Nv i?2 +QHH2+iBQM Q7 MQ/2b /Bz2`2Mi 7`QK v i?i `2 M2B;?#Qm`b Q7 vX  +HB[m2 Bb  bm#b2i Q7 V i?i Bb 2Bi?2`  bBM;H2iQM Q` bm+? i?i Mv irQ /BbiBM+i 2H2K2Mib Q7 Bi `2 M2B;?#Qm`bX h?2`2 Bb  }MBi2 bT+2 Λ +HH2/ i?2 T?b2 bT+2 M/ M 2H2K2Mi x = (x(v) ; v ∈ V ) ∈ ΛV Bb +HH2/  +QM};m`iBQMX h?2 MQiiBQM x(C)r?2`2 C ⊆ S- `2T`2b2Mib i?2 +QM};m`iBQM x `2bi`B+i2/ iQ C, x(C) := {x(v) ; v ∈ C}X  `M/QK 2H2K2Mi X = (X(v), v ∈ V )- r?2`2 X(v) ∈ Λ Bb i?2 `M/QK K`FQ` T?b2- bbQ+Bi2/ iQ i?2 bBi2 UQ` MQ/2V v- Bb +HH2/  `M/QK }2H/ QM V rBi? T?b2b BM ΛX

eXeX L Gh1_LhAo1 JP.1G

k9N

.2}MBiBQM eXeXR  :B##b TQi2MiBH QM ΛV `2HiBp2 iQ ∼ Bb  +QHH2+iBQM {VC }C⊆V Q7 7mM+iBQMb VC : ΛV → R ∪ {+∞} bm+? i?i UBV VC ≡ 0 B7 C Bb MQi  +HB[m2UBBV 7Q` HH x, x ∈ ΛV M/ HH C ⊆ V - x(C) = x (C) ⇒ VC (x) = VC (x )X AM bmKK`v, i?2 7mM+iBQM VC /2T2M/b QMHv QM i?2 T?b2b i i?2 bBi2b BMbB/2 i?2 bm#b2i C M/ Bb i?2 MmHH 7mM+iBQM B7 C Bb MQi  +HB[m2X h?2 7mM+iBQM U : ΛV → R ∪ {+∞}- ?2M+27Q`i? +HH2/ i?2 2M2`;v 7mM+iBQM- Bb bB/ iQ /2`Bp2 7`QK i?2 TQi2MiBH {VC }C⊆V B7 U (x) =



VC (x) .

C

*QMbB/2` i?2 T`Q##BHBiv /Bbi`B#miBQM π(x) =

1 −U (x) e Z

UeXRNV

QM i?2 +QM};m`iBQM bT+2 ΛV - r?2`2 Z Bb i?2 MQ`KHBxBM; +QMbiMi- +HH2/ i?2 T`iBiBQM 7mM+iBQMX h?2 /Bbi`B#miBQM BM UeXRNV Bb +HH2/  :B##b /Bbi`B#miBQMX  `M/QK }2H/ X QM V rBi? T?b2b BM Λ Bb +HH2/  J`FQp `M/QK }2H/ UK`7V rBi? `2bT2+i iQ ∼ B7 7Q` HH MQ/2b v ∈ V - i?2 `M/QK p`B#H2b X(v) M/ X(V \(Nv ∪ {v})) `2 BM/2T2M/2Mi ;Bp2M X(Nv )X AM bvK#QHb, P (X(v) = x(v) | X(V \v) = x(V \v)) = P (X(v) = x(v) | X(Nv ) = x(Nv )) UeXkyV 7Q` HH x ∈ ΛV M/ HH v ∈ V X S`QT2`iv UeXkyV Bb +H2`Hv Q7 i?2 J`FQp ivT2, i?2 /Bbi`B#miBQM Q7 i?2 T?b2 i  bBi2 Bb /B`2+iHv BM~m2M+2/ QMHv #v i?2 T?b2b Q7 i?2 M2B;?#Qm`BM; bBi2bX LQi2 i?i Mv `M/QK }2H/ Bb J`FQpBM rBi? `2bT2+i iQ i?2 i`BpBH `2HiBQM ∼ 7Q` r?B+? i?2 M2B;?#Qm`?QQ/ Q7 Mv bBi2 Bb i?2 r?QH2 b2i V X >Qr2p2`- i?2 BMi2`2biBM; J`FQp }2H/b U7`QK i?2 TQBMi Q7 pB2r Q7 KQ/2HHBM;- bBKmHiBQM M/ QTiBKBxiBQMV `2 i?Qb2 rBi? `2HiBp2Hv bKHH M2B;?#Qm`?QQ/bX h?2 HQ+H +?`+i2`BbiB+ Q7 i?2 K`7 i bBi2 v Bb i?2 7mM+iBQM π v : ΛV → [0, 1] /2}M2/ #v π v (x) := P (X(v) = x(v) | X(Nv ) = x(Nv )). PM2 HbQ r`Bi2b π v (x) = π(x(v) | x(Nv ))X M K`7 Bb bB/ iQ biBb7v i?2 TQbBiBpBiv +QM/BiBQM B7 Bib T`Q##BHBiv /Bbi`B#miBQM Bb bi`B+iHv TQbBiBp2X h?2Q`2K eXeXk hrQ /Bbi`B#miBQMb Q7 M K`7 rBi?  }MBi2 +QM};m`iBQM bT+2 ΛV i?i biBb7v i?2 TQbBiBpBiv +QM/BiBQM M/ ?p2 i?2 bK2 HQ+H bT2+B}+iBQM `2 B/2MiB+HX

k8y *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a S`QQ7X 1MmK2`i2 V b {1, 2, . . . , K}X h?2`27Q`2 x = (x1 , . . . , xK−1 , xK ) ∈ ΛK X h?2 7QHHQrBM; B/2MiBiv π(x) =

K  π(xi | x1 , . . . , xi−1 , yi+1 , . . . , yK ) π(y) π(yi | x1 , . . . , xi−1 , yi+1 , . . . , yK ) i=1

()

?QH/b 7Q` Mv x, y ∈ ΛK X 6Q` i?2 T`QQ7- QM2 +?2+Fb i?i π(x) =

π(xK | x1 , . . . , xK−1 ) π(x1 , . . . , xK−1 , yK ) π(yK | x1 , . . . , xK−1 )

#v bBKTHv `2r`BiBM; i?2 +QM/BiBQMH T`Q##BHBiB2b i?2`2Q7 mbBM; "v2bǶb /2}MBiBQMX aBKBH`Hvπ(xK−1 | x1 , . . . , xK−2 , yK ) π(x1 , . . . , xK−2 , yK−1 , yK ) π(yK−1 | x1 , . . . , xK−2 , yK )

π(x1 , . . . , xK−1 , yK ) =

M/ bQ 7Q`i?X h?2 #Qp2 +H+mHiBQMb KF2 b2Mb2 #2+mb2 Q7 i?2 TQbBiBpBiv +QM/BiBQMX G2i MQr π M/ π  #2 T`Q##BHBiv /Bbi`B#miBQMb rBi? i?2 bK2 HQ+H bT2+B}+iBQMb M/ biBb7vBM; i?2 TQbBiBpBiv +QM/BiBQMX *?QQb2 Mv yX h?2 B/2MiBiv UV b?Qrb i?i 7Q` HH xπ  (x) π  (y) = , π(x) π(y)  +QMbiMi- M2+2bb`BHv 2[mH iQ 1 bBM+2 π M/ π  `2 T`Q##BHBiv /Bbi`B#miBQMbX  :B##b /Bbi`B#miBQMb rBi? M 2M2`;v /2`BpBM; 7`QK  :B##b TQi2MiBH `2HiBp2 iQ  #BM`v bvKK2i`B+ `2HiBQM ∼ `2 /Bbi`B#miBQMb Q7 J`FQp }2H/b `2HiBp2 iQ i?2 bK2 #BM`v `2HiBQMX JQ`2 T`2+Bb2Hv, h?2Q`2K eXeXj A7 X Bb  `M/QK }2H/ QM V rBi? T?b2b BM Λ rBi?  /Bbi`B#miBQM π Q7 i?2 7Q`K UeXRNV- r?2`2 i?2 2M2`;v U /2`Bp2b 7`QK  :B##b TQi2MiBH {VC }C⊆V `2HiBp2 iQ ∼- i?2M X Bb J`FQpBM `2HiBp2 iQ ∼X JQ`2Qp2`- Bib HQ+H bT2+B}+iBQM Bb ;Bp2M #v i?2 7Q`KmH 

e− C v VC (x)  π (x) = , − C v VC (λ,x(V \v)) λ∈Λ e v

r?2`2 i?2 MQiiBQM iBMBM; bBi2 vX

Cv

UeXkRV

K2Mb i?i i?2 bmK 2ti2M/b Qp2` i?2 b2ib C ⊆ V +QM@

S`QQ7X 6B`bi Q#b2`p2 i?i i?2 `B;?i@?M/ bB/2 Q7 UeXkRV /2T2M/b QM x QMHv i?`Qm;? x(v) M/ x(Nv )X AM/22/- VC (x) /2T2M/b QMHv QM (x(w), w ∈ C)- M/ 7Q`  +HB[m2 C- B7 w ∈ C M/ v ∈ C- i?2M 2Bi?2` w = v Q` w ∼ vX h?2`27Q`2- B7 QM2 +M b?Qr i?i P (X(v) = x(v) | X(V \v) = x(V \v)) 2[mHb i?2 `B;?i@?M/ bB/2 Q7 UeXkRVM/ BM T`iB+mH` Bb  7mM+iBQM Q7 x(v) M/ x(Nv ) QMHv- i?2M Ub22 1t2`+Bb2 eXdXeVi?2 J`FQp T`QT2`iv UeXkyV M/ 2[mHBiv UeXkRV rBHH #2 T`Qp2/X "v /2}MBiBQM Q7 +QM/BiBQMH T`Q##BHBiv-

eXeX L Gh1_LhAo1 JP.1G

k8R

P (X(v) = x(v) | X(V \v) = x(S\v)) = "mi π(x) =

π(x) . π(λ, x(V \v)) λ∈Λ

(†)

1 − C v VC (x)+C v VC (x) e , Z

M/ bBKBH`Hvπ(λ, x(S\v)) =

1 − C v VC (λ,x(S\v))−C v VC (λ,x(S\v)) e . Z

A7 C Bb  +HB[m2 M/ v Bb MQi BM C- i?2M VC (λ, x(V \v)) = VC (x) M/ Bb i?2`27Q`2 BM/2T2M/2Mi Q7 ∈ ΛX h?2`27Q`2, λ - i?2 `B;?i@?M/ bB/2 Q7 U†V Bb 7QmM/- 7i2` 7+iQ`BM; Qmi exp x(w) − Cv VC (x) - iQ #2 2[mH iQ i?2 `B;?i@?M/ bB/2 Q7 UeXkRVX  h?2 HQ+H 2M2`;v i bBi2 v Q7 +QM};m`iBQM x Bb  VC (x). Uv (x) = Cv

qBi? i?Bb MQiiBQM- UeXkRV #2+QK2b π v (x) =

e−Uv (x) . −Uv (λ,x(V \v)) λ∈Λ e

h?2Q`2K eXeXj ?b M BKTQ`iMi +QMp2`b2- i?2 >KK2`bH2vĜ*HBzQ`/ i?2Q`2Kr?B+? r2 b?HH MQi T`Qp2 ?2`2,3 h?2Q`2K eXeX9 G2i π #2 i?2 /Bbi`B#miBQM Q7  `M/QK }2H/ i?i Bb J`FQpBM rBi? `2bT2+i iQ ∼ M/ bmTTQb2 i?i π > 0X h?2M π(x) =

1 −U (x) e Z

7Q` bQK2 2M2`;v 7mM+iBQM U /2`BpBM; 7`QK  :B##b TQi2MiBH {VC }C⊆V bbQ+Bi2/ rBi? ∼X h?2 Hi2`MiBp2 TT`Q+? iQ }MBi2 TQBMi T`Q+2bb KQ/2HHBM; +QMbBbib BM }`bi ;2M2`iBM;  bBKTH2 TQBMi T`Q+2bb QM Rm Ui?2 dz7`K2ǴV rBi? }MBi2 K2M K2bm`2 ν QM Rd - M/ i?2M `M/QKHv K`FBM; Ub 2tTHBM2/ #Qp2V i?2 }MBi2 UbBM+2 ν Bb  }MBi2 K2bm`2V MmK#2` Q7 TQBMib #v 2Bi?2` 0 M/ 1X h?2 TQBMib K`F2/ rBi?  1 `2 `2iBM2/ M/ i?2 Qi?2`b `2 /2H2i2/X AM Qi?2` rQ`/b- r?2M i?2 `2HBxiBQM Q7 i?2 SQBbbQM T`Q+2bb ?b n TQBMib v1 , . . . , vn - /2}M2  bBi2 bT+2 V := {v1 , . . . , vn } M/  T?b2 bT+2 Λ := {0, 1}X :Bp2M  #BM`v bvKK2i`B+ `2HiBQM ∼ #2ir22M TQBMib BM Rm - i?2 `2HiBQM ∼ #2ir22M i?2 TQBMib Q7 V Bb i?2 QM2 BM?2`Bi2/ 7`QK BiX q?2M i?2 K`Fb 7Q`K  J`FQpĜ:B##b }2H/ QM  }MBi2 bBi2 bT+2- i?2 }MBi2 TQBMi T`Q+2bb +M #2 bBKmHi2/ #v K2Mb Q7  JQMi2 *`HQ bKTHBM; H;Q`Bi?K 3

a22 7Q` BMbiM+2 ("`ûKm/- kyky)X

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N AM i?2 bBimiBQM /2b+`B#2/ i i?2 2M/ Q7 i?2 T`2pBQmb 2tKTH2- Bi Bb bBKBH` iQ S`2biQMǶb #B`i?@M/@/2i? H;Q`Bi?K Q7 a2+iBQM eX8X

eXdX 1s1_*Aa1a

eXd

k8j

1t2`+Bb2b

1t2`+Bb2 eXdXRX "B`i?@M/@/2i? S`Qp2 7Q`KHHv i?2 bii2K2Mi BM i?2 Hbi #27Q`2 Hbi T`;`T? Q7 i?2 T`QQ7 Q7 h?2Q`2K eX8XRX 1t2`+Bb2 eXdXkX PM i?2 J`FQp T`QT2`iv Q7  TQBMi T`Q+2bb G2i W #2  #QmM/2/ K2bm`#H2 bm#b2i Q7 Rm X G2i P #2  T`Q##BHBiv K2bm`2 QM dP Mpf (W ) /2}M2/ #v i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 dQ (x) = (x) rBi? `2bT2+i iQ f bQK2 T`Q##BHBiv K2bm`2 Q QM Mp (W )X amTTQb2 i?i mM/2` P - i?2 TQBMi T`Q+2bb X Bb J`FQp rBi? `2bT2+i iQ i?2 #BM`v bvKK2i`B+ `2HiBQM ∼X G2i C ⊆ W X a?Qr i?i i?2 +QM/BiBQMH /Bbi`B#miBQM Q7 X(C) ;Bp2M X(C) /2T2M/b QMHv QM X(∂(C)∩C) r?2`2 ∂(C) ∩ C := {u ∈ W ∩ C ; u ∼ v 7Q` bQK2 v ∈ C}X 1t2`+Bb2 eXdXjX GQ+H BMi2`T`2iiBQM Q7 i?2 2tpBbB#H2 BMi2MbBiv G2i X #2  }MBi2 TQBMi T`Q+2bb QM Rm +QMbi`m+i2/ b BM 1tKTH2 eXRX8 M/ bmTTQb2 i?i i?2 `272`2M+2 SQBbbQM T`Q+2bb Bb M ?TT rBi? mMBi BMi2MbBiv- i?i Bb- ν(du) = duX S`Qp2 i?i B7 u → λ(u, X) Bb HKQbi bm`2Hv #QmM/2/ M/ +QMiBMmQmbλ(u, X) = lim ε↓0

1 E [X(B(u, ε)) | X\{u}] .

m (B(u, ε))

1t2`+Bb2 eXdX9X J`F2/ TB`rBb2 BMi2`+iBQM T`Q+2bb *QMbB/2` i?2 7QHHQrBM; KQ/2H Q7  }MBi2 K`F2/ TQBMi T`Q+2bb rBi? #bB+ TQBMib BM  +QKT+i b2i W ⊂ Rm M/ K`Fb BM i?2 b2i K = {1, . . . , M }X h?2 `272`2M+2 /Bbi`B#miBQM Bb i?i Q7 M ?TT QM W rBi? mMBi `i2 M/ BB/ mMB7Q`KHv /Bbi`B#mi2/ K`Fb U`272` iQ 1tKTH2 eXRXd 7Q` i?2 7`K2rQ`F M/ MQiiBQMVX h?2 /2MbBiv rBi? `2bT2+i iQ i?Bb `272`2M+2 K2bm`2 Bb

(x) = α

 (u,k)∈x

βk



γkm (||u − v||) ,

(u,k),(v,m)∈x

r?2`2 α M/ i?2 βk `2 TQbBiBp2 `2H MmK#2`b M/ r?2`2 γkm : [0, ∞) → [0, ∞) U+HH2/ i?2 BMi2`+iBQM 7mM+iBQMbVX bbmK2 i?i γkm ≡ γmk X a?Qr i?i B7 7Q` HH k, m ∈ {1, . . . , M }- γk,m (r) = 1 r?2M r > rkm r?2`2 rkm > 0- M/ B7 7Q` bQK2 }MBi2 TQbBiBp2 MmK#2` R- supk,m rkm ≤ R- i?2M i?2 K`F2/ TQBMi T`Q+2bb X Bb J`FQp rBi? `2bT2+i iQ i?2 `2HiBQM ∼ /2}M2/ #v (u, k) ∼ (v, m) ⇐⇒ ||u − v|| ≤ R .

1t2`+Bb2 eXdX8X MQi?2` ?`/@+Q`2 TQBMi T`Q+2bb 6`QK M ?TT Q7 BMi2MbBiv 1 QM Rm - 2`b2 HH i?2 TQBMib i  /BbiM+2 ≤ R 7`QK MQi?2` TQBMi- r?2`2 R > 0X *QKTmi2 i?2 BMi2MbBiv K2bm`2 Q7 i?2 `2KBMBM; TQBMi T`Q+2bbX

k89 *>Sh1_ eX 1soAaA"G1 ALh1LaAhu P6 6ALAh1 SPALh S_P*1aa1a 1t2`+Bb2 eXdXeX suw G2i X- Y - M/ Z #2 i?`22 /Bb+`2i2 `M/QK p`B#H2b rBi? pHm2b BM E- F - M/ G- `2bT2+iBp2HvX S`Qp2 i?2 7QHHQrBM;, A7 7Q` bQK2 7mM+iBQM g : E × F → [0, 1]P (X = x | Y = y, Z = z) = g(x, y) 7Q` HH x, y, z- i?2M P (X = x | Y = y) = g(x, y) 7Q` HH x, y- M/ X M/ Z `2 +QM/BiBQMHHv BM/2T2M/2Mi ;Bp2M Y X 1t2`+Bb2 eXdXdX .Bb+`2iBx2/ #B`i?@M/@/2i? bBKmHiBQM AM 1tKTH2 eXeXe ;Bp2 i?2 i`MbBiBQM T`Q##BHBiB2b px,y Q7 i?2 JQMi2 *`HQ J`FQp +?BM i?2`2Q7 M/ b?Qr i?i 7Q` HH x, y- i?2 /2iBH2/ #HM+2 2[miBQMb π(x)px,y = π(y)py,x `2 biBb}2/X .2/m+2 7`QK i?Bb i?i π Bb BM/22/ i?2 biiBQM`v /Bbi`B#miBQM Q7 i?Bb J`FQp +?BMX :Bp2 i?2 +QM/BiBQMb QM i?2 2M2`;v 7mM+iBQM U i?i ;m`Mi22 B``2/m+B#BHBiv M/ T2`BQ/B+BivX 1t2`+Bb2 eXdX3X S`QT2`Hv +QHQm`2/ ;`T?b *QMbB/2`  +QM};m`iBQM bT+2 7Q` r?B+? i?2 T?b2 bT+2 Λ +QMbBbib Q7  }MBi2 MmK#2` Q7 dz+QHQm`bǴ H#2H2/ 7`QK 1 iQ qX h?2 K`FBM; Q7 i?2 bBi2b T2`KBib mb iQ /BbiBM;mBb? b2p2`H TQBMi T`Q+2bb2b- 7Q` BMbiM+2 i?2 /Bz2`2Mi ivT2b Q7 i`22b BM  7Q`2biX q2 /2b+`B#2  J`FQp +?BM {Xn }n≥0 iFBM; Bib pHm2b BM i?2 bm#b2i E Q7 ΛV +QMbBbiBM; Q7 i?2 dzT`QT2`Hv +QHQm`2/Ǵ +QM};m`iBQMb- i?i Bb +QM};m`iBQMb x bm+? i?i x(v) = x(w) r?2M2p2` v ∼ wX q2 bi`i 7`QK  T`QT2`Hv +QHQm`2/ +QM};m`iBQM X0 X amTTQb2 i iBK2 n i?2 bii2 Bb xX q2 i?2M +?QQb2 mMB7Q`KHv i `M/QK  bBi2 v- M/ i?2M +?QQb2 mMB7Q`KHv i `M/QK  +QHQm` BM i?2 b2i Q7 +QHQm`b HHQr#H2 i v BM +QM};m`iBQM x- i?i BbAv (x) := {j ∈ {1, 2, . . . , q} ; j = x(w) 7Q` HH w , w ∼ v} . h?2 M2r bii2 i iBK2 n + 1 Bb i?2M y- r?B+? Bb 2[mH iQ x 2t+2Ti 7Q` i?2 M2r +QHQm` j i bBi2 vX o2`B7v i?i i?Bb +?BM Bb B``2/m+B#H2 B7 i?2`2 `2 i H2bi i?`22 +QHQm`br?B+? r2 ?2M+27Q`i? bbmK2X q?i Bb i?2 biiBQM`v /Bbi`B#miBQM Q7 i?Bb +?BM\ 1t2`+Bb2 eXdXNX 1tpBbB#H2 BMi2MbBiv S`Qp2 h?2Q`2K eXjXjX

*?Ti2` d SHK S`Q##BHBiv QM i?2 GBM2 SHK i?2Q`v QM i?2 HBM2 HBMFb irQ ivT2b Q7 biiBQM`Biv 7Q` K`F2/ TQBMi T`Q+2bb2b, iBK2@biiBQM`Biv M/ 2p2Mi@biiBQM`BivX h?2 M2ti irQ 2tKTH2b rBHH +H`B7v i?2b2 MQiBQMbX 1tKTH2 dXyXR, h?2 _2M2rH S`Q+2bbX AM i?Bb ivT2 Q7 TQBMi T`Q+2bb- QM2 /BbiBM;mBb?2b i?2 /2Hv2/ iBK2@biiBQM`v p2`bBQM 7`QK i?2 mM/2Hv2/ p2`bBQM r?Qb2 /Bbi`B#miBQM Bb BMp`BMi rBi? `2bT2+i iQ i?2 b?B7i i?i i`MbHi2b i?2 }`bi 2p2Mi iBK2 iQ i?2 Q`B;BMX q2 ?p2 b22M i?i i?2`2 2tBbib  bBKTH2 `2HiBQM #2ir22M i?2 irQ p2`bBQMb- r?B+? `2 B/2MiB+H 2t+2Ti 7Q` i?2 /Bbi`B#miBQM Q7 i?2 }`bi 2p2Mi iBK2X AM i?2 i2`KBMQHQ;v Q7 SHK i?2Q`v- i?2 mM/2Hv2/ p2`bBQM Bb i?2 SHK p2`bBQM Q7 i?2 iBK2@biiBQM`v p2`bBQMX

1tKTH2 dXyXk, h?2 *QMiBMmQmb@iBK2 J`FQp *?BMX *QMbB/2` M B``2@ /m+B#H2 TQbBiBp2 `2+m``2Mi +QMiBMmQmb@iBK2 ?K+ r?Qb2 BK#2//2/ +?BM Ui?2 +?BM Q#b2`p2/ i i?2 i`MbBiBQM iBK2bV Bb HbQ TQbBiBp2 `2+m``2MiX q?2M bm+?  +?BM Bb UiBK2@VbiiBQM`v- Bi Bb MQi i`m2 BM ;2M2`H i?i i?2 2K#2//2/ +?BM Bb biiBQM`v2p2M r?2M Bi Bb bbmK2/ TQbBiBp2 `2+m``2MiX >Qr2p2`- i?2`2 Bb  bBKTH2 `2HiBQM #2@ ir22M i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 +QMiBMmQmb@iBK2 +?BM M/ i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 BK#2//2/ +?BMX h?2 +QMiBMmQmb@iBK2 +?BM bi`i2/ rBi? i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 BK#2//2/ +?BM Bb i?2 SHK p2`bBQM Q7 i?2 biiBQM`v +QMiBMmQmb@iBK2 +?BMX JQ`2 ;2M2`HHv- SHK i?2Q`v Bb +QM+2`M2/ rBi? DQBMiHv biiBQM`v biQ+?biB+ T`Q+2bb2b M/ TQBMi T`Q+2bb2b BM iBK2 M/ BM bT+2- M/ rBi? i?2 T`Q##BHBbiB+ bBimiBQM i 2p2Mi iBK2b Q` TQBMi HQ+iBQMbX h?2 ivT2 Q7 Bbbm2b i?i Bi //`2bb2b Bb 2bT2+BHHv `2H2pMi BM [m2m2BM; i?2Q`v TTHB2/ iQ b2`pB+2 bvbi2Kb- 7Q` r?B+? i?2`2 `2 irQ /BbiBM+i TQBMib Q7 pB2r- i?i Q7 i?2 dzQT2`iQ`Ǵ- r?Q Bb BMi2`2bi2/ BM i?2 #2?pBQm` Q7  [m2m2 i `#Bi``v iBK2b- M/ i?i Q7 i?2 dz+mbiQK2`Ǵ- r?Q Bb ;2M2`HHv BMi2`2bi2/ BM i?2 bBimiBQM i?i b?2 }M/b mTQM ``BpHX AM  KQ#BH2 +QKKmMB+iBQMb +QMi2ti- r?2`2  TQBMi Kv `2T`2b2Mi i?2 HQ+iBQM Q7  KQ#BH2 T?QM2 Q` Q7 M Mi2MM `2Hv-  bBKBH` /mHBiv 2tBbib #2ir22M i?2 bBimiBQM Q7  ;Bp2M mb2` M/ i?2 bBimiBQM i M `#Bi``v TQBMiX

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_7

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dXR aiiBQM`v SQBMi S`Q+2bb2b h?2 Q`B;BMH 7`K2rQ`F Q7 i?2 KQ/2`M SHK T`Q##BHBivR Bb i?i Q7 biiBQM`v TQBMi T`Q+2bb2bX h?2 b2iiBM; Q7 K2bm`#H2 ~Qrb M/ Q7 +QKTiB#BHBiv rBi?  ~Qr BMi`Q@ /m+2/ BM i?2 +m``2Mi b2+iBQM ?b  MmK#2` Q7 /pMi;2b i?i H`;2Hv +QKT2Mbi2 7Q` Bib `2HiBp2 #bi`+iM2bbX J2bm`#H2 6HQrb q2 }`bi BMi`Q/m+2 i?2 2H2K2Mi`v +QM+2Ti Q7 i`MbHiBQM Q7 7mM+iBQMb M/ Q7 K2@ bm`2b- M/ i?2M T`Q+22/ iQ i?2 KQ`2 #bi`+i MQiBQMb Q7 b?B7i M/ K2bm`#H2 ~QrX h?2 Hii2` MQiBQM rBHH T`QpB/2  TQr2`7mH 7`K2rQ`F 7Q` i?2 /2p2HQTK2Mi Q7 i?2 SHK i?2Q`v BM i?Bb +?Ti2` M/ BM *?Ti2` 3X 1tKTH2 dXRXR, h`MbHi2/ 6mM+iBQMb M/ h`MbHi2/ J2bm`2bX UBV G2i Ω #2 i?2 bT+2 Q7 +QMiBMmQmb 7mM+iBQMb ω : Rm → R M/ H2i F #2 i?2 σ@}2H/ QM Ω ;2M2`i2/ #v i?2 +QQ`/BMi2 7mM+iBQMb y → ω(y) Uy ∈ Rm VX .2}M2  KTTBM; τx : Ω → Ω #v τx (ω)(y) := ω(y + x) (x, y ∈ Rm , ω ∈ Ω) . Uh?Bb KTTBM; τx i`MbHi2b Mv 7mM+iBQM ω ∈ Ω #v −xXV UBBV G2i (Ω, F) = (M (Rm ), M(Rm ))- i?2 bT+2 Q7 HQ+HHv }MBi2 K2bm`2b QM (R , B(Rm ))X G2i θx := Sx - r?2`2 m

(Sx μ)(C) := μ(C + x)

(x ∈ Rm , μ ∈ M (Rm ), C ∈ B(Rm ) .

Uh?2 KTTBM; Sx i`MbHi2b Mv K2bm`2 μ ∈ M (Rm ) #v −xXV

.2}MBiBQM dXRXk  7KBHv {θx }x∈Rm Q7 K2bm`#H2 KTTBM;b 7`QK i?2 K2bm`#H2 bT+2 (Ω, F) BMiQ Bib2H7 Bb +HH2/  b?B7i QM (Ω, F) B7, UV

7Q` HH x ∈ Rm - θx Bb #BD2+iBp2- M/

U#V

7Q` HH x, y ∈ Rm - θx ◦ θy = θx+y X

h?2 b?B7i {θx }x∈Rm QM (Ω, F) Bb +HH2/  K2bm`#H2 ~Qr B7 BM //BiBQM U+V

i?2 KTTBM; (x, ω) → θx (ω) Bb K2bm`#H2 7`QK B(Rm ) ⊗ F iQ FX

AM T`iB+mH`- 7`QK U#V- θ0 Bb i?2 B/2MiBiv M/ θx −1 = θ−x 7Q` HH x ∈ Rm X R

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1tKTH2 dXRX9, J2bm`#H2 6HQr QM  aT+2 Q7 J2bm`2bX h?2 b?B7i {Sx }x∈Rm Q7 1tKTH2 dXRXR UBBV Bb  K2bm`#H2 ~QrX hQ T`Qp2 i?Bb- Bi bm{+2b iQ b?Qr i?i i?2 KTTBM;  (x, μ) → (Sx μ)(f ) := f (y − x)μ(dx) Rm

Bb K2bm`#H2 r?2M2p2` f : R → R Bb  MQM@M2;iBp2 +QMiBMmQmb 7mM+iBQM rBi? +QKT+i bmTTQ`iX h?Bb Bb i?2 +b2 bBM+2 (x, μ) → g(x, μ) := (Sx μ)(f ) Bb +QMiBMmQmb BM i?2 }`bi `;mK2Mi M/ K2bm`#H2

BM i?2 b2+QM/X UAM/22/- g Bb i?2 HBKBi b n ↑ ∞ Q7 i?2 K2bm`#H2 7mM+iBQMb k∈N 1Ck,n (x)g(xk,n , μ)- r?2`2 {xk,n }k∈N Bb M 1 1 m 2MmK2`iBQM Q7 i?2 ;`B/ n−1 Zm M/ Ck,n = xk,n + (− 2n , 2n ] XV m

1tKTH2 dXRX8, a?B7iBM;  J`F2/ SQBMi S`Q+2bbX _272` iQ .2}MBiBQM RXRXky 7Q` i?2 b2iiBM; M/ MQiiBQMX AM i?2 +b2 r?2`2 (E, B(E)) = (Rm , B(Rm ))i?2 b?B7i Sx Ux ∈ Rm V Bb /2}M2/ #v Sx μ (C × L) := μ ((C + x) × L) (C ∈ B(Rm ), L ∈ K) . 6Q` Mv x ∈ Rm i?2 KTTBM; Sx i`MbHi2b i?2 #b2 TQBMi T`Q+2bb #v −x- i?2 K`Fb `2KBMBM; ii+?2/ iQ i?2B` #b2 TQBMi /m`BM; i?2 i`MbHiBQMX q2 MQr BMi`Q/m+2 i?2 7mM/K2MiH MQiBQM Q7 +QKTiB#BHBiv rBi?  ~QrX .2}MBiBQM dXRXe G2i {θx }x∈Rm #2  K2bm`#H2 ~Qr QM (Ω, F)X  biQ+?biB+ T`Q+2bb {Z(x)}x∈Rm /2}M2/ QM (Ω, F) rBi? pHm2b BM i?2 K2bm`#H2 bT+2 (K, K) Bb +HH2/ +QKTiB#H2 rBi? i?2 ~Qr {θx }x∈Rm U7Q` b?Q`i, θx @+QKTiB#H2V B7 Z(x) = Z(0) ◦ θx , i?i BbZ(x, ω) = Z(0, θx (ω))

(ω ∈ Ω, x ∈ Rm ) .

 `M/QK K2bm`2 N QM Rm Bb +HH2/ +QKTiB#H2 rBi? i?2 ~Qr {θx }x∈Rm U7Q` b?Q`i, θx @+QKTiB#H2V B7 N ◦ θx = Sx N , i?i BbN (θx (ω), C) = N (ω, C + x) (ω ∈ Ω, C ∈ B(Rm ), x ∈ Rm ) .

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1tKTH2 dXRX3, G2p2H *`QbbBM;bX AM 1tKTH2 dXRXR UBBV- rBi? m = 1- i?2 `M/QK K2bm`2 N `2+Q`/BM; i?2 +`QbbBM;b Q7 H2p2H a #v i?2 +QQ`/BMi2 T`Q+2bb Bb +QKTiB#H2 rBi? i?2 b?B7i i?2`2Q7X

1tKTH2 dXRXN, .Bb+QMiBMmBiv J2bm`2X G2i Ω #2 i?2 bT+2 Q7 TB2+2rBb2 +QMbiMi +Q`HQH U+QMiBMmQmb QM i?2 `B;?i- rBi? HBKBib QM i?2 H27iV 7mM+iBQMb ω : R → R- M/ H2i F #2 i?2 σ@}2H/ QM Ω ;2M2`i2/ #v i?2 +QQ`/BMi2 7mM+iBQMb X(y) : Ω → R Uy ∈ RV /2}M2/ #v X(y, ω) := ω(y)X .2}M2 7Q` 2+? x ∈ R i?2 KTTBM; θx : Ω → Ω #v θx (ω)(y) := ω(y + x), θx i`MbHi2b  bKTH2 7mM+iBQM #v −xX h?2 `M/QK K2bm`2 N +QmMiBM; i?2 /Bb+QMiBMmBiB2b Q7 i?2 +QQ`/BMi2 T`Q+2bb Bb +QKTiB#H2 rBi? i?2 #Qp2 b?B7iX

1tKTH2 dXRXRy, J`F a2H2+iBQMX AM 1tKTH2 dXRX8- r2 KF2 i?2 7QHHQrBM; +?M;2 Q7 MQiiBQM, p (E)  , F := M p (E)  , θx := Sx . Ω := M h?2 TQBMi T`Q+2bb NL /2}M2/ #v NL (C) := μ (C × L)

(L ∈ K)

Bb θx @+QKTiB#H2X

.2}MBiBQM dXRXRR G2i {θx }x∈Rm #2  K2bm`#H2 ~Qr M/ H2i N #2  bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM Rm +QKTiB#H2 rBi? i?Bb ~Qr- rBi? b2[m2M+2 Q7 TQBMib {Xn }n∈N X G2i {Zn }n∈N #2  b2[m2M+2 Q7 `M/QK p`B#H2b rBi? pHm2b BM bQK2 K2bm`#H2 bT+2 (K, K)X h?2 K`F2/ TQBMi T`Q+2bb (N, Z) Bb bB/ iQ #2 +QKTiB#H2 rBi? i?2 ~Qr NZ (θx (ω), C × L) = NZ (ω, (C + x) × L) (C ∈ B(Rm ), L ∈ K, x ∈ Rm ) . h?2 b2[m2M+2 {Zn }n∈Z Bb i?2M +HH2/  K`F b2[m2M+2 Q7 N +QKTiB#H2 rBi? i?2 ~QrX h?2 K`F Zn Bb ii+?2/ iQ i?2 TQBMi Xn - BM i?2 b2Mb2 i?i Bi dz7QHHQrb Bib TQBMiǴ /m`BM; i?2 i`MbHiBQMb T`QpQF2/ #v i?2 #bi`+i b?B7i θx X q2 bv, ((N, Z), θx ) Bb  K`F2/ TQBMi T`Q+2bb +QKTiB#H2 rBi? i?2 ~QrX

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 rQ`/ #Qmi MQiiBQM 6B`bi- `2+HH i?i #v /2}MBiBQM Q7 +QKTiB#BHBiv- i?2 TQBMi T`Q+2bb N Bb bB/ iQ #2 +QKTiB#H2 rBi? i?2 ~Qr {θx }x∈Rm B7 N ◦ θx := Sx N r?2`2 Sx Bb i?2 i`MbHiBQM QT2`iQ` +iBM; QM K2bm`2b U1tKTH2 dXRXR- UBBVV- i?i Bb(Sx N )(ω) = N (ω) − x := {y − x ; y ∈ N (ω)} . AM bmKK`v- r2 ?p2 i?`22 MQiiBQMb 7Q` i?2 bK2 Q#D2+i N ◦ θx , Sx N , N − x . h?2 Hii2` Bb MQi iQ #2 +QM7mb2/ rBi? N − εx - r?B+? `2T`2b2Mib N \{x} B7 x ∈ N M/ N B7 x ∈ / NX aiiBQM`v 6`K2rQ`Fb G2i (Ω, F, P ) #2  T`Q##BHBiv bT+2 M/ H2i {θx }x∈Rm #2  K2bm`#H2 ~Qr QM (Ω, F)X .2}MBiBQM dXRXRk h?2 T`Q##BHBiv P Bb +HH2/ BMp`BMi rBi? `2bT2+i iQ i?2 ~Qr {θx }x∈Rm U7Q` b?Q`i- θx @BMp`BMiV B7 7Q` HH x ∈ Rm P ◦ θx−1 = P . PM2 i?2M bvb, (P, θx ) Bb  biiBQM`v 7`K2rQ`F QM Rm X 1tKTH2 dXRXRj, aiiBQM`v SQBMi S`Q+2bbX G2i (P, θx ) #2  biiBQM`v 7`K2rQ`F QM Rm X A7 i?2 TQBMi T`Q+2bb N QM Rm Bb +QKTiB#H2 rBi? i?2 b?B7i- Bi Bb biiBQM`vX AM/22/- H2iiBM; A = {ω; N (ω, C1 ) = k1 , . . . , N (ω, Cm ) = km }, r?2`2 C1 , . . . , Cm ∈ B(Rm ) M/ k1 , . . . , km ∈ N- r2 ?p2- #v /2}MBiBQMθx−1 (A) := {ω; θx (ω) ∈ A} = {ω; N (θx (ω), C1 ) = k1 , . . . , N (θx (ω), Cm ) = km } = {ω; N (ω, C1 + x) = k1 , . . . , N (ω, Cm + x) = km }. h?2`27Q`2- bBM+2 P ◦ θx −1 = P P (N (C1 ) = k1 , . . . , N (Cm ) = km ) = P (N (C1 + x) = k1 , . . . , N (Cm + x) = km ) .

AM i?2 bBimiBQM Q7 i?2 T`2pBQmb 2tKTH2- QM2 bvb- 7Q` b?Q`i, (N, θx , P ) Bb  biiBQM`v TQBMi T`Q+2bbX A7 BM .2}MBiBQM dXRXRR- i?2 mM/2`HvBM; T`Q##BHBiv P Bb BMp`BMi rBi? `2bT2+i iQ i?2 ~Qr- r2 bv, ((N, Z), θx , P ) Bb  biiBQM`v K`F2/ TQBMi T`Q+2bbX

key

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

1tKTH2 dXRXR9, aiiBQM`v aiQ+?biB+ S`Q+2bbX G2i (P, θx ) #2  biiBQM@ `v 7`K2rQ`F QM Rm X "v i?2 bK2 `;mK2Mi b BM i?2 2tKTH2 #Qp2-  biQ+?biB+ T`Q+2bb {Z(x)}x∈Rm rBi? pHm2b BM (K, K) i?i Bb θx @+QKTiB#H2 Bb bi`B+iHv biiBQM@ `vX 6Q` b?Q`i, (Z, θx , P ) Bb  biiBQM`v biQ+?biB+ T`Q+2bbX U>2`2 Z biM/b 7Q` {Z(x)}x∈Rm XV h?2 M2ti `2bmHi bvb i?i  biiBQM`v TQBMi T`Q+2bb ?b 2Bi?2` MQ TQBMib i HH Q` M BM}MBiv Q7 TQBMib #Qi? BM R+ M/ R− X h?2Q`2K dXRXR8 G2i (N, θt , P ) #2  biiBQM`v TQBMi T`Q+2bb QM RX h?2M P ({N (R) = 0} ∪ {N ((0, ∞)) = N ((−∞, 0)) = +∞}) = 1. S`QQ7X h?2 b2i Ht := {N ((t, ∞)) = 0} BM+`2b2b rBi? t M/ H :=

∞ 4 n=0

H−n = {N (R) = 0} ⊆ {N ((0, ∞)) < ∞} =

∞ 

Hn := G.

n=0

AM //BiBQM θt Hs = Hs−t 7Q` HH s, t ∈ R- bQ i?i P (Hn ) /Q2b MQi /2T2M/ QM n ∈ ZX >2M+2 P (G) = limn↑∞ P (Hn ) = limn↓−∞ P (Hn ) = P (H)- r?B+?- iQ;2i?2`  s := rBi? H ⊆ G- BKTHB2b i?i P (G − H) = 0X  bBKBH` `;mK2Mi #b2/ QM H {N ((−∞, s)) = 0} H2/b iQ P (G − H) = 0- r?2`2 G = {N ((−∞, 0)) < ∞}X h?2`27Q`2 P ((G ∪ G ) − H) = 0X   bBKBH` T`QQ7 b?Qrb i?i B7 (N, θx , P ) Bb  biiBQM`v TQBMi T`Q+2bb QM Rm i?2M rBi? T`Q##BHBiv 1- 2Bi?2` N (Rm ) = 0 Q` ∞X h?2 MQiBQMb BMi`Q/m+2/ 7Q`  +QMiBMmQmb T`K2i2` ?p2 +QmMi2`T`ib BM i?2 /Bb+`2i2 T`K2i2` +b2X G2i (Ω, F, P 0 ) #2  T`Q##BHBiv bT+2 M/ H2i θ #2  #BD2+iBp2 K2bm`#H2 KT 7`QK Ω iQ Bib2H7 rBi? K2bm`#H2 BMp2`b2X h?2 7KBHv {θn }n∈Z Bb +HH2/  /Bb+`2i2 b?B7i UQ` ~QrV QM (Ω, F) M/  biQ+?biB+ T`Q+2bb {Zn }n∈Z QM (Ω, F) Bb bB/ iQ #2 +QKTiB#H2 rBi? i?2 b?B7i U7Q` b?Q`i, θ@+QKTiB#H2V B7 7Q` HH n ∈ Z M/ HH ω ∈ Ω- Bi ?QH/b i?i Zn = Z0 ◦θ- i?i Bb- Zn (ω) = Z0 (θn (ω))X amTTQb2 i?i P 0 Bb  θ@BMp`BMi T`Q##BHBiv QM (Ω, F)- i?i Bb- P 0 ◦ θ = P 0 X h?2M (P 0 , θ) Bb +HH2/  biiBQM`v 7`K2rQ`FX aBKBH`Hv iQ i?2 +QMiBMmQmb T`K2i2` +b2-  θ@+QKTiB#H2 biQ+?biB+ T`Q+2bb Bb biiBQM`v B7 P 0 Bb θ@BMp`BMiX 1`;Q/B+Biv AM i?Bb bm#b2+iBQM- i?2 biM/`/ `2bmHib +QM+2`MBM; 2`;Q/B+Biv rBHH #2 `2pB2r2/X G2i (P, θx ) #2  biiBQM`v 7`K2rQ`FX _2+HH i?2 7QHHQrBM; /2}MBiBQMb, .2}MBiBQM dXRXRe M 2p2Mi A ∈ F Bb +HH2/ bi`B+iHv θx @BMp`BMi B7 A = θx−1 A 7Q` HH x ∈ Rm X Ai Bb +HH2/ θx @BMp`BMi B7 P (A # θx−1 A) = 0 7Q` HH x ∈ Rm X .2}MBiBQM dXRXRd h?2 ~Qr {θx }x∈Rm Bb +HH2/ P @2`;Q/B+ B7 HH θx @BMp`BMi 2p2Mib `2 i`BpBHX AM Qi?2` rQ`/b, (P, θx ) Bb 2`;Q/B+X

dXRX ahhAPL_u SPALh S_P*1aa1a

keR

.2}MBiBQM dXRXR3  MQM@/2+`2bBM; b2[m2M+2 {An }n≥1 Q7 #QmM/2/ +QMp2t bm#b2ib Q7 Rm Bb +HH2/  +QMp2t p2`;BM; b2[m2M+2 B7 lim sup{r ; B(0; r) ⊂ An } = ∞ ,

n↑∞

r?2`2 B(0; r) Bb i?2 +HQb2/ #HH Q7 `/Bmb r +2Mi2`2/ QM 0X h?2 7mM/K2MiH `2bmHi Q7 2`;Q/B+ i?2Q`v Bb "B`F?QzǶb TQBMirBb2 2`;Q/B+ i?2Q@ `2K, h?2Q`2K dXRXRN G2i (P, θx ) #2 2`;Q/B+X h?2M- 7Q` Mv P @BMi2;`#H2 7mM+iBQM f : (Ω, F) → (R, B(R) 1 f ◦ θx dx = E [f ] . lim m n↑∞ (An ) A n M 2[mBpH2Mi +?`+i2`BxiBQM Q7 2`;Q/B+Biv Bb i?2 7QHHQrBM;,  1 P (A ∩ θx B) = P (A)P (B) (A, B ∈ F ) . lim a↑∞ (2a)m [−a,+a]m

()

AM 7+i- Bi bm{+2b i?i UV ?QH/b 7Q` HH A, B BM M H;2#` A ;2M2`iBM; F 7Q` {θx }x∈Rm iQ #2 P @2`;Q/B+X .2}MBiBQM dXRXky h?2 ~Qr {θx }x∈Rm Bb +HH2/ P @KBtBM; B7 lim P (A ∩ θx B) = P (A)P (B) (A, B ∈ F ) .

|x|↑∞

()

AM Qi?2` rQ`/bc (P, θx ) Bb KBtBM;X AM 7+i- Bi bm{+2b i?i UV ?QH/b 7Q` HH A, B BM M H;2#` A ;2M2`iBM; F 7Q` {θx }x∈Rm iQ #2 P @KBtBM;X *H2`Hv- #v /QKBMi2/ +QMp2`;2M+2 BM UV-  KBtBM; ~Qr Bb 2`;Q/B+X .2}MBiBQM dXRXkR G2i N #2  biiBQM`v `M/QK K2bm`2 QM Rm X Ai Bb +HH2/ 2`;Q/B+ U`2bTX- KBtBM;V B7 (PN , Sx ) Bb 2`;Q/B+ U`2bTX- KBtBM;V- r?2`2 PN Bb i?2 T`Q##BHBiv /Bbi`B#miBQM Q7 N M/ {Sx }x∈Rm Bb i?2 b?B7i Q7 M (Rm )X q2 Kv ?2M+27Q`i? bmTTQb2 i?i N Bb i?2 +MQMB+H `M/QK K2bm`2 QM M (Rm )X h?2 7QHHQrBM; +?`+i2`BxiBQMb Q7 2`;Q/B+Biv M/ KBtBM; Bb H27i b 1t2`+Bb2 dX3XjX G2i LN #2 i?2 GTH+2 i`Mb7Q`K Q7 N X

kek

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

h?2Q`2K dXRXkk UV h?2 biiBQM`v `M/QK K2bm`2 N Bb 2`;Q/B+ B7 M/ QMHv B7  1 lim LN (f + Sx g) dx = LN (f )LN (f ) a↑∞ (2a)m [−a,+a]m 7Q` Mv MQM@M2;iBp2 #QmM/2/ K2bm`#H2 7mM+iBQMb f, g : Rm → R rBi? #QmM/2/ bmTTQ`ibX U#V h?2 biiBQM`v `M/QK K2bm`2 N Bb KBtBM; B7 M/ QMHv B7 lim LN (f + Sx g) = LN (f )LN (f )

|x|↑∞

7Q` Mv MQM@M2;iBp2 #QmM/2/ K2bm`#H2 7mM+iBQMb f, g : Rm → R rBi? #QmM/2/ bmTTQ`ibX 1tKTH2 dXRXkj, h?2 >QKQ;2M2Qmb SQBbbQM S`Q+2bb Bb JBtBM;X h?Bb Bb M BKK2/Bi2 +Q`QHH`v Q7 h?2Q`2K dXRXkkX

dXk  6B`bi GQQF i SHK S`Q##BHBiv G2i mb `2+HH  MQiiBQMX A7 N Bb  TQBMi T`Q+2bb rBi? TQBMi b2[m2M+2 {Xn }n≥1 - N − Xn `2T`2b2Mib i?2 TQBMi T`Q+2bb Q#iBMBM; #v i`MbHiBM; N #v −Xn X AM T`iB+mH` i?2 TQBMi Xn #2+QK2b i?2 Q`B;BMX h?2 TQBMi T`Q+2bb N − εXn Bb i?2 Q`B;BMH N rBi?Qmi i?2 TQBMi Xn X h?2`2 `2 b2p2`H TT`Q+?2b iQ i?2 i?2Q`v Q7 SHK T`Q##BHBivX h?2 QM2 /QTi2/ BM i?Bb +?Ti2` #2;BMb rBi?  M2r HQQF i *KT#2HHǶb 7Q`KmH 7Q` bi@ iBQM`v K`F2/ TQBMi T`Q+2bb2bX h?2 SHK .Bbi`B#miBQM Q7 i?2 J`Fb G2i N #2  bBKTH2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Zn }n∈N #2  b2[m2M+2 Q7 `M/QK p`B#H2b rBi? pHm2b BM i?2 K2bm`#H2 bT+2 (K, K)X 1+? Zn Bb +QMbB/2`2/ b  K`F Q7 i?2 +Q``2bTQM/BM; TQBMi Zn X _2+HH i?i i?2 TQBMi T`Q+2bb M/ Bib b2[m2M+2 Q7 K`Fb `2 `272``2/ iQ b dzi?2 K`F2/ TQBMi T`Q+2bb (N, Z)ǴX Ai +M HbQ #2 `2T`2b2Mi2/ #v  TQBMi T`Q+2bb NZ QM Rm × K /2}M2/ #v  1D (Xn , Zn ) (D ∈ B(Rm ) ⊗ K) . NZ (D) := n∈N

q2 bbmK2 i?i i?2 K`F2/ TQBMi T`Q+2bb (N, Z) Bb biiBQM`v- i?i Bb- 7Q` HH x ∈ Rm - i?2 `M/QK K2bm`2 Sx (NZ ) /2}M2/ #v  Sx (NZ )(D) := 1D (Xn + x, Zn ) n∈N

?b i?2 bK2 /Bbi`B#miBQM b NZ X h?2 BMi2MbBiv Q7 i?2 UbiiBQM`vV TQBMi T`Q+2bb N Bb bbmK2/ #2 TQbBiBp2 M/ }MBi2,

dXkX  6A_ah GPPE h SGJ S_P""AGAhu

kej

0 < λ := E[N ((0, 1]m )] < ∞ . G2i i?2 σ@}MBi2 K2bm`2 νZ QM (Rm × K) #2 /2}M2/ #v (D ∈ B(Rm ) ⊗ K) .

νZ (D) := E [NZ (D)]

_2+HH i?2 7QHHQrBM; MQiiBQM, 7Q` Mv C ⊆ Rm - Mv x ∈ Rm C + x := {y + x ; y ∈ C} . "v biiBQM`BivνZ ((C + x) × L) = νZ (C × L) (x ∈ Rm , C ∈ B(Rm ), L ∈ K) , M/ i?2`27Q`2- 7Q` }t2/ L- i?2 K2bm`2 C → νZ (C × L) Bb BMp`BMi mM/2` i`Mb@ HiBQM bQ i?i- #v h?2Q`2K XRX3- Bi Bb  KmHiBTH2 Q7 i?2 G2#2b;m2 K2bm`2 m QM (Rm , B(Rm )), νZ (C × L) = γ(L) m (C) , 7Q` bQK2 γ(L)X h?2 KTTBM; L → γ(L) Bb  K2bm`2 QM (K, K) i?i Bb }MBi2 bBM+2- rBi? L = K- γ(K) = λ- i?2 }MBi2 TQbBiBp2 BMi2MbBiv Q7 N X AM T`iB+mH`Q0N := λ−1 γ Bb  T`Q##BHBiv K2bm`2 QM (K, K)- M/ νZ (C × L) = λ Q0N (L) m (C) . h?2`27Q`2 Q0N (L)

=

E



1C (Xn )1L (Zn ) . λ m (C)

n∈N

UdXRV

.2}MBiBQM dXkXR h?2 T`Q##BHBiv Q0N QM (K, K) /2}M2/ #v UdXRV Bb i?2 SHK /Bb@ i`B#miBQM Q7 i?2 K`FbX h?2Q`2K dXkXk G2i f : Rm × K → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX h?2M 

   E f (x, z)NZ (dx × dz) = λ f (x, z)Q0N (dz) dx . UdXkV Rm ×K

Rm

K

S`QQ7X 6Q`KmH UdXkV Bb i`m2 7Q` f (x, z) := 1C (x) 1L (z), r?2`2 C ∈ B(Rm ) M/ L ∈ K- bBM+2 Bi i?2M `2/m+2b iQ UdXRVX h?2 ;2M2`H +b2 ;BM 7QHHQrb #v i?2 mbmH KQMQiQM2 +Hbb `;mK2Mi #b2/ QM .vMFBMǶb h?2Q`2K XRXeX  6Q`KmH UdXkV #Qp2 Bb i?2 SHKĜ*KT#2HH 7Q`KmH 7Q` biiBQM`v K`F2/ TQBMi T`Q+2bb2bX

ke9

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

h?2 SHK S`Q##BHBiv hQ Tbb 7`QK i?2 SHK /Bbi`B#miBQM Q7 K`Fb iQ SHK T`Q##BHBiv- Bi Bb +QMp2MB2Mi iQ /Q i?Bb BM i2`Kb Q7 K2bm`#H2 ~Qrb QM #bi`+i T`Q##BHBiv bT+2bX q2 `272` iQ a2+iBQM dXR 7Q` i?2 #bB+ /2}MBiBQMbX G2i ((N, Z), θx , P ) #2  biiBQM`v K`F2/ TQBMi T`Q+2bb QM Rm bm+? i?i N Bb bBKTH2 M/ rBi? }MBi2 TQbBiBp2 BMi2MbBiv λX AM 7+i- i?2 rQ`F M22/2/ 7Q` i?2 /2}MBiBQM Q7 SHK T`Q##BHBiv Bb H`2/v /QM2X Ai bm{+2b iQ +?QQb2 7Q` K`F bT+2 i?2 bKTH2 bT+2 Ω Bib2H7X 6Q` 2+? n ∈ Z- θXn Bb  `M/QK 2H2K2Mi iFBM; Bib pHm2b BM i?2 K2bm`#H2 bT+2 (Ω, F)X hF2 BM UdXRV (K, K) = (Ω, F) M/ Zn = θXn (ω)X .2MQi2 BM i?Bb +b2 Q0N #v PN0 X AM T`iB+mH`- PN0 Bb  T`Q##BHBiv QM (Ω, F)X 6Q`KmH UdXRV i?2M `2/b 7Q` HH C ∈ B(Rm ) Q7 }MBi2 TQbBiBp2 G2#2b;m2 K2bm`2 E n∈N 1C (Xn )1A ◦ θXn 0 . UdXjV PN (A) = λ m (C) .2}MBiBQM dXkXj h?2 T`Q##BHBiv PN0 /2}M2/ #v UdXjV Bb +HH2/ i?2 SHK T`Q#@ #BHBiv bbQ+Bi2/ rBi? P UQ`- KQ`2 T`2+Bb2Hv- rBi? (N, θx , P )VX h?2Q`2K dXkX9 G2i v : Rm ×Ω → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX h?2M



  (v(x) ◦ θx ) N (dx) = λ E0N v(x) dx . UdX9V E Rm

Rm

h?Bb 2[mHBiv Bb Q7i2M ;Bp2M BM i?2 bHB;?iHv KQ`2 2tTHB+Bi 7Q`K,       v(x, θx ω) N (ω, dx) P (dω) = λ v(x, ω) dx PN0 (dω) . Ω

Rm

Ω

Rm

UdX8V

MQi?2` 2tT`2bbBQM 72im`BM; 2tTHB+BiHv i?2 TQBMib Q7 i?2 TQBMi T`Q+2bb QM i?2 `B;?i@ ?M/ bB/2 Bb       v(Xn , θXn ) = λ v(x, ω) dx PN0 (dω) . UdXeV E n∈N

Ω

Rm

S`QQ7X 6Q`KmH UdXjV Bb  bT2+BH +b2 Q7 i?2 MMQmM+2/ 2[mHBiv 7Q` i?2 +?QB+2 v(x, ω) = 1C (x)1A (ω) , 7`QK r?B+? i?2 ;2M2`H +b2 7QHHQrb #v i?2 mbmH KQMQiQM2 +Hbb `;mK2Mi #b2/ QM .vMFBMǶb h?2Q`2K XRXeX  6Q`KmH UdX9V Bb Q7i2M +HH2/ i?2 *KT#2HHĜJ2+F2 7Q`KmH #2+mb2 Bi Bb  bQ@ T?BbiB+i2/ pi` Q7 *KT#2HHǶb 7Q`KmHXk Ai Bb bQK2iBK2b mb2/ BM i?2 Hi2`MiBp2 2[mBpH2Mi 7Q`K 



 E v(x) N (dx) = λE0N (v(x) ◦ θ−x ) dx . UdXdV Rm

k

(_vHH@L`/x2rbFB- RNeR)X

Rm

dXkX  6A_ah GPPE h SGJ S_P""AGAhu

ke8

_2K`F dXkX8 h?2 Tm`TQb2 Q7 i?Bb `2K`F Bb iQ KF2 6Q`KmH dX9 HQQF  HBiiH2 H2bb #bi`+iX G2i h : Rm × M (Rm ) → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX hFBM; v(x, ω) := h(x, N (ω)) BM UdX9V- r2 ?p2   

 0 E h(Xn , N − Xn ) = λEN h(x, N ) dx . Rm

n∈N

aT2+BHBxBM; i?Bb iQ h(x, N ) = g(x)1Γ (N )- r?2`2 Γ ∈ M(Rm )- r2 Q#iBM     E g(Xn )1Γ (N − Xn ) = λPN0 (N ∈ Γ) g(x) dx . Rm

n∈N

qBi? g(x) = 1C (x) r?2`2 C ∈ B(Rm ) Bb #QmM/2/   E 1C (Xn )1Γ (N − Xn ) = λ m (C)PN0 (N ∈ Γ) .

UdX3V

n∈N

 b2i Γ ∈ M(Rm ) `2T`2b2Mib  T`QT2`iv i?i  K2bm`2 μ ∈ M(Rm ) Kv TQbb2bb Q` MQi- M/ N −Xn ∈ Γ K2Mb i?i i?2 TQBMi T`Q+2bb b22M #v M Q#b2`p2` TH+2/ QM i?2 TQBMi Xn Ui?i Bb- T`2+Bb2Hv- N − Xn V biBb}2b i?Bb T`QT2`ivX 6Q` BMbiM+2- rBi? Γ = {μ ; μ(B(0, a)\{0}) = 0}- r?2`2 B(x, a) Bb i?2 QT2M #HH Q7 `/Bmb a ≥ 0 +2Mi2`2/ BM x- {N − Xn ∈

Γ} = {(N − Xn )(B(0, a)\{0}) = 0}- i?i Bb {N (B(Xn , a)\{Xn }) = 0}X h?2 bmK n∈N 1C (Xn )1Γ (N −Xn ) +QmMib i?2 TQBMib Q7 N HvBM; BM C M/ r?Qb2 M2`2bi M2B;?#Qm` Bb i  /BbiM+2 ≥ aX qBi?  ;2M2`H Γ ∈ M(Rm )- i?2 BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb    NΓ (C) := E 1C (Xn )1Γ (N − Xn ) (C ∈ B(Rm )) UdXNV n∈N

Bb- BM pB2r Q7 UdX3V- λΓ = λPN0 (N ∈ Γ)X 1tKTH2 dXkXe,  ǭ`2M2rHǮ 7Q`KmHX G2i (N, θt , P ) #2  biiBQM`v bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R rBi? }MBi2 BMi2MbBiv λ M/ H2i PN0 #2 i?2 bbQ+Bi2/ SHK T`Q##BHBivX h?2M- 7Q` HH a ≥ 0 a 2 E N (0, a] = (2R(t) − 1)λ dt , 0

r?2`2 R(t) := E0N [N (0, t]]

(t ≥ 0) .

Ub r2 b?HH bQQM b22- B7 N Bb  `2M2rH TQBMi T`Q+2bb- R Bb i?2 `2M2rH 7mM+iBQMXV S`QQ7X "v i?2 T`Q/m+i 7Q`KmH Q7 aiB2HiD2b +H+mHmb N (0, a]2 = 2 N ((0, t]) N (dt) − N (0, a] . (0,a]

h?2`27Q`2

kee

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1 E N (0, a]2 = 2E

 R+

N (0, t]1(0,a] (t) N (dt) − λa

AM pB2r Q7 UdXdV rBi? v(t) := N (0, t]1(0,a] (t) UM/ i?2`27Q`2 v(t) ◦ θ−t = N (−t, 0]1(0,a] (t)V  



0 E N (0, t]1(0,a] N (dt) = EN N (−t, 0]1(0,a] (t)λ dt R+ R+  a  E0N [N (−t, 0]] 1(0,a] (t)λ dt = R(t)λ dt . = R+

0



h?2Q`2K dXkXd lM/2` i?2 SHK T`Q##BHBiv- i?2`2 Bb  TQBMi i 0 Ui?2 Q`B;BM Q7 Rm V- i?i BbPN0 (N ({0}) = 1) = 1 . S`QQ7X qBi? g(x) = 1C (x) M/ Γ = {μ ; μ({0}) = 1}- 1[MX UdX3V #2+QK2b- bBM+2 N − Xn ?b Hrvb QM2 TQBMi 2t+iHv i i?2 Q`B;BM   E 1C (Xn ) = λ m (C)PN0 (N ({0}) = 1) . n∈N

aBM+2 i?2 H27i@?M/ bB/2 Bb λ m (C)- i?2 `2bmHi Bb T`Qp2/X



1tKTH2 dXkX3, amT2`TQbBiBQM Q7 BM/2T2M/2Mi biiBQM`v TQBMi T`Q+2bb2bX _2+HH i?i Sx Bb- 7Q` Mv x ∈ Rm - i?2 i`MbHiBQM #v x TTHB2/ iQ K2bm`2b μ ∈ M (Rm ), Sx (μ)(C) = μ(C + x) . G2i P #2  T`Q##BHBiv K2bm`2 QM (M (Rm ), M(Rm )) bm+? i?i P ◦ Sx = P 7Q` HH x ∈ Rm X hFBM; N 2[mH iQ Φ- i?2 B/2MiBiv KT Q7 M (Rm )- r2 Q#@ iBM  biiBQM`v `M/QK (Φ, Sx , P)- r?B+? Bb bB/ iQ #2 BM +MQMB+H 7Q`KX G2i (Mi - Mi - Sx (i) , Φi ) (1 ≤ i ≤ k) #2 `2THB+b Q7 (M (Rm ), M(Rm ), Sx , Φ) M/ H2i Pi #2  T`Q##BHBiv QM (Mi , Mi ) r?B+? Bb Sx (i) @BMp`BMi 7Q` HH x ∈ Rm X amTTQb2 i?i 7Q` HH i U1 ≤ i ≤ kV- Φi Bb Pi @HKQbi bm`2Hv  bBKTH2 TQBMi T`Q+2b rBi? }MBi2 M/ TQbBiBp2 BMi2MbBiv λi X .2}M2 i?2 T`Q/m+i bT+2  k   k k Mi , ⊗i=1 Mi , ⊗i=1 Pi (Ω, F, P ) = i=1

M/- 7Q` 2+? x ∈ Rm - /2}M2 θx := ⊗ki=1 Sx (i) - rBi? i?2 K2MBM; i?i θx (ω) = (Sx (i) μi ; 1 ≤ i ≤ k)- r?2`2 ω = (μi ; 1 ≤ i ≤ k). .2}M2

dXkX  6A_ah GPPE h SGJ S_P""AGAhu Ni (ω) := μi M/ N (ω) :=

ked

k 

μi .

i=1

h?2M (N, θx , P ) Bb  biiBQM`v TQBMi T`Q+2bb- i?2 bmT2`TQbBiBQM Q7 i?2 biiBQM`v TQBMi T`Q+2bb2b (Ni , θx , P ) (1 ≤ i ≤ k). q2 b?HH /2MQi2 #v Pi0 i?2 SHK T`Q##BHBiv bbQ+Bi2/ iQ (Φi , Pi )X Ai rBHH #2 T`Qp2/ #2HQr i?i PN0 =

k    λi  i−1  ⊗j=1 Pj ⊗ Pi0 ⊗ ⊗kj=i+1 Pj , λ i=1

UdXRyV

r?2`2 λ = ki=1 λi X Uh?2 BMi2`T`2iiBQM Q7 UdXRyV Bb i?2 7QHHQrBM;X qBi? T`Q##BHBiv λi i?2 TQBMi i i?2 Q`B;BM BM i?2 SHK p2`bBQM +QK2b 7`QK i?2 i@i? TQBMi T`Q+2bbλ M/ i?2 T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 i@i? T`Q+2bb Bb i?2M Bib SHK T`Q##BHBivr?2`2b i?2 Qi?2` T`Q+2bb2b F22T i?2B` biiBQM`v /Bbi`B#miBQMbX HH i?2 k TQBMi T`Q+2bb2b `2KBM BM/2T2M/2MiXV ! "v /2}MBiBQM- 7Q` A = ki=1 Ai - r?2`2 Ai ∈ Mi 1 E λ  ...



(1A ◦ θx )N (dx)

PN0 (A) = 1 = λ

 M1

(0,1]m



Mk

 k k  

(0,1]m j=1

  k  1 = ... λ M1 Mk j=1

 1Ai ◦

Φj (dx)P1 (dμ1 ) . . . Pk (dμk )

Sx(i)

i=1





k 

(0,1]m

 1Ai ◦

Sx(i)

 Φj (dx) P1 (dμ1 ) . . . Pk (dμk ).

i=1

"mi U6m#BMB M/ i?2 /2}MBiBQM Q7 SHK T`Q##BHBiv Pj0 V 1 λj







k 

... M1

(0,1]m i=1

Mk

 (1Ai ◦

P1 (dμ1 ) . . . Pk (dμk )

Sx(i) ) Φj (dx)

= Pj0 (Aj )

k 

Pi (Ai ),

i=1, i=j

(i)

r?2`2 r2 ?p2 iF2M BMiQ ++QmMi i?2 Sx @BMp`BM+2 Q7 Pi X h?2`27Q`2  PN0

k  i=1

Ai

⎫ ⎪ ⎪ ⎬

⎧ ⎪

 =

k ⎪ ⎨  i=1

 λi 0 Pj (Aj ) , Pi (Ai ) ⎪ ⎪ λ ⎪ ⎪ 1≤j≤k ⎭ ⎩ j=i

r?B+? BKTHB2b UdXRyV- #v h?2Q`2K XRXdX

ke3

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

h?2 *b2 Q7  JQiBQM AMp`BMi S`Q##BHBiv G2i (Ω, F) = (Mp (Rm ), Mp (Rm ) M/ H2i N /2MQi2 i?2 +MQMB+H TQBMi T`Q+2bbX amTTQb2 i?i N ?b  HQ+HHv }MBi2 K2M K2bm`2 M/ Bb KQiBQM@BMp`BMi- i?i Bb7Q` Mv /BbTH+2K2Mi τ - τ N ∈ A M/ N `2 B/2MiB+HHv /Bbi`B#mi2/X Ai Bb BM T`iB+mH` i`MbHiBQM BMp`BMi- M/ i?2 SHK /Bbi`B#miBQM PN0 Bb r2HH /2}M2/X HbQ- i?2 K2M K2bm`2 Q7 N Bb Q7 i?2 7Q`K λ m 7Q`  }MBi2 MQM@M2;iBp2 λ- bbmK2/ TQbBiBp2 UN Bb MQi 2KTivVX h?2Q`2K dXkXN 6Q` Mv `QiiBQM r Q7 Rm `QmM/ i?2 Q`B;BM 0PN0 (A) = PN0 (rA)

(A ∈ F ) .

UAM Qi?2` rQ`/b- KQiBQM@BMp`BM+2 Q7 N mM/2` P BKTHB2b `QiiBQM BMp`BM+2 Q7 PN0 XV S`QQ7X .2MQiBM; #v B(0; r) i?2 +HQb2/ #HH Q7 +2Mi2` 0 M/ `/Bmb R  m 0 λ (B(0; r))PN (A) = 1A (θx ω) N (ω, dx) P (dω) Ω B(0;r)   1A (f (x, ω) N (ω, dx) P (dω) = Ω

B(0;r)

U_2+HH _2K`F dXRXjV   1A (f (x, r−1 ω) N (r−1 ω, dx) P (dω) = Ω

B(0;r)

U#v `QiiBQMH BMp`BM+2 Q7 N mM/2` P )   1A (f (r−1 y, r−1 ω) N (r−1 ω, r−1 dy) P (dω) = Ω

B(0;r)

U+?M;2 Q7 p`B#H2 x = r−1 y)   1rA (f (y, ω) N (ω, dy) P (dω) = Ω

B(0;r)

= λ m (B(0; r))PN0 (rA) .  h?BMMBM; M/ *QM/BiBQMBM; G2i (N, θx , P ) #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM Rm rBi? }MBi2 TQbBiBp2 BMi2MbBivX 6Q` U ∈ F - /2}M2 i?2 TQBMi T`Q+2bb NU #v  NU (ω, C) := 1U (θx (ω))N (ω, dx), C ∈ B(Rm ). UdXRRV C

am+?  TQBMi T`Q+2bb Bb i?2`27Q`2 Q#iBM2/ #v i?BMMBM; Q7 N -  TQBMi x ∈ N (ω) #2BM; `2iBM2/ B7 M/ QMHv B7 θx (ω) ∈ U X 1tKTH2 dXkXRy, J`F b2H2+iBQMX G2i (N, Z, θx , P ) #2  biiBQM`v K`F2/ TQBMi T`Q+2bbX hF2 U = {Z0 ∈ L} 7Q` bQK2 L ∈ KX h?2M- bBM+2 Z0 (θXn (ω)) = Zn (ω)-

dXkX  6A_ah GPPE h SGJ S_P""AGAhu NU (C) =



keN

1L (Zn )1C (Xn ).

n∈N

h?2 TQBMi T`Q+2bb NU Bb Q#iBM2/ #v i?BMMBM; N - QMHv `2iBMBM; i?2 TQBMib Q7 Xn rBi?  K`F Zn 7HHBM; BM LX (NU , θx , P ) Bb Q#pBQmbHv  biiBQM`v TQBMi T`Q+2bb M/ Bi ?b  }MBi2 BMi2MbBiv UbBM+2 NU ≤ N VX A7 i?2 BMi2MbBiv Q7 λU Q7 NU Bb TQbBiBp2- Bib SHK T`Q##BHBiv Bb ;Bp2M #v   1  0 PNU (A) = E (1A ◦ θx )NU (dx) , λU (0,1]m 

r?2`2

(1U ◦ θx ) N (dx) = λPN0 (U ) .

λU = E (0,1]m

AM //BiBQM- r2 ?p2

  (1A ◦ θx ) NU (dx) = E E (0,1]m

(1A ◦ θx )(1U ◦ θx ) N (dx) = λPN0 (A ∩ U ) . (0,1]m

h?2`27Q`2

PN0 (A ∩ U ) = PN0 (A | U ) . PN0 (U ) LQi2 i?i i?2 b2[m2M+2 Q7 K`Fb +QmH/ iF2 Bib pHm2b BM M (Rm )- 7Q` BMbiM+2Zn = N − Xn X _2+HH i?i N − Xn = SXn (N )X hFBM; U := {ω ; , N (ω) ∈ Γ}r?2`2 Γ ∈ M(Rm )- r2 b22 i?i BM i?Bb +b2 NU ≡ NΓ - r?2`2 i?2 Hii2` Bb /2}M2/ #v UdXNVX PN0 U (A) =

1tKTH2 dXkXRR, amT2`TQbBiBQM Q7 biiBQM`v TQBMi T`Q+2bb2bX h?Bb 2t@ KTH2 ;2M2`HBx2b 1tKTH2 dXkX3X G2i Ni (1 ≤ i ≤ k) #2 TQBMi T`Q+2bb2b QM Rm - HH +QKTiB#H2 rBi? i?2 ~Qr {θx }x∈Rm - rBi? TQbBiBp2 }MBi2 BMi2MbBiB2b λi (1 ≤ i ≤ k) `2bT2+iBp2Hv- #mi MQi M2+2bb`BHv BM/2T2M/2MiX *HH N i?2B` bmT2`TQbBiBQM- bbmK2/ bBKTH2X 6`QK "v2bǶb `mH2, PN0 (A) =

k 

PN0 (Ni ({0}) = 1) PN0 (A | Ni ({0}) = 1) .

i=1

"mi PN0

   1 λi 1 . (Ni ({0}) = 1) = E 1(0,1]m (Xn )1{Ni ({Xn })=1} = E [Ni (0, 1]] = λ λ λ n∈N

G2i U = {Ni ({0}) = 1}X aBM+2 r2 ?p2 NU = Ni UrBi? i?2 MQiiBQM Q7 UdXRRVV- r2 Q#iBM PN0 (A | Ni ({0}) = 1) = PN0 i (A) . h?2`27Q`2 PN0 (A) =

k  λi i=1

λ

PN0 i (A) .

kdy

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

dXj SHK h?2Q`v QM i?2 GBM2, "bB+ 6Q`KmHb 6`QK MQr QM BM i?Bb +?Ti2`- ii2MiBQM rBHH #2 `2bi`B+i2/ iQ bBKTH2 biiBQM`v TQBMi T`Q+2bb2b QM i?2 `2H HBM2X h?2 MQiiBQM t BMbi2/ Q7 x rBHH 2KT?bBx2 i?2 7+i i?i QM2 Bb rQ`FBM; QM i?2 `2H HBM2X h?2 SHK T`Q##BHBiv PN0 bbQ+Bi2/ rBi? i?2 biiBQM`v TQBMi T`Q+2bb (N, θt , P ) ?b- b r2 br- Bib Kbb +QM+2Mi`i2/ QM Ω0 := {T0 = 0}X _2+HH i?i i?2 b2[m2M+2 Q7 TQBMib {Tn }n∈Z Bb /2}M2/ BM bm+?  rv i?i Bi Bb bi`B+iHv BM+`2bBM; M/ bm+? i?i T0 ≤ 0 < T1 X T0

T1

T2

t

T3

T−1 ◦θt

T0 ◦θt

0

T1 ◦θt

0

T−2 ◦θt

N

N ◦θt = St N

h?2 KTTBM; θ := θT1 , /2}M2/ 7`QK Ω0 BMiQ Ω0 - Bb  #BD2+iBQM- rBi? BMp2`b2 θ−1 = θT−1 X HbQ- QM Ω0 θTn := θn 7Q` HH n ∈ ZXj h?Bb Bb MQi i`m2 QM Ω U7Q` BMbiM+2- i?2 BMp2`b2 Q7 θT1 Bb MQi θT−1 c 1t2`+Bb2 dX3X9VX h?2 T`QT2`iv BM i?2 M2ti i?2Q`2K Bb `272``2/ iQ b i?2 dz2p2Mi@iBK2 biiBQM`BivǴX h?2Q`2K dXjXR PN0 Bb θ@BMp`BMi. S`QQ7X 6B`bi Q#b2`p2 i?i 7Q` HH A ∈ F 1{θ−1 (A)◦θTn } = 1{θTn ∈θ−1 (A)} = 1{θTn+1 ∈A} = 1A (θTn+1 ) . LQr- rBi? A ∈ F M/ C = (0, t]- 7Q`KmH UdXjV ;Bp2b, & & & & & & 0 1 & 0 −1 &PN (A) − PN (θ (A))& ≤ E & (1A ◦ θTn − 1θ−1 (A) ◦ θTn )1(0,t] (Tn )&& & λt & n∈Z & & & 1 && 2 & = E & (1A ◦ θTn − 1A ◦ θTn+1 )1(0,t] (Tn )& ≤ . & λt λt & n∈Z G2iiBM; t → ∞- r2 Q#iBM PN0 (A) = PN0 (θ−1 (A))X



AM T`iB+mH`- B7 {Z(t)}t∈R Bb θt @+QKTiB#H2 M/ i?2`27Q`2 biiBQM`v mM/2` P i?2 b2[m2M+2 {Z(Tn )}n∈Z Bb- mM/2` PN0 -  biiBQM`v b2[m2M+2X j Ai Bb T2`?Tb rQ`i?r?BH2 iQ 2KT?bBx2 i?2 7+i i?i θTn Bb dzi?2 b?B7i i?i KQp2b i?2 n@i? TQBMi Q7  TQBMi T`Q+2bb iQ i?2 Q`B;BMǴX

dXjX SGJ h>1P_u PL h>1 GAL1, "aA* 6P_JlGa

kdR

1tKTH2 dXjXk, SHKėE?BM+?BM 2[miBQMbX G2i 7Q` k ∈ N M/ t ≥ 0ϕk (t) := PN0 (N (0, t] = k) . q2 ?p2 i?2 SHKĜE?BM+?BM 2[miBQMb,9 

t

P (N (0, t] > k) = λ

ϕk (s) ds. 0

hQ T`Qp2 i?Bb- Q#b2`p2 i?i  1{N (s,t]=k} N (ds),

1N (0,t]>k = (0,t]

M/ /2/m+2 7`QK i?Bb M/ N (s, t] = N (0, t − s] ◦ θs i?i    1(0,t] (s) 1{N (0,t−s]=k} ◦ θs N (ds). 1{N (0,t]>k} = R

"v i?2 J2+F2 7Q`KmH- i?2 2tT2+iiBQM Q7 i?2 `B;?i@?M/ bB/2 rBi? `2bT2+i iQ P Bb 2[mH iQ   t λ 1(0,t] (s)PN0 (N (0, t − s] = k) ds = λ PN0 (N (0, t − s] = k) ds R 0  t = λ PN0 (N (0, s] = k) ds. 0

h?2Q`2K dXjXj G2i (N, θt , P ) #2  biiBQM`v TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞ M/ bm+? i?i P (N (R) = 0) = 0X h?2M λE0N [T1 ] = 1 . S`QQ7X 6`QK i?2 SHKĜE?BM+?BM 2[miBQM rBi? k = 0 t ϕ0 (s) ds . P (N (0, t] = 0) = 1 − λ 0

"mi ϕ0 (s) = PN0 (N (0, s] = 0) = P (T1 > s)- M/ i?2`27Q`2 0 = P (N (R) = 0) = lim P (N (0, t] = 0) t↑+∞  ∞ PN0 (T1 > s) ds = 1 − λE0N [T1 ] . =1−λ 0

 9

(SHK- RN9j) 7Q` k = 0- (E?BM+?BM- RNey) 7Q` k ≥ 1X

kdk

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

AMp2`bBQM 6Q`KmHb >Qr iQ Tbb 7`QK i?2 SHK T`Q##BHBiv iQ i?2 biiBQM`v /Bbi`B#miBQM\ h?2 7Q`KmHb /QBM; i?Bb `2 +HH2/ BMp2`bBQM 7Q`KmHbX h?2Q`2K dXjX9 G2i (N, P, θt ) #2  biiBQM`v bBKTH2 TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞ M/ bm+? i?i P (N (R) = 0) = 0X 6Q` Mv MQM@M2;iBp2 `M/QK p`B#H2 f

 T1

E [f ] = E0N

(f ◦ θs )ds .

0

 T`QQ7 rBHH #2 ;Bp2M pB i?2 7QHHQrBM; +QMb2`piBQM T`BM+BTH2- r?B+? bvb i?i dzi?2 bKQQi? p`BiBQM Q7  biiBQM`v biQ+?biB+ T`Q+2bb Bb #HM+2/ #v i?2 p`BiBQM /m2 iQ DmKTbǴX JQ`2 T`2+Bb2Hv, h?2Q`2K dXjX8 U8 V G2i (N, P, θt ) #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞X G2i {Y (t)}t∈R #2  `2H@pHm2/ biQ+?biB+ T`Q+2bb- `B;?i@ +QMiBMmQmb rBi? H27i@?M/ HBKBib- M/ H2i {Y  (t)}t∈R #2  `2H@pHm2/ biQ+?biB+ T`Q+2bb bm+? i?i  1  Y (1) − Y (0) = Y  (s)ds + (Y (s) − Y (s−))N (ds). UdXRkV 0

(0,1]

amTTQb2 i?i i?2 T`Q+2bb2b Y M/ Y  `2 θt @+QKTiB#H2X amTTQb2 KQ`2Qp2` i?i E[|Y  (0)|] < ∞ M/ E0N [|Y (0) − Y (0−)|] < ∞ .

UdXRjV

E[Y  (0)] + λE0N [Y (0) − Y (0−)] = 0 .

UdXR9V

h?2M

6Q` i?2 T`QQ7- r2 b?HH M22/ i?2 7QHHQrBM; bBKTH2 H2KK, G2KK dXjXe G2i (P 0 , θ) #2  biiBQM`v 7`K2rQ`FX G2i Z #2 MQM@M2;iBp2- P 0 @ XbX }MBi2 `M/QK p`B#H2 bm+? i?i Z − Z ◦ θ ∈ L1 (P 0 )X h?2M E0 [Z − Z ◦ θ] = 0X S`QQ7X 6Q` Mv C > 0 |Z ∧C −(Z ∧C)◦θ| ≤ |Z −Z ◦θ|X h?2M- #v i?2 θ@BMp`BM+2 Q7 P 0 - E[Z ∧C −(Z ∧C)◦θ] = 0 M/ i?2 +QM+HmbBQM 7QHHQrb #v /QKBMi2/ +QMp2`;2M+2H2iiBM; C ↑ ∞ BM i?2 Hbi 2[mHBivX ULQi2 i?i i?2 /2HB+i2 TQBMi ?2`2 Bb i?i r2 /Q MQi bbmK2 i?i Z Bb BMi2;`#H2XV  S`QQ7X P#b2`p2 i?i  1 E[ |Y  (s)| ds] = E[|Y  (0)|] M/ 0  E[ |Y (s) − Y (s−)| N (ds)] = λE0N [|Y (0) − Y (0−)|] . (0,1] 8

(JBvxr- RNN9)X

dXjX SGJ h>1P_u PL h>1 GAL1, "aA* 6P_JlGa

kdj

h?2`27Q`2- +QM/BiBQM UdXRjV ;m`Mi22b i?i Y (1) − Y (0) Bb S@BMi2;`#H2 M/ #v G2KK dXjXe- E [Y (1) − Y (0] = 0X 1[miBM; i?Bb iQ i?2 2tT2+iiBQM Q7 i?2 `B;?i@ 1  ?M/ bB/2 Q7 UdXRkVr2 Q#iBM i?2 MMQmM+2/ `2bmHi bBM+2 E[ Y (s) ds] = E[Y  (0)] 0  0 M/ E[ (0,1] (Y (s) − Y (s−)) N (ds)] = λEN [Y (0) − Y (0−)]X  _2K`F dXjXd G2i mb KF2 KQ`2 T`2+Bb2 i?2 BMi2`T`2iiBQM Q7 i?2 +QMb2`piBQM T`BM+BTH2, E[Y  (0)] Bb i?2 p2`;2 `i2 Q7 BM+`2b2 UT2` mMBi iBK2V /m2 iQ i?2 bKQQi? 2pQHmiBQM Q7 i?2 biQ+?biB+ T`Q+2bb Y #2ir22M DmKTb- r?2`2b λE0N [Y (0) − Y (0−)] Bb i?2 p2`;2 `i2 Q7 BM+`2b2 /m2 iQ i?2 DmKTb Ui /Bb+QMiBMmBiv iBK2bVX h?2`27Q`2 1[MX UdXR9V bii2b i?i i?2 iQiH p2`;2 `i2 Q7 BM+`2b2 Bb MmHH- r?B+? Bb 2tT2+i2/ bBM+2 i?2 T`Q+2bb Bb biiBQM`vX AM i?Bb b2Mb2- UdXR9V Bb  +QMb2`piBQM 2[miBQMX q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K dXjX9X S`QQ7X Ai Bb 2MQm;? iQ T`Qp2 i?2 `2bmHi 7Q` #QmM/2/ f X G2i  T+ (t) Y (t) := (f ◦ θs ) ds, t

r?2`2 T+ (t) = inf{Tn ; Tn > t}VX h?Bb T`Q+2bb biBb}2b i?2 +QM/BiBQMb Q7 h?2Q`2K T dXjX8 rBi? Y  (0) = −f M/ Y (0) − Y (0−) = 0 1 (f ◦ θs ) ds Umb2 i?2 7+i i?i E0N [T1 ] = λ−1 < ∞VX  "v i?2 θ@BMp`BM+2 Q7 P  E [f ] = E0N

Tn+1

(f ◦ θs ) ds ,

UdXR8V

Tn

M/ BM T`iB+mH`- 7Q` f = 1A  +∞ P (A) = PN0 (Tn < s ≤ Tn+1 , θs ∈ A) ds . −∞

UdXReV

M AMi2`T`2iiBQM Q7 i?2 AMp2`bBQM 6Q`KmH h?2 BMp2`bBQM 7Q`KmH `2+2Bp2b M BMi2`2biBM; BMi2`T`2iiBQM r?2M r`Bii2M b

 T1 1 (f ◦ θt ) dt = E0N [λT1 (f ◦ θV )] , E[f ] = E0N λT1 T1 0 r?2`2 V Bb  `M/QK p`B#H2 r?B+?- dz+QM/BiBQMHHv QM 2p2`vi?BM; 2Hb2Ǵ- Bb mMB@ 7Q`KHv /Bbi`B#mi2/ QM [0, T1 ] U7Q` i?2 #Qp2 iQ KF2 b2Mb2- r2 Kmbi Q7 +Qm`b2 2M@ H`;2 i?2 T`Q##BHBiv bT+2VX h?Bb BMi2`T`2iiBQM T`QpB/2b M 2tTHB+Bi +QMbi`m+iBQM Q7 P 7`QK PN0 - b 7QHHQrbX 6B`bi +QMbi`m+i i?2 T`Q##BHBiv P0 #v dP0 = (λT1 ) dPN0 . aBM+2 PN0 (T0 = 0) = 1 M/ P0 Bb #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ PN0 P0 (T0 = 0) = 1X h?2 biiBQM`v T`Q##BHBiv P Bb i?2M Q#iBM2/ #v TH+BM; i?2 Q`B;BM i `M/QK BM i?2 BMi2`pH [0, T1 ]- i?i Bb-

kd9

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1 E[f ] = E0 [f ◦ θV ].

1tKTH2 dXjX3,  aBKmHiBQM H;Q`Bi?KX M TTHB+iBQM Q7 i?2 #Qp2 `2bmHi Bb iQ i?2 2t+i bKTHBM; Q7 i?2 biiBQM`v T`Q##BHBiv P r?2M  bKTH2 Q7 i?2 SHK T`Q##BHBiv PN0 Bb FMQrMX h?Bb +M #2 /QM2 BM i?2 +b2 i?i PN0 (T1 ≤ a) = 1 7Q` bQK2 a < ∞Xe 6Q` i?Bb- 2tKBM2 bm++2bbBp2Hv i?2 BMi2`@2p2Mi iBK2b Sn := Tn − Tn−1 Un ≥ 1V Q7 i?2 PN0 bKTH2 U7Q` r?B+? r2 `2+HH i?i T0 = 0V mMiBH i?2 2p2Mi Un ≤ Sn Bb `2HBx2/- r?2`2 {Un }n≥1 Bb M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b mMB7Q`KHv a /Bbi`B#mi2/ QM [0, 1]X G2i M #2 i?2 `M/QK BM/2t r?2M Bi Bb `2HBx2/X h?2M i?2 TQBMi T`Q+2bb N 7i2` TM −1 Bb  `2HBxiBQM Q7 P0 X A7 U Bb MQi?2` `M/QK p`B#H2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] M/ BM/2T2M/2Mi Q7 2p2`vi?BM; 2Hb2- i?2 TQBMi T`Q+2bb N 7i2` TM −1 + U SM Bb  bKTH2 Q7 P X h?2 /2iBHb Q7 i?2 T`QQ7 `2 `2[mB`2/ BM 1t2`+Bb2 dX3XdX

1tKTH2 dXjXN, J2M pHm2 7Q`KmHbX G2i (N, θt , P ) #2  biiBQM`v bBKTH2 TQBMi T`Q+2bb QM R rBi? }MBi2 TQbBiBp2 BMi2MbBiv M/ bbQ+Bi2/ SHK T`Q##BHBiv PN0 X G2i {Zt }t∈R #2  θt @+QKTiB#H2 biQ+?biB+ T`Q+2bb rBi? pHm2b BM i?2 K2bm`#H2 bT+2 (K, K)X h?2 7QHHQrBM; 7Q`KmHb `2 irQ T`iB+mH` 7Q`Kb Q7 i?2 BMp2`bBQM 7Q`KmH M/ Q7 i?2 /2}MBiBQM Q7 SHK T`Q##BHBivX 6Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb g : (K, K) → (R, B)

E [g(Z0 )] = M/ 



E0N g(Z0 ) =

E0N



T1 0

 g(Zt )dt

E0N [T1 ]

UdXRdV



E

 g(ZTn )1{Tn ∈(0,1]}   . E n∈Z 1{Tn ∈(0,1]} n∈Z

UdXR3V

"+Fr`/ M/ 6Q`r`/ _2+m``2M+2 hBK2b h?2Q`2K dXjXRy G2i (N, P, θt ) #2  biiBQM`v bBKTH2 TQBMi T`Q+2bb QM R rBi? BMi2MbBiv 0 < λ < ∞ M/ bm+? i?i P (N (R) = 0) = 0- M/ rBi? bbQ+Bi2/ SHK T`Q##BHBiv PN0 X G2i F0 #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 T1 mM/2` PN0 , F0 (x) = PN0 (T1 ≤ x). h?2M- 7Q` HH v, w ∈ R+  P (T1 > v, −T0 > w) = λ

∞ v+w

e

(h?Q`BbbQM- kyyy)- b2+iBQM eX9- *?Ti2` 3X

(1 − F0 (u))du .

UdXRNV

dXjX SGJ h>1P_u PL h>1 GAL1, "aA* 6P_JlGa

kd8

S`QQ7X G2i u, v ∈ R+ X qBi? A := {T1 > v, −T0 > w} BM UdXReV r`Bii2M 7Q` n = 0M/ iFBM; BMiQ ++QmMi i?i PN0 @XbX- −T0 ◦ θt = t M/ T1 ◦ θt = T1 − t 7Q` HH t ∈ [0, T1 )- r2 Q#iBM  P (T1 > v, −T0 > w) =

λE0N

= λE0N



T1

1{t>v} 1{T1 −t>w} dt (T1 − (v + w))+ . 0

h?2 `2bmHi i?2M 7QHHQrb 7`QK i?2 7QHHQrBM; +QKTmiiBQM UpHB/ 7Q` Mv MQM@M2;iBp2 `M/QK p`B#H2 X M/ MQM@M2;iBp2 MmK#2` aV,  ∞ E[(X − a)+ ] = P ((X − a)+ > x) dx 0  ∞  ∞ = P (X − a > x) dx = P (X > x) dx . 0

a



λ

hFBM; v = 0 M/ w = 0- r2 Q#iBM i?i P (T1 > 0, −T0 > 0) = 1 bBM+2 ∞ (1 − F0 (u))du = λE0N [T1 ] = 1X AM T`iB+mH` 0

h?2Q`2K dXjXRR lM/2` P - i?2`2 Bb MQ TQBMi Q7 N i i?2 Q`B;BM Q7 iBK2X hFBM; MQr v = 0- r2 Q#iBM 



P (−T0 > w) = λ

(1 − F0 (u)) du.

w

aBKBH`Hv-





P (T1 > v) = λ

(1 − F0 (u)) du.

v

h?mb −T0 M/ T1 `2 B/2MiB+HHv /Bbi`B#mi2/ mM/2` P U#mi MQi BM/2T2M/2Mi BM ;2M2`HVX h?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM QM R+  x F (x) = λ (1 − F0 (u))du 0

Bb i?2 2t+2bb /Bbi`B#miBQM Q7 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F0 X h?2 1t+?M;2 6Q`KmH G2i (N, θt , P ) M/ (N  , θt , P ) #2 irQ biiBQM`v TQBMi T`Q+2bb2b rBi? TQbBiBp2 }MBi2 BMi2MbBiB2b λ M/ λ `2bT2+iBp2HvX LQi2 i?i N M/ N  `2 DQBMiHv biiBQM`v- BM i?2 b2Mb2 i?i i?2B` biiBQM`Biv Bb `2HiBp2 iQ i?2 bK2 [m/`mTH2 (Ω, F, P, θt )X h?2 7QHHQrBM; `2bmHi HBMFb i?2B` SHK /Bbi`B#miBQMb,d d

(L2p2m- RNe3 M/ RNdd)X

kde

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

h?2Q`2K dXjXRk λE0N [f ]







 0 EEN 

(0,T1 ]

(f ◦ θt )N (dt)

UdXkyV

7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb f : (Ω, F) → (R, B(R))X S`QQ7X "v i?2 KQMQiQM2 +QMp2`;2M+2 i?2Q`2K- r2 Kv bbmK2 f #QmM/2/ Ubv #v RVX qBi? bm+? f - bbQ+Bi2 i?2 7mM+iBQM  g := (f ◦ θt )N (dt). (T0  ,T1  ]

6Q` HH t ∈ R+ - r2 ?p2   (f ◦ θs ) N (ds) = (0,t]

(g ◦ θs ) N  (ds) + R(t), (0,t]

r?2`2- /2MQiBM; #v T+ (t) i?2 }`bi TQBMi Q7 N  bi`B+iHv 7i2` t  (f ◦ θs )N (ds) − (f ◦ θs )N (ds). R(t) =  (0)] (0,T+

 (t)] (t,T+

6Q` HH > 0- H2i f := f 1{N (T0 ,T1  ]≤−1} 

M/ g :=

(T0  ,T1  ]

(f ◦ θt ) N (dt) .

6Q` i?2b2 7mM+iBQMb- 2+? i2`K BM R(t) Bb #QmM/2/ #v - M/ i?2`27Q`2 i?2 2tT2+i@ iBQMb `2 }MBi2X JQ`2Qp2`- #v θt @BMp`BM+2 Q7 P - i?2v ?p2 i?2 bK2 2tT2+iiBQMbbQ i?i E[R(t)] = 0X q2 i?2`27Q`2 ?p2 



 E (f ◦ θs ) N (ds) = E (g ◦ θs ) N (ds) . (0,t]

(0,t]

h?Bb T`Qp2b UdXkyV rBi? f = f , g = g X G2iiBM; ;Q iQ BM}MBiv vB2H/b i?2 MMQmM+2/ 2[mHBivX 

dX9 6`QK SHK S`Q##BHBiv iQ aiiBQM`v S`Q##BHBiv h?Bb b2+iBQM3 b?Qrb ?Qr iQ +QMbi`m+i P 7`QK PN0 X G2i {θt }t∈R #2  K2bm`#H2 ~Qr QM (Ω, F)- M/ H2i N #2  TQBMi T`Q+2bb +QKTiB#H2 rBi? i?Bb ~QrX G2i P 0 #2  T`Q##BHBiv K2bm`2 QM (Ω, F) i?i Bb UV +QM+2Mi`i2/ QM Ω0 = {T0 = 0}- i?i Bb- bm+? i?i P 0 (Ω0 ) = 1- M/ U#V θTn @BMp`BMi- i?i Bb- bm+? i?i P 0 (θTn ∈ .) = P 0 (.) 7Q` HH n ∈ ZX 3

(_vHH@L`/x2rbFB- RNeR)- (aHBpMvF- RNek)X

dX9X 6_PJ SGJ S_P""AGAhu hP ahhAPL_u S_P""AGAhu

kdd

h?2Q`2K dX9XR bbmK2 KQ`2Qp2` i?i i?2 7QHHQrBM; i?`22 T`QT2`iB2b `2 biBb}2/, 0 < E0 [T1 ] < ∞, P 0 [T1 > 0] = 1, E0 [N (0, t0 ]] < ∞ 7Q` bQK2 t0 > 0 .

UBV UBBV UBBBV

UdXkRV

h?2M P 0 Bb i?2 SHK T`Q##BHBiv PN0 bbQ+Bi2/ rBi? (N, P ) r?2`2 P Bb i?2 θt @ BMp`BMi T`Q##BHBiv ;Bp2M #v

 T1 1 0 P (A) = 0 E (1A ◦ θt ) dt , (A ∈ B(R)) . UdXkkV E [T1 ] 0 S`QQ7X Ai Kmbi #2 }`bi p2`B}2/ i?i P Bb θt @BMp`BMi 7Q` HH t ∈ R- M/ i?2M i?i PN0 = P 0 X PM Ω0 - 7Q` HH j ∈ Z

T1 ◦θTj T0 ◦θTj



Tj+1

(1A ◦ θs ◦ θTj ) ds =

(1A ◦ θs ) ds,

Tj

M/ i?2`27Q`2- bBM+2 P 0 Bb θTn @BMp`BMiP (A) =

1 1 E0 n E0 [T1 ]



Tn

(1A ◦ θs )ds .

0

HbQ- 7Q` HH t ∈ RP (θt (A)) =

1 1 E0 n E0 [T1 ]



Tn 0



 Tn −t 1 1 0 E (1θt (A) ◦ θs )ds = (1 ◦ θ )ds . A s n E0 [T1 ] −t

h?2`27Q`2 |P (A) − P (θt (A))| ≤

 0 & & Tn 1 1 2t & 0& E . (1 ◦ θ ) ds − (1 ◦ θ ) ds &≤ & A s A s n E0 [T1 ] nE0 [T1 ] −t Tn −t

G2iiBM; n ↑ ∞- Bi 7QHHQrb i?i P (A) = P (θt (A))X q2 MQr T`Qp2 i?i i?2 BMi2MbBiv Q7 N Bb (E0 [T1 ])−1 X 6`QK UdXkkV- r2 Q#iBM  T1

1 E[f ] = 0 E0 (f ◦ θt ) dt , E [T1 ] 0 7Q` HH MQM@M2;iBp2 f X AM T`iB+mH`- 7Q` HH ε > 0

 T1 E[N (0, ε]] 1 1 0 N (t, t + ε] dt . = 0 E ε E [T1 ] ε 0 "v bbmKTiBQM UBBV1 ε→0 ε



T1

N (t, t + ε] dt = 1 .

lim

0

h?2`27Q`2- B7 r2 +M BMi2`+?M;2 HBKBi M/ 2tT2+iiBQM-

kd3

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1 1 lim E0 ε→0 ε

bQ i?i lim

ε→0



T1

N (t, t + ε] dt = 1 ,

0

E[N (0, ε]] 1 = 0 . ε E [T1 ]

h?2 BMi2`+?M;2 Bb DmbiB}2/ #v /QKBMi2/ +QMp2`;2M+2 BM pB2r Q7 i?2 #QmM/  1 T1 N ((t, t + ε])dt ≤ 1 + N ((T1 , T1 + ε]) ε 0 M/ Q7 bbmKTiBQM UBBBVX aBM+2 E[Nε(0,ε]] = λ- r2 Q#iBM λ = E01[T1 ] M/ i?2`27Q`2- BM pB2r Q7 UBV- 0 < λ < ∞X h?2`27Q`2- r2 Kv /2}M2 i?2 SHK T`Q##BHBiv PN0 bbQ+Bi2/ rBi? (N, θt , P )X q2 b?HH +QM+Hm/2 #v T`QpBM; i?i P 0 = PN0 X "v i?2 BMp2`bBQM 7Q`KmH- 7Q` HH A ∈ F

 T1  T1 0 0 (1θ−1 A ◦ θt )dt = λEN 1A dt = λE0N [T1 1A ] , () P (θT0 ∈ A) = λEN 0

T0

0

bBM+2 QM Ω - θT0 ◦ θt Bb i?2 B/2MiBiv 7Q` HH t ∈ (0, T1 ]X 6`QK i?2 /2}MBiBQM Q7 P BM i2`Kb Q7 P 0 , 1 P (θT0 ∈ A) = 0 E0 [T1 1A ]. E [T1 ] 0

h?2`27Q`2- 7Q` HH A ∈ F - E0 [T1 1A ] = E0N [T1 1A ]X 6`QK i?Bb- r2 Q#iBM i?i 7Q` HH MQM@M2;iBp2 `M/QK p`B#H2b f - E0 [T1 f ] = E0N [T1 f ]X aBM+2 T1 > 0 #v +QMbi`m+iBQMr2 +M iF2 f = T1−1 1A iQ Q#iBM P 0 (A) = PN0 (A)X  _2K`F dX9Xk q2 b?HH bBM;H2 Qmi 2[mHBiv UV #2+mb2 Bi b?Qrb v2i MQi?2` +QM@ M2+iBQM #2ir22M i?2 biiBQM`v T`Q##BHBiv M/ i?2 SHK T`Q##BHBiv, 7Q` HH A ∈ F P (θT−1 A) = λE0N [T1 1A ] 0

(A ∈ F) .

1tKTH2 dX9Xj, aiiBQM`BxiBQM Q7  _2M2rH S`Q+2bbX h?2 `2bmHi #2@ HQr- r?B+? ?b H`2/v #22M T`Qp2/ BM i?2 +?Ti2` QM `2M2rH i?2Q`v- rBHH #2 `2pBb@ Bi2/ BM i?2 HB;?i Q7 SHK i?2Q`vX amTTQb2 i?i mM/2` PN0 - i?2 BMi2`@2p2Mi b2[m2M+2 {Sn }n∈Z /2}M2/ #v Sn = Tn+1 − Tn Bb BB/ rBi? }MBi2 K2MX h?2M (N, PN0 ) Bb M mM/2Hv2/ `2M2rH T`Q+2bb M/ (N, P )  biiBQM`v `2M2rH T`Q+2bbX h?2 2tBbi2M+2 Q7 P Bb ;`Mi2/ #v i?2 BMp2`b2 +QMbi`m+iBQMX q2 MQr b?Qr i?i, UV h?2 /Bbi`B#miBQM Q7 i?2 b2[m2M+2 S ∗ = {Sn }n∈Z\{0} Bb i?2 bK2 mM/2` P M/ PN0 X U#V S0 M/ S ∗ `2 P @BM/2T2M/2MiX S`QQ7X UV Ai bm{+2b iQ b?Qr i?i E [g(S ∗ )] = E0N [g(S ∗ )]

dX9X 6_PJ SGJ S_P""AGAhu hP ahhAPL_u S_P""AGAhu

kdN

7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb g : (RZ , (B(R)⊗Z ) → (R, B(R))X "v i?2 BMp2`bBQM 7Q`KmH  T1

∗ 0 ∗ E [g(S )] = λEN g(S (θu ))du . 0

"mi B7 u Bb BM [0, T1 )- i?2M S ∗ (θu ) = S ∗ - bQ i?i 

T1

E [g(S ∗ )] = λE0N

g(S ∗ )du = λE0N [T1 g(S ∗ )]

0

= λE0N [T1 ] E0N [g(S ∗ )] = E0N [g(S ∗ )] , r?2`2 r2 ?p2 mb2/ i?2 BM/2T2M/2M+2 Q7 T1 = S0 M/ S ∗ mM/2` PN0 X U#V aBKBH` +QMbB/2`iBQMb ;Bp2 E [f (S0 )g(S ∗ )] = λE0N [f (S0 )T1 g(S ∗ )] = λE0N [f (S0 )T1 ] E0N [g(S ∗ )] = λE0N [f (S0 )T1 ] E [g(S ∗ )] = E [f (S0 )] E [g(S ∗ )] . 

_2K`F dX9X9 h?2 7Q`KmH E[f (S0 )] = E0N [λS0 f (S0 )] Bb i`m2 7Q`  ;2M2`H biiBQM`v TQBMi T`Q+2bbX Ai bii2b i?i QM i?2 σ@}2H/ ;2M2`i2/ #v S0 - P Bb #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ PN0 - rBi? i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 λS0 X AM T`iB+mH` P (−T0 + T1 ≤ x) = λyF0 (dy), [0,x]

r?2`2 F0 (x) = PN0 (T1 ≤ x)X 1tKTH2 dX9X8, aiiBQM`BxiBQM Q7 b2KB@J`FQp T`Q+2bb2bX h?2 T`2pB@ Qmb 2tKTH2 +M #2 ;2M2`HBx2/ iQ b2KB@J`FQp T`Q+2bb2b QM  /2MmK2`#H2 bii2 bT+2 E Ub22 a2+iBQM 9Xe 7Q`  /2}MBiBQMVX am+? biQ+?biB+ T`Q+2bb Bb +QMbi`m+i2/ b 7QHHQrbX G2i P = {pij }i,j∈E - #2  biQ+?biB+ Ki`Bt QM E- bbmK2/ B``2/m+B#H2 M/ TQbBiBp2 `2+m``2Mi rBi? biiBQM`v /Bbi`B#miBQM πX 6Q` 2+? i, j ∈ E- H2i Gij #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 bQK2 bi`B+iHv TQbBiBp2 M/ T`QT2` `M/QK p`B#H2 rBi? K2M  ∞  ∞ mij = tGij (dt) = (1 − Gij (t))dt < ∞ . 0

0

_2+HH i i?Bb bi;2 i?i B7 U Bb  `M/QK p`B#H2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1] , G−1 ij (U ) Bb  `M/QK p`B#H2 rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Gij U?2`2 G−1 ij Bb i?2 BMp2`b2 Q7 Gij VX

k3y

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

G2i {Xn }n∈Z - #2  biiBQM`v J`FQp +?BM rBi? i`MbBiBQM Ki`Bt P- /2}M2/ QM bQK2 T`Q##BHBiv bT+2 rBi?  T`Q##BHBiv P 0 - M/ H2i {Un }n∈Z - #2  b2[m2M+2 Q7 BB/ `M/QK p`B#H2b- /2}M2/ QM i?2 bK2 bT+2 M/ mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X bbmK2 KQ`2Qp2` i?i i?2 b2[m2M+2b {Un }n∈Z M/ {Xn }n∈Z `2 BM/2T2M/2Mi mM/2` P 0 X .2}M2 Sn = G−1 Xn Xn+1 (Un ). AM T`iB+mH`- +QM/BiBQMHHv QM Xn = i M/ Xn+1 = j- Sn ?b i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Gij X JQ`2Qp2`- +QM/BiBQMHHv QM i?2 r?QH2 b2[m2M+2 {Xn }n∈Z i?2 b2[m2M+2 {Sn }n∈Z 7Q`Kb M BM/2T2M/2Mi 7KBHv Q7 `M/QK p`B#H2bX q2 +M MQr /2}M2  TQBMi T`Q+2bb N #v T0 = 0 , Tn+1 − Tn = Sn

(n ∈ Z)

M/ i?2 b2KB@J`FQp T`Q+2bb {X(t)}t∈R #v X(t) = Xn , if Tn ≤ t < Tn+1

(t ∈ R) .

LQi2 i?2 #b2M+2 Q7 2tTHQbBQM Ui?i Bb- limn→∞ Tn = ∞ M/ limn→−∞ Tn = −∞HKQbi bm`2HvV bBM+2 {Xn }n∈Z Bb  `2+m``2Mi ?K+X Ai rBHH #2 bbmK2/ i?i   E0 [T1 ] = π(i) pij mij < ∞ i∈E

M/ i?i

j∈E

E0 [N (0, t]] < ∞,

(t ∈ R+ ).

G2i (Ω, F) #2 i?2 +MQMB+H bT+2 Q7 +QMiBMmQmb 7mM+iBQMb rBi? HBKBib QM i?2 H27i iFBM; i?2B` pHm2b BM EX G2i {θt }t∈R #2 i?2 mbmH i`MbHiBQM ~Qr +iBM; QM 7mM+@ iBQMb- M/ H2i P 0 = P 0 ◦ X −1 X PM i?Bb +MQMB+H 7`K2rQ`F- N Bb  θt @+QKTiB#H2 K`F2/ TQBMi T`Q+2bb- i?2 K`F Q7 Tn #2BM; (Xn , Un ) rBi? Xn = X(Tn )X G2i Sn = GXn ,Xn+1 (Un )X lM/2` i?2 #Qp2 bbmKTiBQMb- HH i?2 +QM/BiBQMb Q7 i?2 BM@ p2`b2 +QMbi`m+iBQM `2 biBb}2/- bQ i?i i?2`2 2tBbib  θt @BMp`BMi T`Q##BHBiv P QM (Ω, F) 7Q` r?B+? N Bb  biiBQM`v TQBMi T`Q+2bb M/ bm+? i?i PN0 = P 0 X h?2 T`Q##BHBbiB+ bi`m+im`2 Q7 {X(t)}t∈R - mM/2` i?2 biiBQM`v T`Q##BHBiv P Bb i?2 7QHHQrBM;, UV *QM/BiBQMHHv QM X1 = j- i?2 b2[m2M+2 S1 , X2 , S2 , X3 , S3 , . . . ?b i?2 bK2 /Bbi`B#miBQM mM/2` P Q` PN0 X U#V *QM/BiBQMHHv QM X0 = i- i?2 b2[m2M+2 X−1 , S−1 , X−2 , S−2 , . . . ?b i?2 bK2 /Bbi`B#miBQM mM/2` P M/ PN0 X U+V *QM/BiBQMHHv QM X0 = i, X1 = j, −T0 > x, T1 > y- i?2 b2[m2M+2b BM UV M/ U#V `2 BM/2T2M/2MiX U/V JQ`2Qp2`  P (X0 = i , X1 = j , −T0 > x , T1 > y)



= λπ(i)pij

(1 − Gij (t)) dt.

x+y

UdXkjV

dX9X 6_PJ SGJ S_P""AGAhu hP ahhAPL_u S_P""AGAhu

k3R

S`QQ7X h?2 T`QQ7 Q7 UVĜU+V Bb bBKBH` iQ i?i BM 1tKTH2 dX9Xj 7Q` `2M2rH T`Q@ +2bb2bX 6Q` BMbiM+2- 7Q` n ≥ 1  T1

0 1{Xn0 θs =i , Xn+10 θs =j} ds P (Xn = i, Xn+1 = j) = λEN 0

 T1 0 = λEN 1{Xn =i , Xn+1 =j} ds , 0

bBM+2 s ∈ (0, T1 ) BKTHB2b Xn0 θs = Xn 7Q` HH n ∈ ZX h?2`27Q`2 P (Xn = i , Xn+1 = j) = λE0N T1 1{Xn =i} 1{Xn+1 =j} = λE0N G−1 X0 X1 (U0 )1{Xn =i} 1{Xn+1 =j} = λE0N G−1 (U0 )1{X =i} pij X0 X1 n 0 = λEN T1 1{Xn =i} pij , r?2`2 r2 ?p2 mb2/ i?2 ?vTQi?2bBb n ≥ 1- ;m`Mi22BM; i?i Xn+1 Bb +QM/BiBQMHHv BM/2T2M/2Mi Q7 X0 , X1 , U0 ;Bp2M Xn = iX aBKBH`Hv UbmKKBM; mT i?2 Hbi 2[mHBiv BM jV P (Xn = i) = λE0N T1 1{Xn =i} M/ i?2`27Q`2- 7Q` n ≥ 1P (Xn+1 = j|Xn = i) = pij . JQ`2 ;2M2`HHv- Bi +M #2 b?QrM rBi? i?2 bK2 ivT2 Q7 +H+mHiBQMb i?i {Xn }n∈Z Bb mM/2` P  J`FQp +?BM rBi? i`MbBiBQM Ki`Bt PX HbQ- ;BM rBi? i?2 bK2 T`QQ7- mM/2` P - U1 , U2 , . . . `2 BB/ `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]M/ BM/2T2M/2Mi Q7 X1 , X2 , . . .- M/ i?Bb T`Qp2b UVX b 7Q` U#V- Bi bm{+2b iQ `2p2`b2 iBK2- M/ iQ Q#b2`p2 i?i mM/2` PN0 - i?2 b2@ [m2M+2 {X−n }n∈Z Bb HbQ  biiBQM`v M/ 2`;Q/B+ J`FQp +?BM- i?Bb iBK2 rBi? i?2 i`MbBiBQM Ki`Bt Q = {qij }i,j∈E ;Bp2M #v qij = pji

π(j) . π(i)

h?2 T`QQ7 Q7 U+V Bb bBKBH`X Ai `2KBMb iQ ;Bp2 i?2 DQBMi Hr Q7 X0 , X1 , T0 , T1 mM/2` P X "v i?2 BMp2`bBQM 7Q`KmH- i?2 H27i@?M/ bB/2 Q7 UdXkjV 2[mHb

 T1 1{X0 ◦θs =i , X1 ◦θs =j} 1{−T0 ◦θs >x , T1 ◦θs >y} ds . λE0N 0

"mi B7 s ∈ (0, T1 ) , X0 ◦ θs = X0 , X1 ◦ θs = X1 , −T0 ◦ θs = s , T1 ◦ θs = T1 − sX h?2`27Q`2 i?2 Hbi [mMiBiv 2[mHb

 T1 0 λEN 1{X0 =i , X1 =j} 1{s>x} 1{T1 −s>y} ds 0 ∞

= λπ(i)pij E0N E0N 1{T1 >y+s} ds | X0 = i, X1 = j x  ∞ (1 − Gij (t)) dt . = λπ(i)pij x+y

k3k

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

_2+HHBM; mij =

∞ 0

(1 − Gij (t))dt M/ b2iiBM; x = y = 0 BM UdXkjV ;Bp2b,

P (X0 = i , X1 = j) =

π(i)p m

ij ij k∈E l∈E π(k)pkl mkl

M/ i?2`27Q`2 P (−T0 > x , T1 > y | X0 = i , X1 = j) =

1 mij

 (1 − Gij (t))dt. x+y



1tKTH2 dX9Xe, h?2 +Qp`BM+2 Q7  rbb biQ+?biB+ T`Q+2bb 7mM+@ iBQM Q7  b2KB@J`FQp T`Q+2bbX *QMbB/2` i?2 b2KB@J`FQp biiBQM`v T`Q@ +2bb {X(t)}t∈R /2b+`B#2/ #Qp2X G2i f : E → R #2 bQK2 7mM+iBQM bm+? i?i i?2 T`Q+2bb {Y (t)}t∈R - r?2`2 Y (t) := f (X(t))- Bb Q7 i?2 b2+QM/ Q`/2` UM/ i?2`27Q`2 rbbV rBi? +Qp`BM+2 7mM+iBQM CY X q2 ?p2- 7Q` τ ≥ 0, CY (τ ) = E [f (X(0))f (X(τ ))] = E f (X(0))2 1{τ 1) = o(t) M/ U.Q#`mb?BMǶb 2biBKi2V P (N ((0, t]) > 0) = λt + o(t). S`QQ7X G2i PN0 #2 i?2 bbQ+Bi2/ SHK T`Q##BHBivX h?2 BMp2`bBQM 7Q`KmH UdXReVrBi? n = −1- ;Bp2b  ∞ P (N ((0, t]) > 1) = P (T2 ≤ t) = λ PN0 (u < −T−1 , T2 ◦ θ−u ≤ t) du. 0

"mi QM Ω0 - u < −T−1 BKTHB2b T2 ◦ θ−u = T1 + uX h?2`27Q`2  t PN0 (u < −T−1 , T1 ≤ t − u) du P (N ((0, t]) > 1) = λ 0  t ≤λ PN0 (T1 ≤ t) du = λtPN0 (T1 ≤ t). 0

aBM+2

PN0 (T1

> 0) = 1-

limt↓0 PN0 (T1

≤ t) = 0X

6Q` .Q#`mb?BMǶb 2biBKi2- r2 mb2 ;BM i?2 BMp2`bBQM 7Q`KmH UdXReV- i?Bb iBK2 rBi? n = −1X aBM+2 T1 ◦ θ−u = u QM Ω0 ∩ {0 < u ≤ −T−1 } ∞ PN0 (u < −T−1 , T1 ◦ θ−u ≤ t) du P (N ((0, t]) > 0) = P (T1 ≤ t) = λ 0  t = λ PN0 (u < −T−1 ) du 0  t = λt − λ PN0 (−T−1 ≤ u) du = λt + o(t) . 0

 q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K dX8XRX S`QQ7X q2 ?p2 iQ b?Qr i?i P (θT1 ∈ A | T1 ≤ t) =

P (T1 ≤ t, θT1 ∈ A) λt P (T1 ≤ t) λt

i2M/b iQ PN0 (A) b t ↓ 0X hFBM; BMiQ ++QmMi .Q#`mb?BMǶb 2biBKi2lim t→0

λt = 1, P (T1 ≤ t)

Bi bm{+2b iQ b?Qr i?i- mMB7Q`KHv BM A& 1 && P (T1 ≤ t, θT1 ∈ A) − PN0 (A)& → 0 . λt

dX8X GP*G ALh1_S_1hhAPL P6 SGJ S_P""AGAhu

k38

h?2 BMp2`bBQM 7Q`KmH UdXReV UrBi? n = −1V ;Bp2b  ∞ P (T1 ≤ t, θT1 ∈ A) = λ PN0 (u < −T−1 , T1 ◦ θ−u ≤ t, θT1 ◦ θ−u ∈ A) du 0  t = λ PN0 (u < −T−1 , A) du, 0

bBM+2 θT1 ◦ θ−u Bb i?2 B/2MiBiv M/ T1 ◦ θ−u = u QM Ω0 ∩ {0 < u < −T−1 }X h?2`27Q`2`2r`BiBM;  t λtPN0 (A) = λ PN0 (A) du , 0

r2 Q#iBM  & 1 && 1 t 0 P (T1 ≤ t, θT1 ∈ A) − PN0 (A)& = P (u ≥ −T−1 , A) du λt t 0 N  1 t 0 ≤ P (u ≥ −T−1 ) du t 0 N  t 1 ≤ P 0 (t ≥ −T−1 ) du = PN0 (t ≥ −T−1 ) . t 0 N "mi bBM+2 PN0 (−T−1 > 0)- PN0 (t ≥ −T−1 ) → 0 b t → 0X



1tKTH2 dX8Xj,  ;2M2`H bBimiBQMX G2i {X(t)}t∈R #2  θt @+QKTiB#H2 +Q`HQH biQ+?biB+ T`Q+2bb rBi? pHm2b BM Rm M/ H2i f : Rm → R #2  #QmM/2/ +QMiBMmQmb 7mM+iBQMX hFBM; Z := f (X(0−)) BM UdXk8V- r2 ?p2 i?i lim E [f (X(T1 −)) | N (0, h] ≥ 1] = E0N [f (X(0−)] .

h→0

.2}MBM; T+ (t) := t + inf {h > 0; N (t, t + h] = 1} , r2 ?p2 2[mBpH2MiHv lim E [f (X(T+ (t)−) | N (t, t + h] ≥ 1] = E0N [f (X(0−)] .

h→0

aBM+2 i?2 biQ+?biB+ T`Q+2bb {Z(t)}t∈R Bb +Q`HQHlim E [f (X(t) | N (t, t + h] ≥ 1] = E0N [f (X(0−)] .

h→0

_2K`F dX8X9  ivTB+H 2tKTH2 Q7 TTHB+iBQM +M #2 7QmM/ BM [m2m2BM; i?2Q`vr?2M QM2 +QKTmi2b i?2 Hr Q7 i?2 MmK#2` Q7 +mbiQK2`b BM  biiBQM`v bvbi2K ;Bp2M i?i bQK2 2p2Mi U/2T`im`2 Q` ``BpHV Q++m``2/X a22 a2+iBQM RRX9X

k3e

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

dXe h?2 *`Qbb@2`;Q/B+ h?2Q`2K h?Bb b2+iBQM ;Bp2b MQi?2` BMi2`T`2iiBQM Q7 SHK T`Q##BHBiv BM i?2 2`;Q/B+ +b2X q2 }`bi `2+HH  72r 7+ib M/ /2}MBiBQMb Q7 2`;Q/B+ i?2Q`vX 1`;Q/B+Biv M/ JBtBM; h?Bb bm#b2+iBQM +QMiBMb  `2pB2r Q7 i?2 #bB+ `2bmHib Q7 2`;Q/B+ i?2Q`v mb2/ BM i?Bb b2+iBQMXN q2 bi`i rBi? /Bb+`2i2@iBK2 ~QrbX G2i (P 0 , θ) #2  biiBQM`v 7`K2rQ`F QM (Ω, F)X .2}MBiBQM dXeXR M 2p2Mi A ∈ F Bb bB/ iQ #2 bi`B+iHv BMp`BMi B7 θ−1 A = A M/ BMp`BMi B7 P 0 (A # θ−1 A) = 0- r?2`2 # /2MQi2b i?2 bvKK2i`B+H /Bz2`2M+2X .2}MBiBQM dXeXk h?2 /Bb+`2i2 ~Qr {θn }n∈Z Bb +HH2/ P 0 @2`;Q/B+ B7 HH θ@BMp`BMi 2p2Mib `2 i`BpBH UQ7 T`Q##BHBiv 2Bi?2` y Q` RVX q2 b?HH bv, (P 0 , θ) Bb 2`;Q/B+X P#b2`p2 i?i 7Q` HH θ@BMp`BMi 2p2Mib A- i?2 2p2Mi B = lim supn θ−n A Bb bi`B+iHv θ@BMp`BMi M/ bm+? i?i P 0 (A) = P 0 (B)X h?2`27Q`2- 7Q` HH BMp`BMi 2p2Mib- i?2`2 2tBbib  bi`B+iHv BMp`BMi 2p2Mi rBi? i?2 bK2 T`Q##BHBivX AM T`iB+@ mH`- i?2 ~Qr Bb 2`;Q/B+ B7 M/ QMHv B7 HH bi`B+iHv θ@BMp`BMi 2p2Mib `2 Q7 T`Q##BHBiv 2Bi?2` y Q` RX h?2 +QMiBMmQmb T`K2i2` b2i Bb- 7`QK MQr QM BM i?Bb b2+iBQM- RX q2 `2T2i BM i?Bb b2iiBM; i?2 /2}MBiBQMb Dmbi ;Bp2M 7Q` /Bb+`2i2@iBK2 ~QrbX G2i (P, θt ) #2  biiBQM`v 7`K2rQ`F QM (Ω, F)- {θt }t∈R Bb  UK2bm`#H2V ~Qr QM (Ω, F)X .2}MBiBQM dXeXj M 2p2Mi A ∈ F Bb bB/ iQ #2 bi`B+iHv θt @BMp`BMi B7 A = θt−1 A 7Q` HH t ∈ R- θt @BMp`BMi B7 P (A # θt−1 A) = 0 7Q` HH t ∈ RX .2}MBiBQM dXeX9 h?2 ~Qr {θt }t∈R Bb +HH2/ P @2`;Q/B+ B7 HH θt @BMp`BMi 2p2Mib `2 Q7 T`Q##BHBiv 2Bi?2` y Q` RX q2 b?HH bv, (P, θt ) Bb 2`;Q/B+X .2}MBiBQM dXeX8 h?2 /Bb+`2i2 ~Qr {θn }n∈Z Bb +HH2/ P 0 @KBtBM; B7 7Q` HH 2p2Mib A, B ∈ Flim P 0 (A ∩ θ−n B) = P 0 (A)P 0 (B). UdXkeV n

q2 b?HH bv, (P 0 , θ) Bb KBtBM;X LQi2 i?i UdXkeV Bb 2[mBpH2Mi iQ lim P 0 (θ−n B|A) = P 0 (B). n

h?mb- KBtBM; Bb  T`QT2`iv Q7 dz7Q`;2i7mHM2bb Q7 i?2 BMBiBH +QM/BiBQMbǴX N

a22 ("BHHBM;bH2v- RNe8) 7Q` i?2 T`QQ7bX

dXeX h>1 *_Paa@1_:P.A* h>1P_1J

k3d

.2}MBiBQM dXeXe h?2 +QMiBMmQmb ~Qr {θt }t∈R Bb +HH2/ P @KBtBM; B7 7Q` HH 2p2Mib A M/ Blim P (A ∩ θ−t B) = P (A)P 0 (B). t↑∞

q2 b?HH bv, (P, θt ) Bb KBtBM;X A7 A Bb bi`B+iHv BMp`BMi 7Q` i?2 KBtBM; ~Qr θ- i?2M P 0 (A) = P 0 (A)2 - bQ i?i P (A) Bb y Q` RX aBKBH`Hv 7Q`  KBtBM; +QMiBMmQmb ~QrX h?2`27Q`2, 0

h?2Q`2K dXeXd  KBtBM; ~Qr Bb M 2`;Q/B+ ~QrX JBtBM; ~Qrb `2 2`;Q/B+- M/ Bi Bb 2bv iQ +?2+F i?i QM i?2 T`Q/m+i Q7 irQ T`Q##BHBiv bT+2b- 2+? 2M/Qr2/ rBi?  KBtBM; b?B7i- i?2 T`Q/m+i b?B7i Bb KBtBM; M/ ?2M+2 2`;Q/B+X h?2Q`2K dXeX3 (P 0 , θ) Bb 2`;Q/B+ B7 M/ QMHv B7 i?2`2 2tBbib MQ /2+QKTQbBiBQM P 0 = α1 P10 + α2 P20 ,

α1 + α2 = 1, α1 > 0, α2 > 0,

UdXkdV

r?2`2 P10 M/ P20 `2 /BbiBM+i θ@BMp`BMi T`Q##BHBiB2bX >2`2 Bb i?2 +QMiBMmQmb@iBK2 p2`bBQM Q7 h?2Q`2K dXeX3, h?2Q`2K dXeXN (P, θt ) Bb 2`;Q/B+ B7 M/ QMHv B7 i?2`2 2tBbib MQ /2+QKTQbBiBQM P = β1 P1 + β2 P2 ,

β1 + β2 = 1, β1 > 0,

β2 > 0,

UdXk3V

r?2`2 P1 M/ P2 `2 /BbiBM+i θt @BMp`BMi T`Q##BHBiB2b 7Q` HH tX aBKmHiM2Qmb 1`;Q/B+Biv Q7 i?2 SHK M/ aiiBQM`v o2`bBQMb G2i (N, θt , P ) #2  bBKTH2 HQ+HHv }MBi2 biiBQM`v TQBMi T`Q+2bb QM R rBi? }MBi2 TQbBiBp2 BMi2MbBiv λX G2i PN0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? (N, θt , P )X G2i θ := θT1 X h?2Q`2K dXeXRy UV G2i A ∈ F #2 M 2p2Mi BMp`BMi rBi? `2bT2+i iQ i?2 ~Qr {θt }t∈R X h?2M P (A) = 1 B7 M/ QMHv B7 PN0 (A) = 1. U#V G2i A ∈ F #2  θ@BMp`BMi 2p2MiX h?2M PN0 (A) = 1 B7 M/ QMHv B7 P (A) = 1X S`QQ7X UV amTTQb2 i?i PN0 (A) = 1X "v i?2 BMp2`bBQM 7Q`KmH ∞ P (A) = λ PN0 (u < T1 , θ−u A) du 0  ∞ = λ PN0 (u < T1 , A) du (θt @BMp`BM+2 Q7 A) 0 ∞ = λ PN0 (u < T1 ) du (PN0 (A) = 1) = λE00 [T1 ] = 1 . 0

k33

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

*QMp2`b2Hv- bmTTQb2 i?i P (A) = 1X q2 ?p2- ;BM #v i?2 BMp2`bBQM 7Q`KmH ∞ PN0 (u < T1 , A) du , 1 = P (A) = λ 0

∞

M/ i?2`27Q`2- bBM+2 1 = λE0N [T1 ] = λ 0 PN0 (u < T1 ) du ∞ PN0 (u < T1 , A) du = λE0N [T1 1A ] . 0=λ 0

"mi T1 < ∞ XbX bBM+2 E0N [T1 ] = λ−1 < ∞X h?2`27Q`2 E0N [T1 1A ] = 0 BKTHB2b PN0 (A) = 0X U#V A7 P (A) = 1- i?2M 7`QK i?2 /2}MBiBQM Q7 SHK T`Q##BHBiv   1 0 PN (A) = E 1θ−n A 1{0 0) = P20 (T1 > 0) = 1 bBM+2 PN0 (T1 > 0) = 1X AM T`iB+mH`- E01 [T1 ] M/ E02 [T1 ] `2 TQbBiBp2X h?2 BMp2`bBQM 7Q`KmH ;Bp2b  T1 

 1 E01 [T1 ] 0 P (A) = α1 0 E 1A ◦ θt dt EN [T1 ] E01 [T1 ] 1 0  T1 

 1 E0 [T1 ] 0 + α2 02 E 1 ◦ θ dt , A t EN [T1 ] E02 [T1 ] 2 0

dXdX SGJ S_P""AGAhu L. ahP*>ahA* ALh1LaAhu

k3N

i?i Bb P = β1 P1 + β2 P2 , E0 [T ] α1 E01 [T11 ] N

E0 [T ]

M/ β2 = α2 E02 [T11 ] ∈ (0, 1) `2 bm+? i?i β1 +β2 = 1X JQ`2Qp2`N   T P1 M/ P2 - /2}M2/ #v Pi (A) = E01[T1 ] E0i 0 1 1A ◦ θt dt Ui = 1, 2V `2 θt @BMp`BMi i 7Q` HH t ∈ RX h?2`27Q`2- #v h?2Q`2K dXeXN- P +MMQi #2 2`;Q/B+-  +QMi`/B+iBQMX r?2`2 β1 =

h?2 T`QQ7 Q7 i?2 +QMp2`b2 T`i 7QHHQrb i?2 bK2 HBM2 Q7 `;mK2Mi U1t2`+Bb2 dX3XRRVX  h?2Q`2K dXeXRk G2i (N, θt , P ) #2  biiBQM`v TQBMi T`Q+2bb rBi? }MBi2 TQbBiBp2 BMi2MbBiv λ- M/ bmTTQb2 i?i (P, θt ) Bb 2`;Q/B+X G2i PN0 #2 i?2 bbQ+Bi2/ SHK T`Q##BHBivX G2i f #2 BM L1 (P )X h?2M 1 T ↑∞ T



T

f ◦ θt dt = E[f ],

PN0 @XbX

UdXkNV

1 f ◦ θTk = E0N [f ], n↑∞ n k=1

P @XbX

UdXjyV

lim

M/

0

n

lim

S`QQ7X aBM+2 (P, θt ) Bb 2`;Q/B+- i?2 T`Q##BHBiv Q7 2p2Mi , A :=

1 T →∞ T



T

f ◦ θt dt = E[f ]

lim

-

0

Bb 1X JQ`2Qp2` A Bb θt @BMp`BMiX h?2`27Q`2 #v G2KK dXeXRy- PN0 (A) = 1X h?Bb T`Qp2b UdXkNVX aBKBH`Hv- bBM+2 (PN0 , θ) Bb 2`;Q/B+- 7Q` HH f ∈ L1 (PN0 )- i?2 2p2Mi , B :=

1 f ◦ θTk = E0N [f ] n↑∞ n k=1 n

lim

?b T`Q##BHBiv 1X Ai Bb KQ`2Qp2` θ@BMp`BMi- M/ i?2`27Q`2 P (B) = 1 #v G2KK dXeXRyX h?Bb T`Qp2b UdXjyVX  _2K`F dXeXRj h?Bb `2bmHi Bb +HH2/ i?2 +`Qbb@2`;Q/B+ i?2Q`2K 7Q` Q#pBQmb `2@ bQMbX Ai Bb 2bb2MiBH 7Q` [m2m2BM; i?2Q`v BM i?i Bi T`QpB/2b M 2bv BMi2`T`2iiBQM Q7 i?2 +`Qbb@7Q`KmHb- i?i Bb- 7Q`KmHb HBMFBM; PN0 @K2Mb M/ P @K2Mb Q7 QT2`iBQMH [mMiBiB2bX

dXd

SHK S`Q##BHBiv M/ aiQ+?biB+ AMi2MbBiv

_Qm;?Hv bT2FBM;- SHK T`Q##BHBiv /2b+`B#2b r?i ?TT2Mb dzr?2M i?2`2 Bb  TQBMi i iBK2 tǴX aiQ+?biB+ BMi2MbBiv ;Bp2b i?2 2tT2+iiBQM Q7 dzb22BM;  TQBMi i iBK2 tǴ FMQrBM; i?2 Tbi ?BbiQ`v Q7 i?2 TQBMi T`Q+2bbX Ai Bb Mim`H iQ 2tT2+i i?2b2 TQBMib Q7 pB2r iQ #2 +QMM2+i2/X

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.2}MBiBQM dXdXR G2i {θt }t∈R #2  K2bm`#H2 ~Qr QM (Ω, F)X  ?BbiQ`v {Ft }t∈R Bb bB/ iQ #2 θt @+QKTiB#H2 B7 θt Fs = Fs−t 7Q` HH s, t ∈ RX aiQ+?biB+ BMi2MbBiv T`QpB/2b  /2iBH2/ /2b+`BTiBQM Q7 i?2 /vMKB+b Q7  TQBMi T`Q+2bb- M/ r?2M QM2 FMQrb i?2 mM/2`HvBM; T?vbB+b Q7 i?2 ;2M2`iBQM Q7 2p2MibQM2 rBHH BM KQbi +b2b #2 #H2 iQ T`QpB/2 i?2 biQ+?biB+ BMi2MbBiv rBi? `2bT2+i iQ  ?BbiQ`v i?i bmKK`Bx2b i?2 BM7Q`KiBQM pBH#H2 i Mv BMbiMiX Ai Bb i?2`27Q`2 Q7 i?2Q`2iB+H BKTQ`iM+2 iQ FMQr r?2i?2` Q` MQi i?2 biQ+?biB+ BMi2MbBiv Bb i?2 bK2 mM/2` i?2 biiBQM`v T`Q##BHBiv M/ i?2 SHK T`Q##BHBivX Ai im`Mb Qmi i?irBi? bQK2 T`QpBbBQMb iQ #2 bii2/ bQQM- i?Bb Bb i?2 +b2c i?2`27Q`2 i?2 biiBQM`v T`Q##BHBiv M/ i?2 SHK T`Q##BHBiv /2b+`B#2 i?2 bK2 /vMKB+bX JQ`2 T`2+Bb2Hv, h?2Q`2K dXdXk G2i {Ft }t∈R #2  ?BbiQ`v Q7  HQ+HHv }MBi2 bBKTH2 TQBMi T`Q+2bb N - #Qi? θt @+QKTiB#H2X G2i P #2  θt @BMp`BMi T`Q##BHBivX amTTQb2 i?i N ?b  }MBi2 TQbBiBp2 BMi2MbBiv λ M/ H2i PN0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? (N, θt , P )X amTTQb2 i?i N /KBib  θt @+QKTiB#H2 (P, Ft )@BMi2MbBiv {λ(t)}t∈R X PM R+ - N /KBib i?2 (PN0 , Ft )@BMi2MbBiv {λ(t)}t∈R X _2K`F dXdXj LQi2 i?2 //2/ T`2+BbBQM BM i?2 i2`KBMQHQ;v, i?2 /2T2M/2M+2 Q7 biQ+?biB+ BMi2MbBiv QM i?2 mM/2`HvBM; T`Q##BHBiv ?b #22M 2KT?bBx2/X h?Bb /2T2M/2M+2 Bb +imHHv i?2 KBM +QM+2`M Q7 i?2 i?2Q`2KX S`QQ7XRy q2 ?p2 iQ T`Qp2 i?i 7Q` HH (a, b] ⊂ R+ , A ∈ Fa ,  b

λ(t)dt . E0N [1A N (a, b]] = E0N 1A a

"v /2}MBiBQM Q7 SHK T`Q##BHBiv- i?2 H27i@?M/ bB/2 Q7 i?2 #Qp2 2[mHBiv Bb    1 E 1(0,1] (Tn )(1A ◦ θTn )N (a + Tn , b + Tn ] . λ n∈Z P#b2`p2 i?i- B7 n ≥ 1, Tn Bb M Ft @biQTTBM; iBK2, BM/22/ {Tn ≤ t} = {N (0, t] ≥ n} ∈ Ft X U6Q` n ≤ 0, Tn Bb MQi M Ft @biQTTBM; iBK2XV HbQ- bBM+2 a Bb MQM@ M2;iBp2- 1(0,1] (Tn ) Bb Fa+Tn @K2bm`#H2X "2+mb2 {Ft }t∈R Bb θt @+QKTiB#H2 M/ A Bb BM Fa - i?2 `M/QK p`B#H2 1A ◦ θTn Bb Fa+Tn @K2bm`#H2X h?2 T`Q+2bb 1(0,1] (Tn )(1A ◦ θTn )1(a+Tn ,b+Tn ] (t) Bb i?2`27Q`2 Ft @/Ti2/X "2BM; BM //BiBQM H27i@ +QMiBMmQmb- Bi Bb Ft @T`2/B+i#H2X h?2`27Q`2- 7Q` n ≥ 1, a ≥ 0 E 1(0,1] (Tn )(1A ◦ θTn )N (a + Tn , b + Tn ] 

= E 1(0,1] (Tn )(1A ◦ θTn )1(a+Tn ,b+Tn ] (t)N (dt) R

= E 1(0,1] (Tn )(1A ◦ θTn )1(a+Tn ,b+Tn ] (t)λ(t)dt R

 b+Tn λ(t)dt . = E 1(0,1] (Tn )(1A ◦ θTn ) a+Tn Ry

("`ûKm/- RN3N)X

dXdX SGJ S_P""AGAhu L. ahP*>ahA* ALh1LaAhu

kNR

aBM+2 {λ(t)}t∈R Bb θt @+QKTiB#H2 b   b+Tn λ(t)dt = λ(t)dt ◦ θTn , a+Tn

a

M/ i?2`27Q`2 E 1(0,1] (Tn )(1A ◦ θTn )N (a + Tn , b + Tn ]

 b  λ(t)dt ◦ θTn . = E 1(0,1] (Tn )(1A ◦ θTn ) a

amKKBM; mT i?2 Hbi 2[mHBiv rBi? `2bT2+i iQ n = 1, 2, . . .- r2 Q#iBM i?2 MMQmM+2/ `2bmHiX  G2i {Ft }t∈R #2  θt @+QKTiB#H2 ?BbiQ`vX A7 BM //BiBQM Mv Ft @T`2/B+i#H2 T`Q+2bb {H(t)}t∈R ?b i?2 7Q`K H(t, ω) = v(t, θt (ω)), UdXjRV r?2`2 (t, ω) → v(t, ω) Bb B(R) ⊗ 8 F K2bm`#H2 M/ 7Q` HH t ∈ R, ω → v(t, ω) Bb F0− @K2bm`#H2 Ur?2`2 F0− = s aX N,Z @K2bm`#H2- i?2M S`QQ7 Q7 (β), Bi bm{+2b iQ b?Qr i?i B7 W : Ω → R Bb F0− N,Z (t, ω) → W (θt (ω)) Bb P(Ft )@K2bm`#H2X Ai Bb BM im`M 2MQm;? iQ b?Qr i?Bb 7Q` HH `M/QK p`B#H2b W Q7 i?2 7Q`K W = ϕ(N ([a, b) × L)), a ≤ b ≤ 0, L ∈ KX AM i?Bb +b2 {W ◦ θt } Bb  7mM+iBQM Q7  H27i@+QMiBMmQmb FtN,Z @/Ti2/ T`Q+2bb- M/ i?2`27Q`2 Bi Bb FtN,Z @T`2/B+i#H2X 

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G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb M/ H2i {Ft }t∈R #2  ?BbiQ`v Q7 N - #Qi? θt @+QKTiB#H2X amTTQb2 i?i i?2 BMi2MbBiv λ Q7 N Bb }MBi2 M/ TQbBiBp2M/ H2i PN0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? (N, P, θt )X amTTQb2 KQ`2Qp2` i?i {Ft }t∈R ?b  T`2/B+i#H2 bi`m+im`2 /Ti2/ iQ {θt }t∈R X h?2 #bB+ `2bmHi Bb STM;2HQmǶb i?2Q`2K, h?2Q`2K dXdX8 lM/2` i?2 #Qp2 +QM/BiBQMb- N /KBib  (P, Ft )@BMi2MbBiv {λ(t)}t∈R B7 M/ QMHv B7 PN0  P on F0− , UdXjjV M/ BM i?i +b2 {λ(t)}t∈R +M #2 +?Qb2M Q7 i?2 7Q`K λ(t) = (μ ◦ θt )λ ,

UdXj9V

r?2`2 μ Bb i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 QM F0− Q7 PN0 rBi? `2bT2+i iQ P , μ=

dPN0 | dP

F0−

.

UdXj8V

h?2 7QHHQrBM; Bb  +QKT+i 2tT`2bbBQM Q7 i?2 #Qp2 i?2Q`2K, E[f (Z(0))λ(0)] = λE0N [f (Z(0))] ,

UdXjeV

r?2`2 {Z(t)}t∈R Bb  θt @+QKTiB#H2 biQ+?biB+ T`Q+2bb bm+? i?i Z(0) Bb F0− @ K2bm`#H2 UM/ i?2`27Q`2 {Z(t)}t∈R Bb Ft @T`2/B+i#H2 /m2 iQ i?2 bbmKTiBQM i?i i?2 T`2/B+i#H2 bi`m+im`2 Q7 {Ft }t∈R Bb /Ti2/ iQ {θt }t∈R V- M/ f Bb  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM 7`QK i?2 bii2 bT+2 Q7 {Z(t)}t∈R BMiQ RX S`QQ7XRR amTTQb2 i?i UdXjjV ?QH/b- M/ /2}M2 μ #v UdXj8VX G2i MQr {H(t)}t∈R #2  MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bbX aBM+2 {Ft }t∈R ?b  T`2/B+i#H2 bi`m+@ im`2 /Ti2/ iQ {θt }t∈R , H(t, ω) = v(t, θt (ω)) 7Q` bQK2 7mM+iBQM v i?i Bb B ⊗ F @ K2bm`#H2 M/ bm+? i?i 7Q` 2+? t ∈ R, ω → v(t, ω) Bb F0− @K2bm`#H2X 6`QK i?2 Hii2` T`QT2`iv Q7 v M/ i?2 /2}MBiBQM Q7 μE0N [v(t)] = E[μv(t)], 

M/ i?2`27Q`2

R

 E0N [v(t)] dt =

aBM+2 P Bb θt @BMp`BMi 

 E[μv(t)] dt =

R

E[μv(t)] dt. R

R

E[(μ ◦ θt )(v(t) ◦ θt )] dt .

PM i?2 Qi?2` ?M/- #v i?2 J2+F2 7Q`KmH  



 1 1 E0N [v(t)] dt = E (v(t) ◦ θt )N (dt) = E H(t)N (dt) . λ λ R R R RR h?2 7QHHQrBM; T`QQ7 Q7 STM;2HQmǶb Q`B;BMH BM i2`Kb Q7 biQ+?biB+ BMi2MbBiv Bb iF2M 7`QK ("`ûKm/- RN3N)- r?Qb2 Tm`TQb2 rb iQ `2p2H Bib HBMF rBi? i?2 Tbi T`QT2`iv Q7 [m2m2BM; i?2Q`vX a22 a2+iBQM RRX9X

dXdX SGJ S_P""AGAhu L. ahP*>ahA* ALh1LaAhu

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*QK#BMBM; HH i?2 #Qp2- r2 Q#iBM 



E H(t)N (dt) = E (H(t)λμ ◦ θt ) dt , R

R

r?2`2 {H(t)}t∈R Bb Mv `#Bi``v MQM@M2;iBp2 Ft @T`2/B+i#H2 T`Q+2bbX h?2`27Q`2#v h?2Q`2K X9Xj- {λμ ◦ θt }t∈R Bb i?2 Ft @BMi2MbBiv Q7 N X "27Q`2 im`MBM; iQ i?2 T`QQ7 Q7 i?2 +QMp2`b2 T`i- r2 `2K`F i?i {λμ ◦ θt }t∈R Bb M Ft @T`2/B+i#H2 T`Q+2bb #2+mb2 μ Bb F0− @K2bm`#H2 M/ {Ft }t∈R Bb bbmK2/ iQ ?p2  T`2/B+i#H2 bi`m+im`2 /Ti2/ iQ {θt }t∈R X AM Q`/2` iQ T`Qp2 i?2 +QMp2`b2 T`i- r2 bi`i rBi? i?2 ?vTQi?2bBb i?i N /KBib i?2 Ft @BMi2MbBiv {λ(t)}t∈R X q2 FMQr i?i bm+? M BMi2MbBiv +M #2 bbmK2/ Ft @ T`2/B+i#H2 Uh?2Q`2K 8XRX9yV- i?i Bb- BM pB2r Q7 i?2 bbmKTiBQM QM i?2 T`2/B+i#H2 bi`m+im`2 Q7 {Ft }t∈R λ(t, ω) = λ μ(t, θt (ω)), r?2`2 μ  Bb B ⊗ F @K2bm`#H2 M/ 7Q` 2+? t ∈ R, ω → μ (t, ω) Bb F0− @K2bm`#H2X G2i MQr v #2  MQM@M2;iBp2 F0− @K2bm`#H2 `M/QK p`B#H2X "v i?2 bbmKTiBQM QM i?2 T`2/B+i#H2 bi`m+im`2 Q7 {Ft }t∈R - i?2 biQ+?biB+ T`Q+2bb {v ◦ θt }t∈R Bb Ft @ T`2/B+i#H2X h?2`27Q`2- #v h?2Q`2K X9Xj- 7Q` HH (a, b] ⊂ R 



E (v ◦ θt )(λ μ(t) ◦ θt ) dt = E (v ◦ θt )N (dt) . (a,b]

(a,b]

"v i?2  θt @BMp`BM+2 Q7 P - i?2 H27i@?M/ bB/2 Q7 i?Bb 0 2[mHBiv Bb 2[mH iQ λE[v (a,b] μ (t) dt]- r?2`2b i?2 `B;?i@?M/ bB/2 Bb λ(b − a)EN [v]X h?2`27Q`2- 7Q` HH MQM@M2;iBp2 `M/QK p`B#H2b v i?i `2 F0− @K2bm`#H2

 b 1 μ (t) dt . E0N [v] = E v b−a a h?Bb b?Qrb i?i PN0  P QM F0− UrBi? μ = /2`BpiBp2

0 dPN | VX dP F0−

1 b−a

b a

μ (t) dt b i?2 _/QMĜLBFQ/ɷK

q2 `2K`F i?i i?2 T`QQ7 BM i?2 }`bi T`i +M #2 mb2/ iQ b?Qr i?i {λμ ◦ θt }t∈R Bb  T`2/B+i#H2 Ft @BMi2MbBiv Q7 N X AM T`iB+mH`- i?Bb BMi2MbBiv Bb θt @+QKTiB#H2 Ur2 /B/ MQi bi`i rBi? i?Bb b  ?vTQi?2bBbVX  h?2 7QHHQrBM; Bb M BKK2/Bi2 +QMb2[m2M+2 Q7 h?2Q`2K dXdX8 M/ Q7 qiM#2Ƕb +?`+i2`BxiBQM Q7 SQBbbQM T`Q+2bb2bX *Q`QHH`v dXdXe lM/2` i?2 +QM/BiBQMb bii2/ BM i?2 T`;`T? T`2+2/BM; h?2Q`2K dXdX8- i?2 TQBMi T`Q+2bb N Bb  ?QKQ;2M2Qmb Ft @SQBbbQM T`Q+2bb B7 M/ QMHv P = PN0 QM F0− X aBM+2 i?2 ``Qr Q7 iBK2 +M #2 BMp2`i2/ rBi?Qmi +?M;BM; i?2 /Bbi`B#miBQM Q7 M ?TT-

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*Q`QHH`v dXdXd A7 N Bb M ?TT- i?2M i?2 PN0 @/Bbi`B#miBQM Q7 N − ε0 M/ i?2 P @/Bbi`B#miBQM Q7 N `2 i?2 bK2X S`QQ7X h?Bb 7QHHQrb 7`QK h?2Q`2K dXdXe rBi? Ft ≡ FtN Ut ∈ RV- M/ Q#b2`pBM; i?i F0− = σ{N (C) ; C ∈ B(R) , C ⊂ (−∞, 0)}X  AM Qi?2` rQ`/b- B7 N Bb mM/2` P M ?TT Q7 BMi2MbBiv λ- mM/2` i?2 SHK /Bbi`B#miBQMN Bb  SQBbbQM T`Q+2bb iQ r?B+? ?b #22M //2/  TQBMi i 0X M 2H2K2Mi`v 7Q`K Q7 i?Bb `2bmHi TT2`b BM 1t2`+Bb2 jX3XkX

dX3 1t2`+Bb2b 1t2`+Bb2 dX3XRX J2bm`#BHBiv Q7 i?2 b?B7i a?Qr i?i- 7Q` }t2/ x ∈ Rm - i?2 KTTBM;b θx Q7 1tKTH2 dXRXR `2 K2bm`#H2X 1t2`+Bb2 dX3XkX θt @+QKTiB#BHBiv G2i {θx }t∈R #2  K2bm`#H2 ~Qr M/ H2i N #2  bBKTH2 θt @+QKTiB#H2 TQBMi T`Q+2bb QM RX G2i {Tn }n∈Z #2 Bib b2[m2M+2 Q7 TQBMibX S`Qp2 i?2 7QHHQrBM;, {Zn }n∈Z Bb  θx @ +QKTiB#H2 T`Q+2bb Q7 K`Fb Q7 N B7 M/ QMHv B7 i?2`2 2tBbib  biQ+?biB+ T`Q+2bb {Z(t)}t∈R +QKTiB#H2 rBi? i?2 ~Qr M/ bm+? i?i Z(Tn ) = Zn 7Q` HH n ∈ NX 1t2`+Bb2 dX3XjX S`QQ7 Q7 h?2Q`2K dXRXkk S`Qp2 h?2Q`2K dXRXkkX 1t2`+Bb2 dX3X9X θT1 _272` iQ i?2 /Bb+mbbBQM #27Q`2 h?2Q`2K dXjXRX :Bp2  bBKTH2 2tKTH2 b?QrBM; i?i BM ;2M2`H- QM Ω- i?2 BMp2`b2 Q7 θT1 Bb MQi θT−1 X 1t2`+Bb2 dX3X8X 6B`bi TQbBiBp2 2p2Mi iBK2 Q7  bmT2`TQbBiBQM hF2 Rm = R BM i?2 1tKTH2 dXkX3X G2i Fi M/ Fi0 #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 }`bi TQBMi Q7 μi mM/2` Pi M/ Pi0 `2bT2+iBp2HvX S`Qp2 i?i- mM/2` i?2 SHK T`Q##BHBiv PN0 - i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM G0 Q7 i?2 }`bi TQBMi bi`B+iHv iQ i?2 `B;?i Q7 i?2 Q`B;BM Q7 i?2 bmT2`TQbBiBQM T`Q+2bb N Bb ⎛ ⎞ G0 (x) = 1 −

k   ⎜ λi ⎟ ⎜ (1 − Fi0 (x)) (1 − Fj (x))⎟ ⎝λ ⎠. i=1

1≤j≤k j=i

1t2`+Bb2 dX3XeX JQiBQM@BMp`BMi biiBQM`v TQBMi T`Q+2bb2bX G2i P #2  T`Q##BHBiv QM (Mp (Rm ), Mp (Rm )) i?i KF2b Q7 i?2 +QQ`/BMi2 T`Q+2bb N  KQiBQM@BMp`BMi biiBQM`v TQBMi T`Q+2bb Q7 }MBi2 MQM@MmHH BMi2MbBivX JQiBQM BMp`BM+2 K2Mb i?i 7Q` Mv /BbiM+2@T`2b2`pBM; i`Mb7Q`KiBQM τ Q7 Rm - M/ Mv Γ ∈ M(Rm )- P (τ N ∈ Γ) = P (N ∈ Γ)X a?Qr i?i i?2 SHK T`Q##BHBiv P0N Bb BMp`BMi mM/2` `QiiBQM Q7 +2Mi2` 0- i?i Bb 7Q` Mv bm+? `QiiBQM r M/ Mv Γ ∈ Mp (Rm )- P0N (rN ∈ Γ) = P0N (N ∈ Γ)X

dX3X 1s1_*Aa1a

kN8

1t2`+Bb2 dX3XdX P @bKTH2 7`QK  PN0 @bKTH2 S`QpB/2 i?2 /2iBHb BM 1tKTH2 dXjX3X 1t2`+Bb2 dX3X3X 62HH2`ǰb #mb T`/QtX lM/2` i?2 +QM/BiBQMb T`2pBHBM; BM 1tKTH2 dX9Xj rBi? i?2 //BiBQMH +QM/BiBQM i?i E0N [S0 2 ] < ∞- T`Qp2 i?i E[S0 ] =

E0N [S0 2 ] , E0N [S0 ]

r?2`2 S0 := T1 − T0 X .2/m+2 7`QK i?Bb i?i 7Q` S0 iQ ?p2 i?2 bK2 /Bbi`B#miBQM mM/2` P M/ mM/2` PN0 Bi Bb M2+2bb`v M/ bm{+B2Mi i?i i?2 BMi2``2M2rH iBK2b #2 /2i2`KBMBbiB+X 1t2`+Bb2 dX3XNX h?2 `2HB#BHBiv TQBMi T`Q+2bb *QMbB/2`  MQM@/2Hv2/ `2M2rH T`Q+2bb QM R UMQi Dmbi R+ V 7Q` r?B+? i?2 BMi2``2@ M2rH b2[m2M+2 {Sn }n∈Z Bb Q7 i?2 7Q`K Sn = Un + Vn - r?2`2 {Un }n∈Z M/ {Vn }n∈Z `2 BM/2T2M/2Mi BB/ b2[m2M+2b Q7 `M/QK p`B#H2b rBi? }MBi2 K2MX .2b+`B#2 i?2 iBK2@biiBQM`v p2`bBQM Q7 i?2 TQBMi T`Q+2bb r?Qb2 2p2Mi iBK2b `2 U0 , U0 + V0 , U0 + V0 + U1 , U0 + V0 + U1 + V1 , . . .

1t2`+Bb2 dX3XRyX  +QMiBMmQmb@iBK2 ?K+ M/ Bib 2K#2//2/ +?BM *QMbB/2`  +QMiBMmQmb@iBK2 ?K+ {X(t)}T ∈R rBi? bii2 bT+2 E i?i Bb TQbBiBp2 `2@ +m``2Mi M/ biiBQM`vX :Bp2 M 2tT`2bbBQM BM i2`Kb Q7 i?2 BM}MBi2bBKH ;2M2`iQ` Q M/ Q7 i?2 biiBQM`v /Bbi`B#miBQM π Q7 i?2 p2`;2 BMi2MbBiv Q7 i?2 biiBQM`v TQBMi T`Q+2bb +QmMiBM; i?2 i`MbBiBQM iBK2b Q7 i?2 +?BMX bbmKBM; i?Bb p2`;2 BMi2M@ bBiv }MBi2- ;Bp2 i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 /Bb+`2i2@iBK2 ?K+ {X(Tn −)}n∈N r?2`2 Tn Bb i?2 n@i? i`MbBiBQM iBK2 Q7 i?2 +?BMX 1t2`+Bb2 dX3XRRX 1`;Q/B+Biv :Bp2 i?2 /2iBHb Q7 i?2 T`QQ7 Q7 h?2Q`2K dXeXRR- i?i Bb- T`Qp2 i?i B7 (PN0 , θ) Bb 2`;Q/B+- i?2M (P, θt ) Bb 2`;Q/B+X 1t2`+Bb2 dX3XRkX 1`;Q/B+Biv M/ i?2 biiBQM`v `2M2rH T`Q+2bb Ab  biiBQM`v `2M2rH T`Q+2bb 2`;Q/B+\ CmbiB7v vQm` Mbr2`X 1t2`+Bb2 dX3XRjX JBtBM; M/ MQ KBtBM; *QMbB/2` i?2 TQBMi T`Q+2bb rBi? 2[mBbT+2/ TQBMib Ubv, Tn+1 − Tn = 1 7Q` HH n ∈ ZVX *H2`Hv- 7Q` i?2 SHK p2`bBQM- (PN0 , θT1 ) Bb KBtBM;X a?Qr i?i i?2 biiBQM`v p2`bBQM (P, θt ) Bb MQi KBtBM;X

kNe

*>Sh1_ dX SGJ S_P""AGAhu PL h>1 GAL1

1t2`+Bb2 dX3XR9X o`BiBQM /BbiM+2 #2ir22M i?2 biiBQM`v M/ mM@ /2Hv2/ `2M2rH T`Q+2bb2b a?Qr i?i r?2M i?2 bBKTH2 HQ+HHv }MBi2 N Bb mM/2` PN0 M mM/2Hv2/ `2M2rH TQBMi T`Q+2bb rBi? M BMi2`@`2M2rH +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F /KBiiBM;  /2MbBiv f rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2dV (P, PN0 ) = dV (F, G) , r?2`2 G Bb  +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM rBi? /2MbBiv g(x) = λ(1 − F (x)) Ui?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 `2bB/mH BMi2`@`2M2rH /Bbi`B#miBQMVX U>2`2 P M/ PN0 `2 `2bi`B+i2/ iQ R+ XV 1t2`+Bb2 dX3XR8X E0N



λ λ(0)

 =1

h?2 b2iiBM; Bb i?i Q7 h?2Q`2K dXdX8X a?Qr i?i E0N /B+i#H2 p2`bBQM Q7 i?2 biQ+?biB+ BMi2MbBivVX



λ λ(0)

 = 1 UmbBM; i?2 T`2@

1t2`+Bb2 dX3XReX .BbiM+2 BM p`BiBQM #2ir22M i?2 SHK M/ biiBQM@ `v p2`bBQMb AM i?2 bBimiBQM Q7 h?2Q`2K dXdX8- b?Qr i?i i?2 /BbiM+2 #2ir22M i?2   p`BiBQM `2bi`B+iBQMb Q7 P M/ PN0 iQ F0− Bb 2[mH iQ 12 E |λ(0)−λ| X S`Qp2 i?i Bi Bb 2[mH iQ λ   |λ(0)−λ| 1 0 UmbBM; i?2 T`2/B+i#H2 p2`bBQM Q7 i?2 biQ+?biB+ BMi2MbBivVX E 2 N λ(0) 1t2`+Bb2 dX3XRdX J2+F2ǰb +?`+i2`BxiBQM Q7  SQBbbQM T`Q+2bb G2i {θt }t∈R #2  K2bm`#H2 ~Qr QM RX G2i N #2  TQBMi T`Q+2bb QM R M/ H2i {Ft }t∈R #2  ?BbiQ`v Q7 N - #Qi? θt @+QKTiB#H2X amTTQb2 i?i N ?b  }MBi2 TQbBiBp2 p2`;2 BMi2MbBiv λ- M/ H2i PN0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? (N, θt , P )X S`Qp2 UV pB STM;2HQmǶb i?2Q`2K M/ U#V pB qiM#2Ƕb i?2Q`2K- i?i  M2+2b@ b`v M/ bm{+B2Mi +QM/BiBQM 7Q` N iQ #2  ?QKQ;2M2Qmb Ft @SQBbbQM T`Q+2bb Ui?i Bb- bm+? i?i- 7Q` HH (a, b] ⊂ R- N (a, b] Bb  SQBbbQM p`B#H2 rBi? K2M λ × (b − a) M/ Bb P @BM/2T2M/2Mi Q7 Fa V Bb i?i P ≡ PN0 QM F0− X 1t2`+Bb2 dX3XR3X a2H2+i2/ i`MbBiBQMb Q7  biiBQM`v ?K+ G2i (P, θt ) #2  biiBQM`v 7`K2rQ`F M/ H2i {X(t)}t∈R #2  θt @+QKTiB#H2 biQ+?b@ iB+ T`Q+2bb rBi? pHm2b BM i?2 /2MmK2`#H2 bT+2 E i?i Bb KQ`2Qp2` M 2`;Q/B+ +QMiBMmQmb@iBK2 ?K+ rBi? BM}MBi2bBKH ;2M2`iQ` Q = {qij }i,j∈E M/ biiBQM`v /Bbi`B#miBQM πX G2i N (C) #2 i?2 MmK#2` Q7 /Bb+QMiBMmBiB2b Q7 i?Bb +?BM BM C ⊆ EX h?Bb /2}M2b  biiBQM`v T`Q+2bb (N, θt , P )X G2i 7Q` H ∈ E × E − /B;(E × E)- NH #2 i?2 TQBMi T`Q+2bb +QmMiBM; i?2 H@ i`MbBiBQMb Q7 i?2 +?BM- i?i Bb,  NH (C) :=

1H (Xs− , Xs )N (ds) . C

UdXjdV

dX3X 1s1_*Aa1a

kNd

h?2M (NH , θt , P ) Bb  biiBQM`v TQBMi T`Q+2bbX AM T`iB+mH`- 7Q` H = E × E − /B;(E ×E)- r2 ?p2 NH = N - i?2 #bB+ TQBMi T`Q+2bb Q7 i?2 +?BM- r?B+? +QmMib HH i`MbBiBQMbX :Bp2 i?2 2tT`2bbBQMb Q7 i?2 [mMiBiB2b PN0 H (X0− = k) M/ PN0 H (X0 = k) BM i2`Kb Q7 i?2 BM}MBi2bBKH ;2M2`iQ` M/ Q7 i?2 biiBQM`v /Bbi`B#miBQM πX

*?Ti2` 3 SHK S`Q##BHBiv BM aT+2 h?2 SHK i?2Q`v BM bT+2 Bb #b2/ QM J2+F2Ƕb K2bm`2R M/ HbQ TTHB2b iQ MQM@ biiBQM`v TQBMi T`Q+2bb2bX Ai `2iBMb bQK2 72im`2b Q7 i?2 i?2Q`v `2bi`B+i2/ iQ i?2 HBM2X h?2 KBbbBM; `2bmHi Bb i?2 dz2p2Mi@iBK2 biiBQM`BivǴ, Bi Bb MQi i`m2 BM ;2M2`H i?i ;Bp2M M 2MmK2`iBQM {Xn }n∈N Q7 i?2 TQBMib Q7 N - i?2 /Bbi`B#miBQM Q7 N M/ i?i Q7 N − Xn rBi? `2bT2+i iQ i?2 SHK T`Q##BHBiv `2 i?2 bK2X

3XR

h?2 oQ`QMQB *2HH M/ i?2 AMp2`bBQM 6Q`KmH

h?2Q`2K dXjX9 ;Bp2b 7Q` biiBQM`v TQBMi T`Q+2bb2b QM i?2 HBM2 M BMp2`bBQM 7Q`KmH KFBM; Bi TQbbB#H2 iQ +QKTmi2 2tT2+iiBQMb rBi? `2bT2+i iQ i?2 biiBQM`v T`Q#@ #BHBiv BM i2`Kb Q7 2tT2+iiBQMb rBi? `2bT2+i iQ i?2 SHK T`Q##BHBivX h?2 MHQ;Qmb BMp2`bBQM 7Q`KmH BM i?2 bTiBH +b2 Bb BM i2`Kb Q7 oQ`QMQś +2HHb-  MQiBQM i?i r2 MQr BMi`Q/m+2X .2}MBiBQM 3XRXR G2i N #2  TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N X h?2 UQT2MV oQ`QMQB +2HH Q7 N i x ∈ Rm Bb i?2 b2i Vx (N ) := {y ∈ Rm ; ||y − x|| < inf {||y − Xn || ; Xn = x}} . n∈N

R

(J2+F2- RNek)- +QKKQMHv `272``2/ iQ b i?2 *KT#2HHĜJ2+F2 K2bm`2X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_8

kNN

jyy

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

1tKTH2 3XRXk, h?2 oQ`QMQB +2HH QM i?2 HBM2X A7 N Bb  TQBMi T`Q+2bb QM R- V0 (N ) = ( 12 T0 , 12 T1 ) P @XbX- M/ V0 (N ) = ( 12 T−1 , 12 T1 ) PN0 @XbX

h?2Q`2K 3XRXj G2i N #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb rBi? }MBi2 TQbBiBp2 BMi2MbBiv λX 6Q` HH MQM@M2;iBp2 7mM+iBQMb f : Ω → R 

E [f ] = λE0N (f ◦ θx ) 1V0 (N ) (x) dx . Rm

S`QQ7X G2i h(x, ω) := 1{N (ω,B(0,||x||))=0} , r?2`2 B(0, ||x||) Bb i?2 QT2M #HH Q7 `/Bmb ||x|| +2Mi2`2/ i 0X "v G2KK 3XRX9 #2HQr- i?2`2 Bb HKQbi bm`2Hv MQ TB` Q7 TQBMib Q7 N i i?2 bK2 /BbiM+2 7`QK  ;Bp2M a ∈ Rm M/ r2 Kv i?2`27Q`2 bb2`i i?i HKQbi bm`2Hv  h(x, ω) N (ω, dx) = 1 . Rm

UAM/22/- i?2 #Qp2 BMi2;`H +QmMib i?2 TQBMib Xn bm+? i?i i?2`2 Bb MQ TQBMi Q7 N BMbB/2 i?2 QT2M #HH Q7 +2Mi2` 0 M/ `/Bmb ||Xn ||- M/ i?2`2 Bb QM2 M/ QMHv QM2 bm+? TQBMi- MK2Hv i?2 TQBMi +HQb2bi iQ 0XV LQr 



E [f ] = E f h(x) N (dx) = E f h(x) N (dx) m Rm  R

=E (f ◦ θ−x ◦ θx ) (h(x) ◦ θ−x ◦ θx ) N (dx) Rm

 = λE0N (f ◦ θ−x ) (h(x) ◦ θ−x ) dx . Rm

h?2`27Q`2- Q#b2`pBM; i?i h(x) = h(−x) M/ i?i mM/2` i?2 SHK T`Q##BHBivx ∈ Rm #2HQM;b iQ i?2 oQ`QMQś +2HH V0 (N ) B7 M/ QMHv B7 i?2`2 Bb MQ TQBMi Q7 N i /BbiM+2 < ||x|| 7`QK x  E [f ] = λE0N

Rm



(f ◦ θx ) (h(x) ◦ θx ) dx = λE0N

Rm

(f ◦ θx ) 1V0 (N ) (x) dx . 

G2KK 3XRX9 6Q`  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM Rm - i?2`2 HKQbi bm`2Hv /Q2b MQi 2tBbi  TB` Q7 TQBMib i i?2 bK2 /BbiM+2 7`QK i?2 Q`B;BMX S`QQ7X .2MQi2 #v F i?2 2p2Mi i?i i?2`2 2tBbi i H2bi irQ TQBMib i i?2 bK2 /BbiM+2 7`QK i?2 Q`B;BMX q2 ?p2

3XRX h>1 oP_PLPA *1GG L. h>1 ALo1_aAPL 6P_JlG  P (F ) ≤ E

R

jyR

m

1{N ({y ; y=x,||y||=||x||})≥1} N (dx)

1{(N −x)({y ; y =0,||y +x||=||x||})≥1} N (dx) Rm

 = λE0N 1{N ({y ; y =0,||y +x||=||x||})≥1} dx m R  

0 ≤ λEN 1{||y +x||=||x||} N (dy  ) dx Rm Rm \{0}   

1{||y +x||=||x||} dx N (dy  ) ≤ λE0N Rm \{0} Rm



m ({x ; ||y  + x|| = ||x||}) N (dy  ) = 0 . = λE0N

=E

Rm \{0}

 _2K`F 3XRX8 h?Bb H2KK HHQrb mb iQ /2}M2 HKQbi@bm`2Hv mMK#B;mQmbHv i?2 TQBMi R1 +HQb2bi 7`QK i?2 Q`B;BMX G2i N #2 i?2 +QQ`/BMi2 T`Q+2bb Q7 i?2 +MQMB+H K2bm`#H2 bT+2 Q7 TQBMi K2bm`2b (Mp (Rm ), Mp (Rm ))X h?2 BMp2`bBQM 7Q`KmH Q7 h?2Q`2K 3XRXj i?2M iF2b i?2 7QHHQrBM; 7Q`K, 7Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM g : M (Rm ) → R

 0 g(N − x) 1V0 (N ) (x) dx . E [g(N )] = λEN Rm

1tKTH2 3XRXe, h?2 QM2@/BK2MbBQMH +b2X AM i?2 mMB/BK2MbBQMH  +b2 M/ mM/2` i?2 SHK T`Q##BHBiv- i?2 oQ`QMQś +2HH i 0 = T0 Bb i?2 BMi2`pH 12 T−1 , 12 T1 M/ i?2`27Q`2 i?2 BMp2`bBQM 7Q`KmH `2/b   0 E [f ] = λEN f ◦ θt dt . [ 12 T−1 , 12 T1 ]

h?2 1t+?M;2 6Q`KmH 6Q` bTiBH TQBMi T`Q+2bb2b- i?2`2 2tBbib M 2t+?M;2 7Q`KmH bBKBH` iQ i?2 QM2 BM i?2 QM2@/BK2MbBQMH +b2X _2+HH i?2 MQiiBQM ∂C 7Q` i?2 #QmM/`v rBi? `2bT2+i iQ i?2 1m+HB/2M iQTQHQ;v Q7  b2i C ⊂ Rm X h?2Q`2K 3XRXd G2i (P, θx ) #2  biiBQM`v 7`K2rQ`F M/ H2i N M/ N  #2 irQ θx @+QKTiB#H2 UM/ i?2`27Q`2V DQBMiHv biiBQM`v TQBMi T`Q+2bb2b QM Rm rBi? TQbBiBp2 }MBi2 BMi2MbBiB2b λ M/ λ `2bT2+iBp2HvX 6Q` Mv MQM@M2;iBp2 7mM+iBQM f 7`QK (Mp (E) × Mp (E), Mp (E) ⊗ Mp (E)) iQ (R, B(R))- mM/2` i?2 +QM/BiBQM i?i E0N  [N (∂V0 (N )] = 0

 0   0   λEN [f (N, N )] = λ EN  f (N − x, N − x)1V0 (N )(x) N (dx) . Rm

jyk

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

S`QQ7X lbBM; i?2 +HbbB+H KQMQiQM2 +QMp2`;2M+2 `;mK2Mi- r2 Kv bbmK2 f Bb #QmM/2/- bv #v 1X "v J2+F2Ƕb 7Q`KmH, 

 E f (x, N − x, N  − x) N (dx) = λ E0N [f (x, N, N  )] dx . U3XRV Rm

Rm

"v /2}MBiBQM Q7 SHK T`Q##BHBiv M/ U3XRV 

0   f (N − x, N − x) N (dx) λEN [f (N, N )] = E [0,1]m 

  f (N − x, N  − x)1Vy (N  ) (x) N (dx) N  (dy) + A − B , E [0,1]m

r?2`2

Rm





 

A := E Rm \[0,1]m

M/

[0,1]m

f (N − x, N − x)1Vy

(N  )

(x) N (dx)



B := E [0,1]m

N (dy)









Rm \[0,1]m

f (N − x, N − x)1Vy

(N  )

(x)N (dx)



N (dy)

`2 bm+? i?i A − B = 0- b rBHH #2 T`Qp2/ Hi2`X h?2 T`QQ7 Q7 i?2 i?2Q`2K Bb i?2M ;Bp2M BM i?2 7QHHQrBM; +H+mHiBQMb,  

 f (N − x, N  − x)1Vy (N  ) (x) N (dx) N  (dy) E [0,1]m Rm  

   f (N − x, N − x)1Vy−y (N  −y) (x − y) N (dx) N (dy) =E [0,1]m Rm 

  f (N − x − y, N  − x − y)1V0 (N  −y) (x) (N − y)(dx) N  (dy) =E [0,1]m Rm 

f (N − x, N  − x)1V0 (N  )(x) N (dx) . = λ E0N  Rm

Ai `2KBMb iQ b?Qr i?i A − B = 0X q2 }`bi MQi2 i?i  

 1Vy (N  ) (x) N  (dy) N (dx) = E [N ([0, 1]m )] < ∞ , A≤E [0,1]m

Rm \[0,1]m

 bBM+2 Rm \[0,1]m 1Vy (N  ) (x) N  (dy) ≤ Rm 1Vy (N  ) (x) N  (dy) = 1X h?2`27Q`2- BM Q`/2` iQ T`Qp2 i?i A − B = 0- Bi bm{+2b iQ b?Qr i?i A = B- r?B+? r2 MQr T`Q+22/ iQ /QX h?2 bT+2 Rm #2BM; i?2 +QmMi#H2 mMBQM ∪r ([0, 1]m + r)- r2 ?p2 i?i

   E f (N − x, N  − x)1Vy (N  ) (x) N (dx) N  (dy) A= 

r=0

M/

[0,1]m +r

[0,1]m

3XkX h>1 GP*G ALh1_S_1hhAPL B=

  E r=0

 [0,1]m

[0,1]m +r

jyj

f (N − x, N  − x)1Vy (N  ) (x) N (dx) N  (dy) .

6BMHHv- 7Q` 2+? r = 0 

 E f (N − x, N  − x)1Vy (N  ) (x) N (dx) N  (dx) [0,1]m +r [0,1]m

  f (N − x, N  − x)1Vy+r (N  ) (x) N (dx) (N  − r)(dy) =E [0,1]m [0,1]m

    f (N − x, N − x)1Vy (N  −r) (x − r) N (dx) (N − r)(dy) =E [0,1]m [0,1]m

  f (N − x − r, N  − x − r)1Vy (N  −r) (x) (N − r)(dx) (N  − r)(dy) =E [0,1]m [0,1]m

  f (N − x, N  − x)1Vy (N  ) (x) N (dx) N  (dy) . =E [0,1]m

[0,1]m +r



3Xk

h?2 GQ+H AMi2`T`2iiBQM

h?2Q`2K dX8XR Bb MQr ;2M2`HBx2/ iQ TQBMi T`Q+2bb2b QM Rm X q2 bi`i rBi? i?2 ;2M2`HBxiBQM Q7 i?2 EQ`QHvmFĜ.Q#`mb?BM 2biBKi2b Q7 h?2Q`2K dX8XkX h?2Q`2K 3XkXR G2i {Cn }n≥1 #2  MQM@BM+`2bBM; b2[m2M+2 Q7 +QKT+i b2ib BM Rm bm+? i?i ∩n≥1 Cn = ∅X h?2M lim

P (N (Cn ) = 1) =λ

m (Cn )

lim

P (N (Cn ) > 1) = 0.

m (Cn )

n↑∞

M/ n↑∞

S`QQ7X G2i C ∈ B(Rm )X TTHv 7Q`KmH UdXdV rBi? v(x, ω) = N (ω, C)1{N (ω,C)≥2} iQ Q#iBM 



 0 E N (C)1{N (C)≥2} = E 1{N (C)≥2} N (dx) = λEN 1{N (C−x)≥2} dx . () C

C

P#b2`p2 i?i 7Q` HH x ∈ C- N (C − x) ≤ N (C − C) Ur?2`2 C − C := {x − y ; x ∈ C, y ∈ C}VX h?2`27Q`2- #v UV E N (C)1{N (C)≥2} ≤ λ m (C)PN0 (N (C − C) ≥ 2) . qBi? C = Cn b BM i?2 bii2K2Mi Q7 i?2 i?2Q`2K- r2 ?p2 i?i {N (Cn −Cn ) ≥ 2} ↓ ∅ M/ i?2`27Q`2- #v /QKBMi2/ +QMp2`;2M+2- bBM+2 E [N (Cn )] ≤ E [N (C1 )] < ∞-

jy9

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1 ( m (Cn ))−1 E N (Cn )1{N (Cn )≥2} → 0 .

AM T`iB+mH` LQr

( m (Cn ))−1 E 1{N (Cn )≥2} → 0 . E [N (Cn )] = E N (Cn )1{N (Cn )≥2} + P (N (Cn ) = 1)

M/ i?2`27Q`2

E N (Cn )1{N (Cn )≥2} E [N (Cn )] − P (N (Cn ) = 1) = → 0.

m (Cn )

m (Cn )

AM T`iB+mH`- bBM+2 E [N (Cn )] = λ m (Cn )lim ( m (Cn ))−1 P (N (Cn ) = 1) = λ .

n↑∞

 G2i R1 /2MQi2 i?2 HKQbi bm`2Hv mMB[m2 Ub22 _2K`F 3XRX8V TQBMi Q7 N +HQb2bi iQ i?2 Q`B;BMX h?2Q`2K 3XkXk G2i {Cn }n≥1 #2  /2+`2bBM; b2[m2M+2 Q7 +HQb2/ bT?2`2b +2Mi2`2/ i i?2 Q`B;BM M/ bm+? i?i ∩n Cn = ∅X G2i f : Mp (Rm ) → R #2  #QmM/2/ K2bm`#H2 7mM+iBQMX h?2M lim E [f (SR1 (N )) | N (Cn ) > 0] = E0N [f (N )] .

n↑∞

S`QQ7X q`Bi2 & & &E [f (SR1 (N )) | N (Cn ) > 0] − E0N [f (N )]& & & & & E f (S (N ))1 R1 {N (Cn )>0} & & 0 − EN [f (N )]& =& & & P (N (Cn ) > 0) & & & E f (S (N ))1 & m 0 R1 {N (Cn )>0} − λ (Cn )EN [f (N )] & & ≤& & & & λ m (Cn ) & E f (SR1 (N ))1{N (Cn )>0} && −1 + λ − m (Cn )P (N (Cn ) > 0)−1 & . m

(Cn ) "v h?2Q`2K 3XkXR- i?2 KQ/mHmb BM i?2 Hbi i2`K Q7 i?2 `B;?i@?M/ bB/2 i2M/b iQ 0 b n ↑ ∞ r?BH2 E f (SR1 (N ))1{N (Cn )>0} P (N (Cn ) > 0) ≤ (sup f ) → λ(sup f ) < ∞ .

m (Cn )

m (Cn ) h?2`27Q`2- i?2 Hbi i2`K Q7 i?2 `B;?i@?M/ bB/2 i2M/b iQ 0X aBM+2 #v /2}MBiBQM Q7 SHK T`Q##BHBiv    m 0 f (SXi (N )1{Xi ∈Cn } , λ (Cn )EN [f (N )] = E i∈N

3XjX 1_:P.A*Ahu

jy8

Bi Dmbi `2KBMb iQ T`Qp2 i?i & & & &  1 & & E f (S (N )1 − f (S (N )1 & R1 Xi {N (Cn )>0} {Xi ∈Cn } & → 0 . & &

m (Cn ) i∈N

h?Bb [mMiBiv Bb MmHH B7 N (Cn ) = 0X A7 N (Cn ) > 0- R1 Bb QM2 Q7 i?2 Xi Ƕb BM Cn h?2`27Q`2 i?Bb [mMiBiv Bb #QmM/2/ #v (sup f )

P (N (Cn ) ≥ 2) → 0.

m (Cn ) 

L2`2bi L2B;?#Qm` M/ 1KTiv aT+2 .BbiM+2b b BM i?2 +b2 Q7 TQBMi T`Q+2bb2b QM i?2 HBM2- i?2 HQ+H BMi2`T`2iiBQM i2HHb mb i?i 7Q`  biiBQM`v TQBMi T`Q+2bb i?2 SHK /Bbi`B#miBQM /2b+`B#2b i?2 bBimiBQM i  dzivTB+HǴ TQBMiX lM/2` i?2 SHK /Bbi`B#miBQM- i?2 Q`B;BM `2T`2b2Mib bm+?  ivTB+H TQBMiX 6Q` BMbiM+2 i?2 [mMiBiv G(r) := PN0 (N (B(0, r) > 1) r?2`2 B(0, r) Bb i?2 +HQb2/ #HH Q7 `/Bmb r +2Mi2`2/ i i?2 Q`B;BM Bb BMi2`T`2i2/ b i?2 T`Q##BHBiv i?i i?2`2 Bb MQ TQBMi i /BbiM+2 ≤ r 7`QK  ivTB+H TQBMi Q7 i?2 TQBMi T`Q+2bbX PM i?2 Qi?2` ?M/- i?2 [mMiBiv F (r) := P (N (B(0, r) > 0) Bb i?2 T`Q##BHBiv i?i i?2`2 Bb MQ TQBMi Q7 i?2 biiBQM`v TQBMi T`Q+2bb i /BbiM+2 ≤ r 7`QK M `#Bi``v HQ+iBQMX h?2 oM GB2b?QmiĜ"//2H2v +Hmbi2`BM; BM/2t J(r) :=

1 − G(r) 1 − F (r)

+QKT`2b M2`2bi@M2B;?#Qm` M/ 2KTiv bT+2 /BbiM+2bX dzAi iF2b pHm2b H2bb i?M 1 r?2M2p2` i?2 2KTiv bT+2b i2M/ iQ #2 H`;2` i?M i?2 /BbiM+2 #2ir22M M2`2bi M2B;?#Qm` TB`b- i?mb BM/B+iBM; +Hmbi2`BM;- r?2`2b pHm2b 2t+22/BM; 1 bm;;2bi  KQ`2 `2;mH` Tii2`MXǴk AM T`iB+mH`- 7Q`  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb- J(r) = 1 7Q` HH r > 0X

3Xj

1`;Q/B+Biv

q2 MQr ;Bp2  T`iBH MHQ;m2 Q7 UdXjyV BM h?2Q`2K dXeXRkX G2i (P, θx , N ) #2  biiBQM`v 2`;Q/B+ TQBMi T`Q+2bb QM Rm rBi? }MBi2 TQbBiBp2 BMi2MbBiv λ M/ H2i PN0 #2 i?2 bbQ+Bi2/ SHK T`Q##BHBivX k

6`QK (oM GB2b?Qmi- kyyy)- TX jdc (oM GB2b?Qmi M/ "//2H2v- RNNe)X

jye

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

_2+HH i?2 /2}MBiBQM Q7  +QMp2t p2`;BM; b2[m2M+2 U.2}MBiBQM dXRXR3VX  MQM@ /2+`2bBM; b2[m2M+2 {An }n≥1 Q7 #QmM/2/ +QMp2t bm#b2ib Q7 Rm Bb +HH2/  +QMp2t p2`;BM; b2[m2M+2 B7 lim sup{r ; B(0; r) ⊂ An } = ∞ .

n↑∞

h?2Q`2K 3XjXR G2i {An }n≥1 #2  +QMp2t p2`;BM; b2[m2M+2 Ub22 .2}MBiBQM dXRXR3VX A7 f Bb  PN0 @BMi2;`#H2 `M/QK p`B#H2 1 f ◦ θx N (dx) = λE0N [f ] P @XbX lim m n (An ) A n AM T`iB+mH`lim

n↑∞

N (An ) = λ.

m (An )

S`QQ7X ai2T RX f ≡ 1 Ui?2 T`iB+mH` +b2 BM i?2 i?2Q`2KVX 6Q` Mv K2bm`#H2 b2i A Q7 Rm M/ Mv ε > 0- /2}M2 i?2 ε@TT`QtBKiBQMb Q7 A 7`QK #Qp2 M/ 7`QK #2HQr `2bT2+iBp2Hv #v Aε := ∪x∈A B(x, ε) M/ Aε := {x ∈ A ; B(x, ε) ⊂ A} . G2i MQr gε : Rm → R+ #2  MQM@M2;iBp2 +QMiBMmQmb 7mM+iBQM BMi2;`iBM; iQ 1 Qp2` Rm M/ pMBb?BM; QmibB/2 B(0, ε) M/ H2i  hε := gε (x) N (dx) . Rm

AM T`iB+mH`- #v *KT#2HHǶb 7Q`KmH E[hε ] = λ

gε (x) dx = λ . Rm

P#b2`p2 7Q` 7mim`2 mb2 i?i



hε ◦ θt =

Rm

gε (x − t) N (dx) .

()

qBi? A ≡ An - r2 ?p2 i?i  gε (y − t) dt = 1

(y ∈ A)

gε (y − t) dt = 0

(y ∈ A) .





M/



h?2`27Q`2



 gε (y − t) dt ≤ 1A (y) ≤ Aε

gε (y − t) dt . Aε

()

3XjX 1_:P.A*Ahu

jyd

AMi2;`iBM; rBi? `2bT2+i iQ N     gε (y − t) dt N (dy) ≤ N (A) ≤ Rm



aBM+2 r?i2p2` A ∈ B(Rm )     gε (y − t) dt N (dy) = Rm

A

A





Rm

 gε (y − t) dt

(  ) 

Rm

N (dy) .

gε (y − t) N (dy)

 hε ◦ θt dt ,

dt = A

i?2 #QmM/b BM U  V #2+QK2- KFBM; mb2 Q7 UV  hε ◦ θt dt ≤ N (A) ≤ hε ◦ θt dt . Aε



_2im`MBM; iQ i?2 7mHH MQiiBQM A = An lim sup n

N (An )

m ((An )ε ) 1 ≤ lim sup m m

(An )

(An ) m ((An )ε ) n

 hε ◦ θt dt . (An )ε

"v 2`;Q/B+Biv- i?2 b2+QM/ 7+iQ` Q7 i?2 `B;?i@?M/ bB/2 +QMp2`;2b iQ λX h?2 }`bi 7+iQ` +QMp2`;2b iQ 1 U?2`2 ;BM r2 mb2 i?2 /2}MBiBQM Q7  +QMp2t p2`;BM; b2@ [m2M+2VX h?2`27Q`2 N (An ) lim sup m ≤ λ. n (An ) aBKBH` `;mK2Mib vB2H/ lim inf n

N (An )

m ((An )ε ) 1 ≥ lim inf m m n

(An )

(An ) m ((An )ε )

M/ i?2M lim inf n

 hε ◦ θt dt (An )ε

N (An ) ≥ λ.

m (An )

ai2T kX f BMi2;`#H2X h?2 `;mK2Mib 7QHHQr +HQb2Hv i?Qb2 Q7 ai2T RX h?Bb iBK2- H2i  gε (x)(f ◦ θx ) N (dx) . hε := Rm

AM T`iB+mH` UJ2+F2Ƕb 7Q`KmHV  E0N [gε (x)f ] dx = λE0N [f ] E[hε ] = λ Rm

Rm

P#b2`p2 7Q` 7mim`2 mb2 i?i   gε (x)(f ◦ θx ◦ θt ) (N ◦ θt )(dx) = hε ◦ θ t = Rm

gε (x) dx = λE0N [f ] .

Rm

gε (x − t)(f ◦ θx ) N (dx) .

JmHiBTHvBM; #Qi? bB/2b Q7 UV #v f ◦ θy M/ BMi2;`iBM; rBi? `2bT2+i iQ N -

jy3

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1 

 Rm

  gε (y − t) dt (f ◦ θy ) N (dy) ≤ (f ◦ θy ) N (dy) Aε A    gε (y − t) dt (f ◦ θy ) N (dy) . ≤ Rm



aBM+2 7Q` Mv A ∈ B(Rm )   gε (y − t) dt (f ◦ θy ) N (dy) Rm A     = gε (y − t)(f ◦ θy ) N (dy) dt = hε ◦ θt dt , A

r2 ?p2 i?i

Rm







hε ◦ θt dt ≤ Aε

h?2`27Q`2  lim sup n

An

A

(f ◦ θy ) N (dy)

m (An )

(f ◦ θy ) N (dy) ≤

hε ◦ θt dt .

(†)



A



m ((An )ε ) 1 hε ◦ θt dt

m (An ) m ((An )ε ) (An )ε n 

m ((An )ε ) 1 ≤ lim sup m lim sup m hε ◦ θt dt ε

(An ) n n ((An ) ) (An )ε ≤ lim sup

= λE0N [f ] . aBKBH`Hv- bi`iBM; 7`QK i?2 `B;?i@?M/ BM2[mHBiv BM U†V (f ◦ θy ) N (dy) ≥ λE0N [f ] . lim inf An m n

(An )   :2M2`HBx2/ _2M2rH h?2Q`2K G2i (N, θx , P ) #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM Rm Q7 i?2 b2+QM/ Q`/2`- Q7 BMi2MbBiv λ- M/ H2i PN0 #2 i?2 bbQ+Bi2/ SHK T`Q##BHBivX G2i σ(C) := E0N [N (C)] UC ∈ B(Rm )VX .2}MBiBQM 3XjXk h?2 T`Q##BHBiv P Bb +HH2/ N @KBtBM; B7 7Q` HH G2#2b;m2 BMi2@ ;`#H2 7mM+iBQMb a, b : Rm → R pMBb?BM; QmibB/2  #QmM/2/ b2i    

a(x)N (ω, dx) b(x)N (θx ω, dx) lim E ||x||↑∞ m Rm  R   

=E a(x)N (ω, dx) E b(x)N (ω, dx) m m  R   R = λ2 a(x) dx b(x) dx . Rm

Rm

_2K`F 3XjXj A7 P Bb KBtBM;- Bi Bb  7Q`iBQ`B N @KBtBM;X

3XjX 1_:P.A*Ahu

jyN

h?2Q`2K 3XjX9 Uj V A7 P Bb N @KBtBM;- i?2M 7Q` HH b : Rm → R pMBb?BM; QmibB/2  #QmM/2/ b2i  2 lim b(x)Sx σN (dx) = λ b(x) dx . ||x||↑∞

Rm

Rm

Uh?2 #Qp2 `2bmHi bvb- BM Qi?2` i2`Kb- i?i i?2 K2bm`2 Sx σ +QMp2`;2b p;m2Hv b ||x|| ↑ ∞ iQ λ2 iBK2b i?2 G2#2b;m2 K2bm`2 QM Rm XV S`QQ7X q2 b?HH mb2 J2+F2Ƕb 7Q`KmH BM i?2 7Q`K,  





f (−x, ω, θx ω) PN0 (dω) dx ,

f (x, θx ω, ω)N (ω, dx)P (dω) = λ Ω

Rm

Rm

r?B+? ;Bp2b- rBi? i?2 MQiiBQM N (ω, a) :=

Rm

 

  Rm

a(x)N (ω, dx) M/ i?2 HBF2

a(x)N (ω, dx) Ω



Ω

b(x)N (θx ω, dx) P (dω) R  

m

a(x)N (θx ω, b)N (ω, dx) P (dω)

= Ω Rm



=λ a(−x)N (θx+t ω, b) dx PN0 (dω) m Ω R   = N (θx ω, b ∗  a) PN0 (dω) = (b ∗  a)(x)Sx σ(dx) , Rm

Ω

r?2`2  a(x) := a(−x)X h?2`27Q`2 

 Rm

(b ∗  a)(x)Sx σ(dx) → λ



2

a(x) dx Rm

b(x) dx , Rm

7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrb  #v  biM/`/ `;mK2Mi mbBM;  b2[m2M+2 Q7 7mM+iBQMb an : R → R+ bm+? i?i Rm an (x) dx = 1 M/ b ∗ an → bX 

_2K`F 3XjX8 aBM+2  biiBQM`v `2M2rH i?2Q`2K rBi? MQM@HiiB+2 /Bbi`B#miBQM Bb P @KBtBM;- i?2 +HbbB+H `2M2rH i?2Q`2K +M #2 b22M b  T`iB+mH` +b2 Q7 i?2 #Qp2 `2bmHiX >Qr2p2`- i?Bb Bb  bQK2r?i KBbH2/BM; bii2K2Mi- bBM+2 i?2 T`QQ7 i?i  biiBQM`v MQM@HiiB+2 `2M2rH TQBMi T`Q+2bb Bb KBtBM; Bb MQi i HH 2bv- M/ Bi ?b H`2/v #22M MQi2/9 i?i i?2 T`QQ7 Q7 i?2 KBtBM; T`QT2`iv 7Q` `2M2rH T`Q+2bb2b 7QHHQrb `;mK2Mib i?i `2 p2`v +HQb2 iQ i?Qb2 mb2/ BM i?2 T`QQ7 Q7 "H+Fr2HHǶb i?2Q`2KX

j 9

ii`B#mi2/ iQ L2p2m BM (.2HbM2`B2- RNdd)X (.H2v M/ o2`2@CQM2b- RN33)- TX 93NX

jRy

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

3X9 h?2 J2+F2 J2bm`2 G2i E #2 M HX+X/X#X bT+2 M/ H2i B(E) #2 i?2 bbQ+Bi2/ "Q`2H σ@}2H/X G2i N #2  `M/QK K2bm`2 QM E rBi? HQ+HHv }MBi2 K2M K2bm`2 νX G2KK 3X9XR h?2 b2i 7mM+iBQM C : B(E) × F → R+ /2}M2/ QM i?2 `2+iM;H2b A × B UA ∈ B(E), F ∈ F V #v C(A × F ) := E[N (A)1F ]

U3XkV

+M #2 mMB[m2Hv 2ti2M/2/ iQ  σ@}MBi2 K2bm`2 C : B(E) ⊗ F → [0, ∞]X S`QQ7X h?2 +Hbb Q7 b2ib S := {A × F ; A ∈ B(E), F ∈ F} 7Q`Kb  b2KB@H;2#`X h?2 7mM+iBQM C : S → [0, ∞] /2}M2/ #v U3XkV Bb +H2`Hv +QmMi#Hv //BiBp2X A7 Bi +M #2 b?QrM iQ #2 σ@}MBi2- i?2 `2bmHi 7QHHQrb 7`QK *`i?ûQ/Q`vǶb 2ti2MbBQM i?2Q`2K Uh?2Q`2K XRXRkV bBM+2 B(E) ⊗ F = σ(S)X hQ T`Qp2 σ@}MBi2M2bb- +QMbB/2`  +QmMi#H2 +Qp2`BM; Q7 E #v `2HiBp2Hv +QKT+i b2ib Am Um ∈ NV M/ H2i Fmn := {ω; N (ω, Am ) ≤ n} Um, n ∈ NVX h?2 7KBHv {Am × Fm,n ; m, n ∈ N} Bb  +Qp2`BM; Q7 E × F X AM/22/- 7Q` Mv (x, ω) ∈ E × F - i?2`2 2tBbib M Am bm+? i?i x ∈ Am - M/ 7Q` bm+? }t2/ Am - i?2`2 2tBbib M n ∈ N bm+? i?i N (Am ) ≤ n- bBM+2 N (Am ) < ∞X h?2`27Q`2 (x, ω) ∈ Am × Fmn X JQ`2Qp2` C(Am × Fmn ) = E[N (Am )1Fmn ] ≤ nE[1Fmn ] ≤ n < ∞ r?B+? T`Qp2b σ@}MBi2M2bbX  .2}MBiBQM 3X9Xk h?2 K2bm`2 C /2}M2/ BM G2KK 3X9XR Bb +HH2/ i?2 J2+F2 K2bm`2 Q7 i?2 `M/QK K2bm`2 N X h?2Q`2K 3X9Xj 6Q` HH MQM@M2;iBp2 K2bm`#H2 g : E × Ω → [0, ∞]    g(x, ω)N (ω, dx)P (dω) = g(x, ω) C(dx × dω) . Ω

E

E

U3XjV

Ω

S`QQ7X "v i?2 mbmH KQMQiQM2 +Hbb `;mK2Mi- Bi bm{+2b iQ T`Qp2 i?2 Hii2` 2[mH@ Biv 7Q` 7mM+iBQMb Q7 i?2 ivT2 g(x, ω) = 1B (x, ω) (B ∈ B(E) ⊗ F) . AM/22/ i?2 +QHH2+iBQM D Q7 b2ib B ∈ B(E) ⊗ F 7Q` r?B+? 2[mHBiv U3XjV Bb i`m2 7Q` g = 1B Bb  .vMFBM bvbi2K M/ +QMiBMb i?2 π@bvbi2K S = {A × F ; A ∈ B(E)F ∈ F} #v /2}MBiBQM Q7 i?2 K2bm`2 CX h?2`27Q`2 D ⊇ σ(S) = B(E) ⊗ F X h?2 `2bmHi Bb i?2M Q#iBM2/ 7`QK i?2`2- #v iFBM; i?2 mbmH `Qmi2 7`QK BM/B+iQ` 7mM+iBQMb iQ bBKTH2 7mM+iBQMb M/ i?2M iQ MQM@M2;iBp2 K2bm`#H2 7mM+iBQMbX  *Q`QHH`v 3X9X9 G2i g : E × Ω → R #2  K2bm`#H2 7mM+iBQM bm+? i?i 2Bi?2` QM2 Q7 i?2 irQ 7QHHQrBM; +QM/BiBQMb Bb biBb}2/,     |g(x, ω)|N (ω, dx)P (dω) < ∞ Q` |g(x, ω)|C(dx × dω) < ∞ . Ω

E

E

Ω

h?2M i?2 Qi?2` +QM/BiBQM Bb biBb}2/ M/ 7Q`KmH U3XjV TTHB2bX

3X9X h>1 J1*E1 J1al_1

jRR

h?2 +QM/BiBQMb T`2pBHBM; #Qp2 `2 KBMiBM2/X 6Bt F ∈ F- M/ /2MQi2 #v CF i?2 K2bm`2 QM (E, B(E)) /2}M2/ #v CF (A) = C(A × F )X *H2`Hv CF  ν- M/ F i?2`27Q`2- i?2`2 2tBbib  _/QMĜLBFQ/ɷK /2`BpiBp2 dC /2}M2/ 7Q` ν@HKQbi HH dν x ∈ E,  dCF C(A × F ) = (x) ν(dx) . U3X9V A dν G2i dCF Px (F ) := (x) . dν h?Bb ν@HKQbi@2p2`vr?2`2 /2}M2/ 7mM+iBQM i?2`27Q`2 biBb}2b  C(A × F ) = Px (F ) ν(dx) . U3X8V A

AM T`iB+mH`-

 ν(A) = C(A × Ω) =

Px (Ω) ν(dx) A

M/ i?2`27Q`2 Px (Ω) = 1- i?i Bb- Px Bb ν@X2X  T`Q##BHBiv K2bm`2X .2}MBiBQM 3X9X8 h?2 T`Q##BHBiv Px bQ /2}M2/ Uν@X2XV Bb +HH2/ i?2 SHK T`Q#@ #BHBiv Q7 N i x ∈ EX  KQ`2 +QKTH2i2 MQiiBQM rBHH bQK2iBK2b #2 mb2/ BM Q`/2` iQ bT2+B7v i?2 TQBMi T`Q+2bb BMpQHp2/, PN,x X q2 rQmH/ HBF2 iQ #2 #H2 iQ +?QQb2 i?2 T`Q##BHBiB2b Px Ux ∈ EV BM bm+?  rv i?i 7Q` ν@HKQbi HH x ∈ E F → Px (F ) Bb  T`Q##BHBiv K2bm`2 QM (Ω, F) M/ 7Q` }t2/ F

x → Px (F ) Bb B(E)@K2bm`#H2 .

h?2 2tBbi2M+2 Q7 bm+?  `2;mH` p2`bBQM Bb ;m`Mi22/ mM/2` +2`iBM +QM/BiBQMb- 7Q` BMbiM+2- #v i?2 K2bm`2 /BbBMi2;`iBQM i?2Q`2K Uh?2Q`2K XRXRjV- B7 i?2 bT+2 Ω Bb  +QKTH2i2 b2T`#H2 K2i`B+ bT+2 M/ F Bb i?2 "Q`2H }2H/ bbQ+Bi2/ iQ i?2 iQTQHQ;v BM/m+2/ #v Bib K2i`B+X h?Bb Bb i?2 +b2 BM T`iB+mH` B7 (Ω, F) ≡ (M (E), M(E)) #2+mb2 i?2`2 2tBbib  /BbiM+2 QM M (E) KFBM; Q7 Bi  +XbXKXbX h?2 T`Q##BHBiv F2`M2H 7`QK E × Ω iQ (Ω, F)) /2}M2/ #v (x, A) → Px (A) Bb +HH2/ i?2 SHK F2`M2HX 6`QK MQr- i?2 2tBbi2M+2 Q7  `2;mH` p2`bBQM Bb bbmK2/X AM pB2r Q7 U3X8V- i?2 7QHHQrBM; `2bmHi Bb Dmbi  `2r`BiBM; Q7 7Q`KmH U3XjVX h?2Q`2K 3X9Xe 6Q` HH MQM@M2;iBp2 K2bm`#H2 g : E × Ω → [0, ∞]     g(x, ω)N (ω, dx)P (dω) = g(x, ω)Px (dω) ν(dx) . Ω

E

E

U3XeV

Ω

6Q`KmH U3XeV HbQ ?QH/b i`m2 7Q` Mv K2bm`#H2 g : E × Ω → R bm+? i?i 2Bi?2` QM2 Q7 i?2 7QHHQrBM; BM2[mHBiB2b Bb i`m2,     |g(x, ω)|N (ω, dx)P (dω) < ∞ Q` |g(x, ω)|Px (dω) ν(dx) < ∞ Ω

E

E

Ω

Ui?2 Qi?2` QM2 i?2M #2BM; miQKiB+HHv biBb}2/VX

jRk

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

.B`2+iBQMH .2`BpiBp2 Q7 i?2 GTH+2 h`Mb7Q`K q2 MQr BMi`Q/m+2  mb27mH iQQH 7Q` i?2 /2i2`KBMiBQM Q7 i?2 GTH+2 i`Mb7Q`K mM/2` i?2 SHK T`Q##BHBiv i  ;Bp2M TQBMi xX .2}MBiBQM 3X9Xd G2i N #2  HQ+HHv }MBi2 TQBMi T`Q+2bb QM E rBi? BMi2MbBiv K2@ bm`2 ν M/ GTH+2 7mM+iBQMH LN X G2i ϕ M/ f #2 irQ MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb 7`QK E iQ RX h?2 [mMiBiv ∂LN LN (ϕ + εf ) − LN (ϕ) (ϕ, f ) := lim ε↓0 ∂f ε Bb +HH2/- B7 Bi 2tBbib- i?2 /B`2+iBQMH /2`BpiBp2 i ϕ Q7 i?2 GTH+2 i`Mb7Q`K BM i?2 /B`2+iBQM f X h?2Q`2K 3X9X3 A7 f ∈ L1E (ν)- i?2 /2`BpiBp2 i ϕ Q7 i?2 GTH+2 i`Mb7Q`K BM i?2 /B`2+iBQM f 2tBbib M/ Bb ;Bp2M #v i?2 7Q`KmH  ∂LN (ϕ, f ) = − f (x)Lx (ϕ)ν(dx) , U3XdV ∂f E    exp − ϕ(y) N (ω, dy) Px (dω)



r?2`2 Lx (ϕ) :=

Ω

E

Bb i?2 GTH+2 /Bbi`B#miBQM Q7 N mM/2` i?2 T`Q##BHBiv K2bm`2 Px X S`QQ7X .Bz2`2MiBiBQM rBi? `2bT2+i iQ ε i i?2 pHm2 ε = 0 Q7   

LN (ϕ + εf ) = E exp − (ϕ(x) + εf (x)) N (dx)  E    

= E exp − ϕ(x) N (dx) exp −ε f (x) N (dx) E

E

;Bp2b   

∂LN (ϕ, f ) = E exp − ϕ(y) N (dy) f (x) N (dx) ∂f E E    

=E f (x) exp − ϕ(y) N (dy) N (dx) . E

E

Ai bm{+2b MQr iQ TTHv 7Q`KmH U3XeV rBi?    g(x, ω) := f (x) exp − ϕ(y) N (ω, dy) E

iQ Q#iBM ∂LN (ϕ, f ) = ∂f





    exp − ϕ(y) N (ω, dy) Px (dω) ν(dx) ,

f (x) E

r?B+? Bb i?2 MMQmM+2/ `2bmHiX

Ω

E



3X9X h>1 J1*E1 J1al_1

jRj

1tKTH2 3X9XN, SHK F2`M2H Q7  SQBbbQM T`Q+2bbX A7 N Bb mM/2` P  SQBbbQM T`Q+2bb Q7 HQ+HHv }MBi2 BMi2MbBiv K2bm`2 ν   −ϕ(x)  LN (ϕ) = exp e − 1 ν(dx) E

M/ i?2`27Q`2





 e−ϕ(x)−εf (x) − 1 ν(dx)

LN (ϕ + εf (x)) = exp



E

7`QK r?B+? Bi 7QHHQrb i?i     1 −ϕ(x) 1 −εf (x) LN (ϕ + εf ) − LN (ϕ) = LN (ϕ) exp (e − 1) ν(dx) e ε ε E M/ i?2`27Q`2- H2iiBM; ε → 0 ∂LN (ϕ, f ) = − f (x)LN (ϕ)e−ϕ(x) ν(dx) . ∂f E h?2`27Q`2- 7`QK U3XdV  f (x)Lx (ϕ)ν(dx) = f (x)LN (ϕ)e−ϕ(x) ν(dx) . E

E

h?Bb #2BM; i`m2 7Q` HH #QmM/2/ f Lx (ϕ) = LN (ϕ)e−ϕ(x)

(ν@X2X) .

−ϕ(x)

Bb i?2 GTH+2 7mM+iBQMH Q7 i?2 bBKTH2 /2i2`KBMBbiB+ TQBMi T`Q+2bb "mi ϕ → e rBi?  bBM;H2 TQBMi x- M/ i?2`27Q`2 ϕ → LN (ϕ)e−ϕ(x) Bb i?2 GTH+2 i`Mb7Q`K Q7  SQBbbQM TQBMi T`Q+2bb Q7 BMi2MbBiv K2bm`2 ν iQ r?B+? QM2 //b  TQBMi i xX h?Bb b?Qrb i?i- ν@HKQbi@2p2`vr?2`2- N ?b mM/2` Px i?2 bK2 /Bbi`B#miBQM b N + εx mM/2` P X 1tKTH2 3X9XRy, G2i X1 , . . . , Xk #2 BB/ `M/QK 2H2K2Mib Q7 (E, B(E)) M/ H2i N :=

k 

εX j .

j=1

h?2 GTH+2 i`Mb7Q`K Q7 i?Bb TQBMi T`Q+2bb Bb k LN (ϕ) = E e−ϕ(X1 ) M/ M 2H2K2Mi`v +QKTmiiBQM ;Bp2b  k−1 ∂LN (ϕ, f ) = −E e−ϕ(X1 ) f (x)e−ϕ(x) kP (X1 ∈ dx) . ∂f E h?2`27Q`2 k−1 −ϕ(x) Lx (ϕ) = E e−ϕ(X1 ) e , r?B+? Bb i?2 GTH+2 i`Mb7Q`K Q7  TQBMi T`Q+2bb Nx :=

k−1  j=1

ε X j + εx .

jR9

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h?2 aiiBQM`v *b2 h?2Q`2K 3X9XRR G2i N #2- mM/2` P -  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM Rm Q7 }MBi2 BMi2MbBivX h?2M Px (N ∈ Γ) = PN0 (N + x ∈ Γ) U3X3V 7Q` HKQbi HH x rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 QM Rm X

S`QQ7X >2`2 ν(dx) = λ dxX TTHvBM; U3XeV rBi? g(x, ω) := 1C (x)1Γ (N (ω) + x) (C ∈ B(Rm ) , Γ ∈ M(Rm ) ,  r2 Q#iBM 7Q` i?2 `B;?i@?M/ bB/2 λ C Px (N + x ∈ Γ) dx M/ 7Q` i?2 H27i@?M/ bB/2mbBM; J2+F2Ƕb 7Q`KmH 7Q` biiBQM`v TQBMi T`Q+2bb2b E







h?2`27Q`2

PN0 (N + x ∈ Γ) dx .

1C (Xn )1Γ (N (ω) + Xn ) = λ C

n∈N



 Px (N ∈ Γ) dx = C

PN0 (N + x ∈ Γ) dx . C

aBM+2 C Bb `#Bi``v BM B(Rm )- i?2 `2bmHi 7QHHQrbX Ai bvb i?i 7Q` HKQbi HH x UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2V- i?2 /Bbi`B#miBQM Q7 N mM/2` Px Bb i?2 BK;2 Q7 i?2 /Bbi`B#miBQM Q7 N mM/2` PN0 #v i?2 KTTBM; N → N + xX  *Q`QHH`v 3X9XRk A7 N Bb mM/2` P  bBKTH2 biiBQM`v TQBMi T`Q+2bb QM E Q7 }MBi2 BMi2MbBiv- Px (N ({x} = 1) = 1 7Q` HKQbi HH x rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 QM EX S`QQ7X hF2 Γ := {μ ; μ({x}) = 1} BM U3X3VX AM T`iB+mH`- {N ∈ Γ} ≡ {N ({x}) = 1} M/ {N + x ∈ Γ} ≡ {N ({0}) = 1}X h?2`27Q`2 U3X3V `2/b 7Q` i?Bb +?QB+2 Q7 ΓPx (N ({x}) = 1) = PN0 (N ({0}) = 1) , i?i Bb- #v h?2Q`2K dXkXd- Px (N ({x}) = 1) = 1X



h?2Q`2K 3X9XRR Bb bii2/ BM i2`Kb Q7 i?2 /Bbi`B#miBQM Q7 N X >2`2 Bb i?2 #bi`+i 7Q`K Q7 Bi BM i?2 θx @7Q`KHBbK Ui?Bb Bb KQ`2 ;2M2`H b Bi HHQrb mb iQ +QMbB/2` `M/QK 2H2K2Mib Qi?2` i?M i?2 TQBMi T`Q+2bb N Bib2H7VX h?2Q`2K 3X9XRj G2i (N, θx , P ) #2  biiBQM`v TQBMi T`Q+2bb QM Rm X h?2M- 7Q` HH F ∈ F - 7Q` HKQbi HH UrX`XiX G2#2b;m2 K2bm`2V x ∈ Rm Px (F ) = PN0 (θx (F )) .

()

3X9X h>1 J1*E1 J1al_1

jR8

−1 S`QQ7X q2 T`Qp2 i?2 2[mBpH2Mi 7Q`K, Px (F ) = PN0 (θ−x (F )) 7Q` HKQbi HH UrX`XiX m G2#2b;m2 K2bm`2V x ∈ R X 6Q` i?Bb- r2 TTHv J2+F2Ƕb 7Q`KmH UdX9V rBi? v(x, ω) := 1A (x)1F (θx (ω)) iQ Q#iBM     −1  λ PN0 θ−x (F ) dx = 1A (x)1F ((θx ◦ θ−x )(ω)) N (ω, dx) P (dω) A Ω E = 1A (x)1F (ω) N (ω, dx) P (dω) = C(A × F ) .





−1 θ−x (F )

h?2`27Q`2 λ A PN0 B(Rm )- i?2 `2bmHi 7QHHQrbX



Ω

E

dx = λ

 A

Px (F ) dxX h?Bb #2BM; i`m2 7Q` HH A ∈ 

h?2 SQBbbQM *b2, aHBpMvFǶb i?2Q`2K h?2 7QHHQrBM; `2bmHi ;2M2`HBx2b *Q`QHH`v dXdXdX h?2Q`2K 3X9XR9 G2i P #2  T`Q##BHBiv i?i KF2b Q7 i?2 TQBMi T`Q+2bb N QM Rm  bBKTH2 SQBbbQM T`Q+2bb rBi? BMi2MbBiv K2bm`2 νX h?2M- 7Q` HH Γ ∈ M(Rm )Px (N ∈ Γ) = P (N + εx ∈ Γ) ,

dν@X2X

U3XNV

S`QQ7X aBM+2 #Qi? Px (N ∈ ·) M/ P (N + εx ∈ ·) `2 T`Q##BHBiv /Bbi`B#miBQMb Q7 bBKTH2 TQBMi T`Q+2bb2b- Bi bm{+2b #v i?2 JƺM+?Ĝ_ûMvB i?2Q`2K Uh?2Q`2K RXjXk8V iQ T`Qp2 i?i i?2B` `2bT2+iBp2 pQB/M+2 T`Q##BHBiv 7mM+iBQMb vx M/ v +QBM+B/2X 6Q` HH A- B ∈ B(Rm )  C(A × {N (B) = 0}) = Px (N (B) = 0) ν(dx) = vx (B) ν(dx) . A

A

"v i?2 BM/2T2M/2M+2 T`QT2`iv Q7 SQBbbQM T`Q+2bb2b C(A × {N (B) = 0}) = E N (A)1{N (B)=0} = E N (A\B)1{N (B)=0}  v(B) ν(dx) . = ν(A\B) v(B) = A\B

h?2`27Q`2-



 vx (B) ν(dx) =

v(B) ν(dx) .

A

"mi 

A\B

 v(B) ν(dx) =

A\B

v(B)1{x∈B} ν(dx) / 

A

 P (N (B) = 0)1{x∈B} ν(dx) = /

= A

P ((N + εx )(B) = 0) ν(dx) . A

h?2`27Q`2 7Q` HH A- B ∈ B(Rm )   Px (N (B) = 0) ν(dx) = P ((N + εx )(B) = 0) ν(dx) A

A

jRe

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M/ i?Bb BKTHB2b Px (N (B) = 0) = P ((N + εx )(B) = 0)- ν@X2X



AM i?2 biiBQM`v +b2- iFBM; BMiQ ++QmMi U3XNV- r2 ?p2 i?i 7Q` HKQbi HH x UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2VPN0 (N + x ∈ Γ) = P (N + εx ∈ Γ) .  dz?QKQ;2M2Qmb SQBbbQM T`Q+2bb Bb i?2 bK2 mM/2` i?2 SHK T`Q##BHBiv 2t+2Ti 7Q`  TQBMi //2/ i i?2 Q`B;BMǴX JQ`2 T`2+Bb2Hv, h?2Q`2K 3X9XR8 U8 V G2i P #2  T`Q##BHBiv i?i KF2b Q7 i?2 TQBMi T`Q+2bb N  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λX h?2M- 7Q` HH Γ ∈ Mp (Rm )PN0 (N ∈ Γ) = P (N + ε0 ∈ Γ) . S`QQ7X *QK#BM2 h?2Q`2K 3X9XR9 M/ 1[MX U3X3V iQ Q#iBM i?i 7Q` i H2bi QM2 a UBM 7+i 7Q` HKQbi HH aV PN0 (N + a ∈ Γ) = P (N + εa ∈ Γ) . _2TH+2 N #v N  := N + aX aBM+2 N  M/ N `2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb2b rBi? i?2 bK2 BMi2MbBiv- P (N + εa ∈ Γ) = P (N + a + εa ∈ Γ) = P (N + ε0 ∈ Γ)X HbQ PN0  (N  ∈ Γ) = PN0 (N ∈ Γ) b QM2 +M +?2+F 7`QK i?2 /2}MBiBQM Q7 i?2 SHK T`Q##BHBivX  1tKTH2 3X9XRe, h?2 Jiö`M ?`/@+Q`2 KQ/2HXe G2i N #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv λX G2i {Xn }n∈N #2 Bib b2[m2M+2 Q7 TQBMibX h?2  Q#iBM2/ #v i?BMMBM; N BM bm+?  rv i?i HH Jiû`M KQ/2H Bb  TQBMi T`Q+2bb N TB`b Q7 TQBMib Q7 i?2 i?BMM2/ TQBMi T`Q+2bb `2 i H2bi i  /BbiM+2 R > 0 T`iX h?2 T`Q#H2K Bb, ?Qr iQ +?QQb2 i?2 TQBMiUbV iQ 2`b2  TB` Q7 TQBMib i KmimH /BbiM+2 ≤ R Ur2 /Q MQi rMi i?2 2bv bQHmiBQM i?i +QMbBbib BM 2`bBM; #Qi? TQBMibV\ h?2 B/2 BM i?Bb KQ/2H Bb iQ BMi`Q/m+2 dzT`BQ`Biv K`FbǴX JQ`2 T`2+Bb2HvH2i {Un }n∈N #2 M BB/ b2[m2M+2 Q7 `2H `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM  i?2 BMi2`pH [0, 1]- BM/2T2M/2Mi Q7 N X  TQBMi Xn Q7 N Bb `2iBM2/ b  TQBMi Q7 N B7 M/ QMHv B7 Un < Uk 7Q` HH k = n bm+? i?i Xk ∈ B(Xn ; R) ,  r?2`2 B(x, ; R) /2MQi2b i?2 +HQb2/ #HH Q7 +2Mi2` x M/ `/Bmb RX P#pBQmbHv- N Bb  biiBQM`v TQBMi T`Q+2bbX AM Q`/2` iQ +QKTmi2 Bib BMi2MbBiv K2bm`2 ν- }`bi +QM/BiBQM rBi? `2bT2+i iQ N , 6Q` HH C ∈ B(Rm ) 



 N   ν(C) := E . 1C (x) N (dx) = E E 1C (x) N (dx) | F Rn

AM pB2r Q7 i?2 BM/2T2M/2M+2 Q7 N M/ i?2 Un Ƕb, 8 e

(aHBpMvF- RNek)X (Jiû`M- RNey)- (Jiû`M- RN3e)X

Rn

3X9X h>1 J1*E1 J1al_1

jRd

 E

 (dx) | F N 1C (x) N Rn    N =E 1C (Xn )1(Un < Uk 7Q` HH k = n bm+? i?i Xk ∈ B(Xn ; r)) | F n∈N

=E

 

 1C (Xn ) P (Un < Uk 7Q` HH k = n bm+? i?i Xk ∈ B(Xn ; R) | F ) . N

n∈N

LQr P (Un < Uk 7Q` HH k = n bm+? i?i Xk ∈ B(Xn ; R) | F N )   min Uk | F N = P Un < k=n ; Xk ∈B(Xn ;R)   1  = P min Uk > u | F N du 

k=n ; Xk ∈B(Xn ;R)

0 1

(1 − u)N (B(Xn ;R))−1 du .

= 0





h?2`27Q`2 ν(C) = E

1C (x)g(x, N ) N (dx) , Rm

1 r?2`2 g(x, N ) := 0 (1 − u)N (B(x;R))−1 du M/ #v J2+F2Ƕb 7Q`KmH M/ i?2 7+i i?i BM i?2 SQBbbQMBM +b2 Px (N ∈ ·) = P (N + εx ∈ ·)

  1C (x)g(x, N ) N (dx) = λ 1C (x)E [ g(x, N + εx )] dx E Rm Rm  1 

 =λ 1C (x)E (1 − u)N (B(x;R)) du dx 0 Rm  1     =λ 1C (x) E uN (B(x;R)) du dx. Rm

0

aBM+2 N (B(x; R)) Bb  SQBbbQM p`B#H2 rBi? K2M λVm Rm - r?2`2 Vm := Bb i?2 pQHmK2 Q7 i?2 mMBi m@/BK2MbBQMH bT?2`2  m E uN (B(x;R)) = e−λ(1−u)Vm R M/ i?2`27Q`2 

1

 E u

N (B(x;R))





1

du =

0

0 1 =

e−λ(1−u)Vm R du m

1 − e−λV R . λV m Rm m

e−λuVm R du = m

0

 r?2`2 6BMHHv ν(C) = m (C)λ−λV R = 1−e λ . m V Rm m

m

m

m

π2 Γ(1+ m 2 )

jR3

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h?Bb BMi2MbBiv Bb M BM+`2bBM; 7mM+iBQM Q7 λ M/ Bib HBKBi b λ ↑ ∞ Ubim`i2/ KQ/2HV Bb V m1Rm X 1tKTH2 3X9XRd,  +Qp2`;2 T`Q#H2K BM +QKKmMB+iBQMbXd G2i N #2 mM/2` T`Q##BHBiv P  biiBQM`v ?TT QM R2 Q7 BMi2MbBiv λ rBi? TQBMi b2[m2M+2 {Xn }n∈N X *QMbB/2` i?2 7QHHQrBM; dz`2+2TiBQM +2HHǴ3 bbQ+Bi2/ rBi? TQBMi Xn ,  C(Xn , N ) :=

F D(|y − Xn |)

n >T y∈R ; W + k∈N ; k=n Fk D(|y − Xk |) 2

 ,

r?2`2 W ≥ 0- T > 0- D : R+ → R+ Bb  MQM@M2;iBp2 /2+v 7mM+iBQM- i?i Bb-  7mM+iBQM i?i /2+`2b2b iQ 0 b u → ∞- M/ bm+? i?i 7Q` HH a > 0 ∞ uD(u) du < ∞ , U3XRyV a

7Q` BMbiM+2 D(u) = u−β

(β > 2) ,

U3XRRV

M/ {Fn }n∈N Bb M BB/ b2[m2M+2 Q7 2tTQM2MiBH `M/QK p`B#H2b rBi? K2M μ−1 X *QM/BiBQM U3XRyV ;m`Mi22b i?i i?2 /2MQKBMiQ` BM i?2 /2}MBiBQM Q7 i?2 `2+2TiBQM

+2HH Bb }MBi2X AM/22/- i?2 UH`;2`V bmK k∈N Fk D(|y − Xk |) +M #2 b2T`i2/ BM  bmK +Q``2bTQM/BM; iQ TQBMib BM i?2 #HH B(0; a)- a > 0- r?B+? `2 }MBi2 BM MmK#2` M/ /Bz2`2Mi 7`QK 0- M/ i?2 `2bi Q7 i?2 bmK r?Qb2 2tT2+iiBQM Bb }MBi2 b i?2 7QHHQrBM; +QKTmiiBQM b?Qrb,     E Fk D(|y − Xk |)1B(0;a) (Xk ) = E [F0 ] 1B(0;a) (x)D(|y − x|)λ dx 

k∈N

= E [F0 ]

R2

1B(0;a) (x)D(|x|)λ dx  ∞ = E [F0 ] 2π uD(u)λ du < ∞ . R2

a

h?2 +2HH C(Xn , N ) Bb i?2 b2i Q7 TQBMib rBi?  dz;QQ/Ǵ `2+2TiBQM 7`QK i?2 Mi2MM Ui?2 2KBii2`V HQ+i2/ i Xn - i?i Bb- 7Q` r?B+? i?2 dzbB;MH@iQ@BMi2`72`2M+2@MQBb2@ `iBQǴ UbBM`V Bb H`;2` i?M i?2 i?`2b?QH/ T X h?2 bBM` Bb i?2 `iBQ Q7 i?2 bB;MH `2+2Bp2/ Fn D(|y − Xn |) 7`QK i?2 biiBQM iQ i?2 dzMQBb2Ǵ- +QMbBbiBM; Q7 bQK2 dz`2@ +2TiBQM 2H2+i`QMB+ MQBb2Ǵ W M/ Q7 i?2 dzBMi2`72`2M+2 MQBb2Ǵ 7`QK Qi?2` biiBQMb

k∈N ; k=n Fk D(|y − Xk |)X G2i PN0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? P M/ N X h?2 [mMiBiv PN0 (x ∈ C(0, N )) K2bm`2b i?2 [mHBiv Q7 `2+2TiBQM Q7  ;Bp2M `2+2TiQ` HQ+i2/ i /BbiM+2 |x| 7`QK bQK2 ;Bp2M Mi2MMX "v aHBpMvFǶb i?2Q`2K d

("++2HHB- "ƈbx+xvbxvM M/ JɃ?H2i?H2`- kyyj)X h?Bb KQ/2H rb BMi`Q/m+2/ BM  +QKKmMB+iBQMb +QMi2ti #v ("++2HHB M/ "ƈbx+xvbxvMkyyR)X 3

3X9X h>1 J1*E1 J1al_1

jRN 

PN0 (x ∈ C(0, N )) = P

W+

F0 D(|x|) >T k∈N Fk D(|x − Xk |)



= P (F0 > D(|x|)−1 T (W + I)) ,

r?2`2 I = k∈N Fk D(|x − Xk |)X LQr- bBM+2 F0 Bb M 2tTQM2MiBH `M/QK p`B#H2 rBi? K2M μ−1 M/ BM/2T2M/2Mi Q7 i?2 Fk Ƕb rBi? k ≥ 1  −1 P (F0 > D(|x|)−1 T (W + I)) = E e−μD(|x|) T (W +I)   −1 −1 = e−μD(|x|) T W E e−μD(|x|) T I . aBM+2 N Bb M ?TT E e−γI = E e−γ k Fk D(|x−Xk |)    −γF D(|x−y|)  E e − 1 λ dy = exp 2 R   −γF D(|y|)  = exp E e − 1 λ dy 2 R  = exp (LF (γD(|y|)) − 1) λ dy , R2

r?2`2 LF Bb i?2 GTH+2 i`Mb7Q`K Q7 F0 X SbbBM; iQ TQH` +QQ`/BMi2b

−γI

E e









= exp −2πλ

 u (LF (γD(u)) − 1) du

.

0

h?2`27Q`2- rBi? γ = μD(|x|)−1 T PN0 (x ∈ C(0, N )) = e−μD(|x|)

−1 T W

  exp −2πλ



 u (LF (D(|x|)−1 T D(u)) − 1) du

0

1pHmiBM; i?2 GTH+2 i`Mb7Q`K LF PN0 (x

∈ C(0, N )) = e

−μD(|x|)−1 T W







exp −2πλ 0



u du 1 + D(|x|)/(T D(u))

.

qBi? i?2 /2+v KQ/2H U3XRRV M/ i?2 ?vTQi?2bBb W = 0- i?2 +QKTmiiBQMb H2/ iQ   PN0 (x ∈ C(0, N )) = exp −λ|x|2 T 2/β K(β) , r?2`2 K(β) =

2πΓ(2/β)Γ(1 − 2/β) 2π 2 = . β β sin(2π/β)

.

jky

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

SHK E2`M2H Q7  SQBbbQM *Hmbi2` S`Q+2bb *QMbB/2` i?2 +Hmbi2` TQBMi T`Q+2bb Q7 1tKTH2 RX8Xk rBi? i?2 7QHHQrBM; bT2+BHBx@ iBQM, i?2 ;2`K T`Q+2bb N0 Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ0 X h?2Q`2K 3X9XR3 UN V lM/2` i?2 SHK K2bm`2 PN0 - i?2 /Bbi`B#miBQM Q7 i?2 +Hmbi2` TQBMi T`Q+2bb N Bb i?i Q7 i?2 biiBQM`v +Hmbi2` TQBMi T`Q+2bb iQ r?B+? QM2 ?b //2/  +Hmbi2` i 0 r?Qb2 /Bbi`B#miBQM Bb i?2 dzSHKǴ F2`M2H πZ0 i 0 Q7 i?2 ;2M2`B+ +Hmbi2` Z, 

1 πZ0 (Γ) = E 1 (Z − y) Z(dy) (Γ ∈ Mp (Rm )) . Γ νZ (Rm ) m R "27Q`2 i?2 T`QQ7- r2 b?HH ;Bp2 KQ`2 /2iBHb +QM+2`MBM; πZ0 X 6B`biHv- MQi2 i?i Bb MQi 2[mH iQ PZ0 - i?2 SHK F2`M2H i 0 Q7 ZX h?2 T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 MmK#2` T Q7 TQBMib Q7 Z mM/2` i?Bb T`Q##BHBiv Bb

πZ0

πZ0 (T = n) =

1 nP (Z(Rm ) = n) E[T ]

(n ≥ 0) .

UP#b2`p2 i?i πZ0 (T = 0) = 0- b 2tT2+i2/ 7`QK  dzSHKǴ F2`M2HXV hQ T`Qp2 i?BbBi bm{+2b iQ TTHv i?2 /2}MBiBQM Q7 πZ0 rBi? Γ := {μ ; μ(Rm ) = n}, 

1 m E 1 Z(dy) πZ0 (Z(Rm ) = n) = {(Z−y)(R )=n} νZ (Rm ) m R

1 m )=n} Z(dy) = E 1 {(Z)(R νZ (Rm ) Rm 1 1 = E 1{(Z)(Rm )=n} Z(Rm )) = E 1{(Z)(Rm )=n} n . m m νZ (R ) νZ (R ) .2MQi2 #v x1 , . . . , xn i?2 TQBMib Q7 Z r?2M T = nX q`Bi2 ∞  n    1 0 πZ (Γ) = E 1T =n E 1Γ (Z − xi ) | T = n E[T ] n=0 i=1  n  ∞  1 1 n 1T =n E 1Γ (Z − xi ) | T = n E = E[T ] n i=1 n=0  n  ∞  n 1 P (T = n)E 1Γ (Z − xi ) | T = n = E[T ] n i=1 n=0  n  ∞  1 0 πZ (T = n)E 1Γ (Z − xi ) | T = n . = n i=1 n=0 h?Bb +M #2 BMi2`T`2i2/ b 7QHHQrbX AM Q`/2` iQ ;2M2`i2  bKTH2 Q7 Z mM/2` T`Q##BHBiv πZ0 - +?QQb2 n rBi? T`Q##BHBiv πZ0 (T = n)- i?2M +`2i2 i?2 n TQBMib Q7 Z mM/2` i?2 T`Q##BHBiv P (Z ∈ · | T = n)X h?2M +?QQb2 QM2 Q7 i?2b2 TQBMib i `M/QK- bv U - M/ +QMbi`m+i i?2 /2bB`2/ bKTH2 b {x1 − U, x2 − U, . . . , xn − U 'X h?2 7Q`i?+QKBM; T`QQ7 Q7 h?2Q`2K 3X9XR3 KF2b mb2 Q7 i?2 7QHHQrBM; i2+?MB+H H2KKX N

(K#`ixmKBM- RNee)X

3X9X h>1 J1*E1 J1al_1

jkR

G2KK 3X9XRN G2i LZ,x #2 i?2 GTH+2 7mM+iBQMH Q7 i?2 HQ+HHv }MBi2 TQBMi T`Q+2bb Z mM/2` i?2 SHK F2`M2H PZ,x X .2MQi2 #v νZ Bib BMi2MbBiv K2bm`2- M/ #v LZ,x i?2 GTH+2 i`Mb7Q`K Q7 Z mM/2` PZ,x - i?i Bb   LZ,x (ϕ) := e− Rm ϕ(t) Z(dt) P (dω) . Ω

h?2M- 7Q` HH b ∈ Rm - HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb ϕ : Rm → R M/ HH B ∈ B(Rm )νZ @X2X LZ,x (ϕ) = LZ+b,x−b (ϕ(· − b)) , S`QQ7XRy qBi? i?2 +?M;2 Q7 p`B#H2b x → x + b LZ+b,x−b (ϕ(· − b)) 1B (x)νZ (dx) Rm  = LZ+b,x−b+b (ϕ(· − b)) 1B (x − b) νZ+b (dx) m R = LZ+b,x (ϕ(· − b)) 1B+b (x) νZ+b (dx) m R   = e Rm ϕ(t−b) (Z(ω)+b)(dt) PZ+b,x P (dω)1B+b (x) νZ+b (dx) . Rm

Ω

AM pB2r Q7 U3XeV- i?2 Hbi i2`K Q7 i?2 #Qp2 b2[m2M+2 Q7 2[mHBiB2b Bb 2[mH iQ

  e− Rm ϕ(t−b) (Z+b)(dt) 1B+b (x) (Z + b)(dx) E Rm 

 − Rm ϕ(t) Z(dt) =E e 1B (x) Z(dx) m  R = LZ,x (ϕ) 1B (x) νZ (dx) . Rm

h?2`27Q`2- 7Q` HH B ∈ B(Rm )  LZ+b,x−b (ϕ(· − b))νZ (dx) = LZ,x (ϕ) νZ (dx) , B

B

r?B+? BKTHB2b i?2 MMQmM+2/ `2bmHiX q2 `2 MQr `2/v 7Q` i?2 T`QQ7 Q7 h?2Q`2K 3X9XR3 S`QQ7X _2+HH 7Q`KmH URXkyV,  LN (ϕ) = exp Rm

      E e− Rm ϕ(y+x) Z(dy) − 1 λ0 dx .

1H2K2Mi`v +QKTmiiBQMb vB2H/ Ry

h?Bb T`QQ7 rb +QKKmMB+i2/ iQ K2 #v "`i2F "ƈbx+xvbxvMX



jkk −

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1 ∂LN (ϕ, f ) ∂f





= LN (ϕ)





f (y + x)Z(dy)

E Rm



ϕ(y + x)Z(dy)

exp

Rm

TTHvBM; 7Q`KmH U3XeV rBi?

  g(y, ω) := f (y + x) exp −

;Bp2b



 E

λ0 dx .

Rm

f (y + x)Z(dy) Rm  =

 ϕ(y + x) Z(ω, dy) Rm

  exp −



ϕ(y + x)Z(dy) Rm

f (y + x) LZ+y,y+x (ϕ)νZ (dy) .

Rm

h?2`27Q`2   ∂LN − (ϕ, f ) = LN (ϕ) f (y + x) LZ+x,y+x (ϕ)νZ (dy) λ0 dx ∂f m Rm  R  = LN (ϕ) f (y + x) LZ+x,y+x (ϕ)λ0 dx νZ (dy) . Rm

Rm

qBi? i?2 +?M;2 Q7 p`B#H2b x → u − y- M/ mbBM; i?2 2tT`2bbBQM λ = λ0 νZ (Rm ) 7Q` i?2 BMi2MbBiv Q7 N - r2 Q#iBM BM pB2r Q7 G2KK 3X9XRN    ∂LN − (ϕ, f ) = LN (ϕ) f (u) LZ+u−y,u (ϕ)λ0 du νZ (dy) ∂f m Rm

 R 1 = LN (ϕ) f (u) LZ+u−y,u (ϕ) νZ (dy) λ du ν (Rm ) Rm m Z

 R 1 = LN (ϕ) f (u) L (ϕ(· − y + u)) ν (dy) λ du Z,y Z ν (Rm ) Rm m Z

 R 1 = LN (ϕ) f (u) EZy eϕ(t−y+u)Z(dt) νZ (dy) λ du . m) ν (R m m Z R R "v 7Q`KmH U3XeV ;BM−

∂LN (ϕ, f ) = LN (ϕ) ∂f

 f (u) Rm

1 E νZ (Rm )

 e−

 Rm

ϕ(t−y+u)Z(dt)

Z(dy) λ du ,

Rm

7`QK r?B+? r2 /2/m+2- b BM 1tKTH2 3X9XN- i?i 7Q` G2#2b;m2 HKQbi HH u ∈ Rm 

 1 − Rm ϕ(t−y+u)Z(dt) LN,u (ϕ) = LN (ϕ) E e Z(dy) . νZ (Rm ) Rm AM T`iB+mH`- i?2`2 2tBbib M a ∈ Rm bm+? i?i i?2 #Qp2 Bb i`m2 7Q` u = a- M/ #v biiBQM`Biv 

 1 − Rm ϕ(t−y)Z(dt) LN,0 (ϕ) = LN (ϕ) E e Z(dy) . νZ (Rm ) Rm 

3X9X h>1 J1*E1 J1al_1

jkj

>B;?2` P`/2` J2+F2 J2bm`2b G2i E #2 M HX+X/X#X bT+2 rBi? "Q`2H σ@}2H/ B(E)X G2i (Ω, F) := (Mp (E), Mp (E))X G2i N #2  TQBMi T`Q+2bb QM E /2}M2/ QM (Ω, F)X Ai Bb bbmK2/ iQ #2 M m@i? Q`/2` bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM E- rBi? m@i? KQK2Mi K2bm`2b Mm X aBKBH`Hv iQ i?2 +b2 m = 1- r2 +M b?Qr i?i i?2 b2i 7mM+iBQM Cm (A × F ) := E N (m) (A)1F (A ∈ B(E)⊗m, F ∈ F ) mMB[m2Hv /2}M2b  σ@}MBi2 K2bm`2 Cm QM (E m × Ω, B(E)⊗m ⊗ F)- +HH2/ i?2 m@i? Q`/2` J2+F2 K2bm`2X .2MQi2 #v Cm,F i?2 K2bm`2 F → Cm (· × F )X aBKBH`Hv iQ i?2 mMBp`Bi2 +b2rBi? i?2 bK2 T`QQ7- QM2 b?Qrb i?2 2tBbi2M+2 Q7  _/QMĜLBFQ/ɷK /2`BpiBp2 dCm,F Px1 ,...,xm (F ) = (x1 , . . . , xm ) dMm /2}M2/ 7Q` Mm @HKQbi HH (x1 , . . . , xm ) ∈ E m X Ai biBb}2b  Px1 ,...,xm (F ) Mm (dx1 × · · · × dxm ) U3XRkV C(A × F ) = A

M/ BM T`iB+mH`-



Mm (A) = C(A × Ω) =

Px1 ,...,xm (Ω)Mm (dx1 × · · · × dxm ) . A

h?2`27Q`2 Px1 ,...,xm (Ω) = 1- Mm @X2X h?2 T`Q##BHBiv Px1 ,...,xm bQ /2}M2/ UMm @X2XV Bb +HH2/ i?2 m@i? Q`/2` SHK T`Q##BHBiv Q7 N i (x1 , . . . , xm ) ∈ E m X h?2 T`Q##BHBiB2b Px1 ,...,xm U(x1 , . . . , xm ) ∈ E m V +M #2 +?Qb2M BM bm+?  rv i?i 7Q` Mm @HKQbi HH (x1 , . . . , xm ) ∈ E m F → Px1 ,...,xm (F ) Bb  T`Q##BHBiv K2bm`2 QM (Ω, F) M/ 7Q` }t2/ F ∈ F (x1 , . . . , xm ) → Px1 ,...,xm (F ) Bb B(E)@K2bm`#H2 . h?2 T`Q##BHBiv F2`M2H 7`QK E m × Ω iQ (Ω, F)) /2}M2/ #v ((x1 , . . . , xm ), A) → Px1 ,...,xm (A) Bb +HH2/ i?2 m@i? Q`/2` SHK F2`M2H Q7 N X h?2Q`2K 3X9Xky 6Q` HH MQM@M2;iBp2 K2bm`#H2 g : E m × Ω → [0, ∞]   g(x1 , . . . , xm , ω)N (m) (ω, dx1 × · · · × dxm )P (dω) Ω En   = g(x1 , . . . , xm , ω)Px1 ,...,xm (dω) Mm (dx1 × · · · × dxm ) . En

Ω

U3XRjV h?2 T`QQ7 Q7 i?2 2ti2MbBQM Q7 h?2Q`2K 3X9XR9 #2HQr 7QHHQrb 7`QK i?2 bK2 `;mK2MibX h?2Q`2K 3X9XkR G2i P #2  T`Q##BHBiv i?i KF2b Q7 i?2 TQBMi T`Q+2bb N QM Rm  bBKTH2 SQBbbQM T`Q+2bb rBi? m@i? Q`/2` KQK2Mi K2bm`2 Mm X h?2M- 7Q` HH Γ ∈ M(Rm )Px1 ,...,xm (N ∈ Γ) = P (N + εx1 + · · · + εxm ∈ Γ) ,

Mm @X2X

U3XR9V

jk9

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

3X8 h?2 _2/m+2/ J2+F2 J2bm`2 AM i?Bb b2+iBQM- r2 rQ`F QM i?2 +MQMB+H K2bm`#H2 bT+2 (Mp (E), Mp (E)) QM r?B+? Bb ;Bp2M  T`Q##BHBiv K2bm`2 P X .2MQi2 #v N i?2 +QQ`/BMi2 KTTBM;, N (μ) := μX bbmK2 i?i N Bb bBKTH2 rBi?  σ@}MBi2 BMi2MbBiv K2bm`2 νX 6Q` Mv b2ib A ∈ B(E) M/ Γ ∈ Mp (E)- H2i 

C ! (A × Γ) := E 1Γ (N − εx ) N (dx) . U3XR8V A

_2+HH i?i i?2 MQiiBQM μ − εa - r?2`2 μ ∈ Mp (E) Bb bBKTH2 M/ a ∈ E- `2T`2b2Mib i?2 K2bm`2 μ B7 μ({a}) = 0 M/ i?2 K2bm`2 μ KBMmb i?2 mMBi Kbb i a Qi?2`rBb2X h?2`27Q`2- B7 {Xn }n∈N /2MQi2b i?2 TQBMi b2[m2M+2 Q7 N    ! 1A (Xn )1Γ (N \{Xn }) . () C (A × Γ) := E n∈N

b BM i?2 T`QQ7 Q7 G2KK 3X9XR- Bi +M #2 b?QrM i?i i?2 b2i 7mM+iBQM C ! /2}M2/ #v U3XR8V 7Q` b2ib Q7 i?2 7Q`K A × Γ- r?2`2 A ∈ B(E) M/ Γ ∈ Mp (E)+M #2 mMB[m2Hv 2ti2M/2/ iQ  σ@}MBi2 K2bm`2 C ! : B(E) ⊗ Mp (E) → [0, ∞]X h?2 K2bm`2 C ! : B(E) ⊗ Mp (E) → [0, ∞] bQ /2}M2/ Bb +HH2/ i?2 U}`bi@Q`/2`V `2/m+2/ J2+F2 K2bm`2X qBi?  T`QQ7 bBKBH` iQ i?i Q7 h?2Q`2K 3X9Xj- r2 ?p2 i?2 `2/m+2/ J2+F2 7Q`KmH, h?2Q`2K 3X8XR 6Q` HH MQM@M2;iBp2 K2bm`#H2 g : E × Mp (E) → [0, ∞]     g(x, μ − εx ) N (μ, dx) P (dμ) = g(x, μ) C ! (dx × dμ) . Mp (E)

E

E×Mp (E)

U3XReV AM i?2 MQiiBQM Q7 UV- i?2 H27i@?M/ bB/2 Q7 1[MX U3XReV Bb    g(Xn , N \{Xn }) . E n∈N

Ai 7QHHQrb 7`QK h?2Q`2K 3X8XR i?i 7Q` HH K2bm`#H2 g : E × Mp (E) → R bm+? i?i 2Bi?2` QM2 Q7 i?2 irQ 7QHHQrBM; +QM/BiBQMb,    |g(x, μ − εx )| N (μ, dx) P (dμ) < ∞ Mp (E)

Q`

E

  |g(x, μ)|C ! (dx × dμ) < ∞ , E×Mp (E)

Bb biBb}2/- i?2M i?2 Qi?2` Bb HbQ biBb}2/- M/ i?2 7Q`KmH BM h?2Q`2K 3X8XR Bb i`m2X

3X8X h>1 _1.l*1. J1*E1 J1al_1

jk8

6Q` 2+? Γ ∈ Mp (E)- /2MQi2 #v CΓ! i?2 K2bm`2 QM B(E) /2}M2/ #v CΓ! (A) = C (A × Γ)X b 7Q` i?2 J2+F2 K2bm`2 Ua2+iBQM 3X9V- r2 +M bb2`i i?2 2tBbi2M+2 Q7 i?2 _/QMĜLBFQ/ɷK /2`BpiBp2!

Px! (Γ) :=

dCΓ! (x) dν

U3XRdV

/2}M2/ 7Q` ν@HKQbi HH x ∈ EX Ai biBb}2b  ! Px! (Γ)ν(dx) C (A × Γ) =

U3XR3V

A

M/ Px! (Mp (E)) = 1- ν@X2X .2}MBiBQM 3X8Xk h?2 T`Q##BHBiv bQ /2}M2/ ν@X2X Bb +HH2/ i?2 `2/m+2/ SHK T`Q##BHBiv i x ∈ EX aBKBH`Hv iQ r?i rb /QM2 BM a2+iBQM 3X9 7Q` J2+F2Ƕb K2bm`2- r2 +M +?QQb2 7Q` HH Γ  p2`bBQM Q7 i?2 +QM/BiBQMH T`Q##BHBiv Px (Γ) bm+? i?i ν@HKQbi HH x ∈ E Px! (·) Bb  T`Q##BHBiv K2bm`2 QM (Mp (E), Mp (E)) M/ 7Q` }t2/ Γ

x → Px! (Γ) Bb B(E)@K2bm`#H2 .

h?2 `2/m+2/ J2+F2 7Q`KmH Q7 h?2Q`2K 3X8XR MQr `2/b       ! g(x, μ − εx ) μ(dx) P (dμ) = g(x, μ) Px (dμ) ν(dx) , Mp (E)

E

E

Mp (E)

7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQM g : E × Mp (E) → RX

U3XRNV

h?2Q`2K 3X8Xj h?2 `2/m+2/ J2+F2 K2bm`2 Px! M/ i?2 MQM@`2/m+2/ J2+F2 K2@ bm`2 Px `2 HBMF2/ #v i?2 7QHHQrBM; `2HiBQM, 7Q` HH Γ ∈ Mp (E)Px (Γ) = Px! ({μ ; μ + εx ∈ Γ})

ν@X2X

U3XkyV

S`QQ7X TTHv 6Q`KmH U3XRNV rBi? g(x, μ) := 1C (x)1Γ (μ + εx )- r?2`2 C ∈ B(E)X h?2 H27i@?M/ bB/2 2[mHb     1C (x)1Γ (μ) μ(dx) P (dμ) = Px (Γ) ν(dx) , Mp (E)

E

C

r?2`2b i?2 `B;?i@?M/ bB/2 2[mHb    1Γ (μ + εx ) Px! (dμ) C

 Px! ({μ ; μ + εx ∈ Γ}) ν(dx) .

ν(dx) =

Mp (E)

b Bi Bb i`m2 Q7 HH C ∈ B(E)- Bi BKTHB2b U3XkyVX

C



jke

*>Sh1_ 3X SGJ S_P""AGAhu AL aS*1

h?2Q`2K 3X8X9 6Q` i?2 T`Q##BHBiv P iQ KF2 Q7 i?2 TQBMi T`Q+2bb N  bBKTH2 SQBbbQM T`Q+2bb rBi? σ@}MBi2 MQM@iQKB+ BMi2MbBiv K2bm`2 ν- Bi Bb M2+2bb`v M/ bm{+B2Mi i?i 7Q` HH Γ ∈ Mp (Rm )Px! (N ∈ Γ) = P (N ∈ Γ)

dν@X2X

U3XkRV

S`QQ7X h?2 T`QQ7 Bb bBKBH` iQ i?i Q7 h?2Q`2K eXjXdX L2+2bbBiv, *QK#BM2 h?2@ Q`2Kb 3X8Xj M/ 3X9XR9X 6Q` i?2 bm{+B2M+v- Bi bm{+2b iQ T`Qp2 i?i 7Q` 2p2`v #QmM/2/ "Q`2H b2i A ⊂ Rm - N (A) Bb  SQBbbQM p`B#H2 rBi? K2M ν(A)X qBi? Γ := {μ ; μ(A) = n} Un ≥ 0V- i?2 H27i@?M/ bB/2 Q7 U3XkyV Bb  C (A × Γ) = E !



 1A (Xn )1{N (A)=n+1} = (n + 1)P (N (A) = n + 1) .

n∈N

h?2 `B;?i@?M/ bB/2 Q7 U3XR3V Bb- #v ?vTQi?2bBb U3XkRV P (N ∈ Γ)ν(dx) = ν(A)P (N ∈ Γ) = ν(A)P (N (A) = n) . A

h?2`27Q`2 (n + 1)P (N (A) = n + 1) = ν(A)P (N (A) = n) 7Q` HH n ≥ 0- 7`QK r?B+? r2 ?p2 ν(A)n P (N (A) = n) = P (N (A) = 0) . n! LQ`KHBxiBQM ;Bp2b P (N (A) = 0) = e−ν(A) X



*Q`QHH`v 3X8X8 G2i P #2  T`Q##BHBiv i?i KF2b Q7 i?2 TQBMi T`Q+2bb N M ?TT rBi? BMi2MbBiv λX h?2M- 7Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM f : Rm × Mp (Rm ) → R  E Rm

 f (x, N − εx )N (dx) = λ

E [f (x, N )] dx. Rm

1tKTH2 3X8Xe, _@BbQHi2/ SQBMib BM  SQBbbQM S`Q+2bbX G2i N #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM Rm Q7 BMi2MbBiv λ > 0X  TQBMi Xn ∈ N Bb +HH2/ R@BbQHi2/ B7 i?2`2 Bb MQ Qi?2` TQBMi Q7 N i /BbiM+2 7`QK Xn H2bb i?M Q` 2[mH iQ R > 0X AM Qi?2` i2`Kb- i?2 TQBMi T`Q+2bb N1 Q7 R@BbQHi2/ TQBMib Q7 N Bb /2}M2/ #v  N1 (A) := 1N (B(Xn ;R))=1 (A ∈ B(Rm )) . n∈N

q2 +QKTmi2 Bib BMi2MbBiv K2bm`2 ν1 ,

3X8X h>1 _1.l*1. J1*E1 J1al_1  ν1 (A) = E



jkd 

1A (Xn )1N (B(Xn ;R))=1

n∈N



R

1A (x)1μ(B(x;R))=1 N (dx)

=E m

1A (x)1μ(B(x;R))=1 N (dx) m  R  =λ 1A (x) 1μ(B(x;R))=1 Px (dμ) dx Rm M (Rm )  1A (x)P ((N + εx )(B(x; R)) = 0) dx =λ =E

Rm

= λ m (A)e−λ

m (B(0;R))

.

1tKTH2 3X8Xd, JmimH L2`2bi L2B;?#Qm`bXRR hrQ TQBMib Q7  bBKTH2 TQBMi T`Q+2bb N QM Rm `2 +HH2/ KmimH M2`2bi M2B;?#Qm`b UKMMV B7 2+? QM2 Bb i?2 +HQb2bi TQBMi Q7 i?2 Qi?2`X h?2 TB`b Q7 KmimH M2`2bi M2B;?#Qm`b `2 Q#pBQmbHv /BbDQBMiX  TQBMi Q7 N Bb +HH2/ BbQHi2/ UrBi? `2bT2+i iQ i?2 KMM T`QT2`ivV B7 Bi /Q2b MQi #2HQM; iQ M KMM TB`X

q2 MQr bmTTQb2 i?i N Bb mM/2` P M ?TT QM R2 Q7 BMi2MbBiv λX .2MQi2 #v x ↔ y i?2 bvKK2i`B+ `2HiBQM U/2T2M/BM; QM N V bvBM; i?i i?2`2 Bb MQ TQBMi Q7 N BM i?2 mMBQM Q7 QT2M #HHb B(x, |y − x|) ∪ B(y, |y − x|)X AM T`iB+mH`- 7Q` k = n- Xk ↔ Xn K2Mb i?i (Xk , Xn ) Bb  TB` Q7 KmimH M2`2bi M2B;?#Qm`bX q2 b22F M 2tT`2bbBQM 7Q`         E g(Xn , Xk )1Xn ↔Xk = E g(Xn , y)1Xn ↔y (N − εXn )(dy) . n

q`Bi2

n

k;k=n

 Rm

RR

Rm

g(Xn , y)1Xn ↔y (N − εXn )(dy) := f (Xn , N − εXn ) .

(:BQpMB/Bb- Hp`2x *Q``H2b M/ .2+`2mb27QM/- kyR9)X

jk3

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"v *Q`QHH`v 3X8X8 E







f (Xn , N − εXn = λ

E [f (x, N )] dx . R2

n



q2 +M r`Bi2 f (x, N ) =

R2

g(x, y)1x↔y (N − εy )(dy)

M/ i?2`27Q`2- #v *Q`QHH`v 3X8X8 ;BM 

 E [f (x, N )] = E g(x, y)1x↔y dy = λ R2

R2

g(x, y)E [1x↔y ] dy .

LQr 2

E [1x↔y ] = P (x ↔ y) = P (N (B(x, |y − x|) ∪ B(y, |y − x|) = 0) = e−Cλ(y−x) , √

r?2`2 C = 43 + 2π3 X 6BMHHv     E g(Xn , Xk )1Xn ↔Xk = n

k;k=n

 R2

2

g(x, y)e−Cλ(y−x) λ2 dx dy . R2

>B;?2` P`/2` _2/m+2/ J2+F2 J2bm`2 q2 rQ`F QM i?2 +MQMB+H K2bm`#H2 bT+2 (Mp (E), Mp (E)) QM r?B+? Bb ;Bp2M  T`Q##BHBiv K2bm`2 P X .2MQi2 #v N i?2 +QQ`/BMi2 TTHB+iBQM, N (μ) := μX bbmK2 i?i N Bb  bBKTH2 m@i? Q`/2` TQBMi T`Q+2bb rBi?  σ@}MBi2 m@i? KQK2Mi ! K2bm`2 Mm X 6Q` Mv b2ib A ∈ B(E)⊗m M/ Γ ∈ Mp (E)- H2i 

! (m)! Cm (A × Γ) := E 1Γ (N − εx1 − · · · − εxm ) N (dx1 × · · · × dxm ) . U3XkkV A ! b BM i?2 +b2 r?2`2 m = 1- Bi +M #2 b?QrM i?i i?2 b2i 7mM+iBQM Cm /2}M2/ ⊗m #v U3XkkV 7Q` b2ib Q7 i?2 7Q`K A × Γ- r?2`2 A ∈ B(E) M/ Γ ∈ Mp (E)- +M ! #2 mMB[m2Hv 2ti2M/2/ iQ  σ@}MBi2 K2bm`2 Cm : B(E)⊗m ⊗ Mp (E) → [0, ∞]X ! ⊗m h?2 K2bm`2 Cm : B(E) ⊗ Mp (E) → [0, ∞] bQ /2}M2/ Bb +HH2/ i?2 m@i?@Q`/2` `2/m+2/ J2+F2 K2bm`2X qBi?  T`QQ7 bBKBH` iQ i?i Q7 h?2Q`2K 3X8XR- r2 ?p2 i?2 `2/m+2/ m@i?@Q`/2` J2+F2 7Q`KmH,

h?2Q`2K 3X8X3 6Q` HH MQM@M2;iBp2 K2bm`#H2 g : E m × Mp (E) → [0, ∞]   (m)! g(x, μ − εx1 − · · · − εxm ) N (μ, dx1 × · · · × dxm ) P (dμ) Mp (E) E   ! g(x1 , . . . , xm , μ) Cm (dx1 × · · · × dxm × dμ) . = E m ×Mp (E)

3X8X h>1 _1.l*1. J1*E1 J1al_1

jkN

! 6Q` 2+? Γ ∈ Mp (E)- /2MQi2 #v Cm,Γ i?2 K2bm`2 QM B(E) /2}M2/ #v CΓ! (A) = C (A×Γ)X b BM i?2 +b2 m = 1- r2 +M bb2`i i?2 2tBbi2M+2 Q7 i?2 _/QMĜLBFQ/ɷK /2`BpiBp2!

Px! 1 ,...,xm (Γ) :=

! dCm,Γ (x1 , . . . , xm ) dM!m

U3XkjV

! @HKQbi HH x1 , . . . , xm ∈ EX Ai biBb}2b /2}M2/ 7Q` Mm  ! (A × Γ) = Px! 1 ,...,xm (Γ)M!m (dx1 × · · · × dxm ) Cm

U3Xk9V

A

M/ Px! 1 ,...,xm (Mp (E)) = 1- M!m @X2X .2}MBiBQM 3X8XN h?2 T`Q##BHBiv Px! 1 ,...,xm QM (Mp (E), Mp (E)) bQ /2}M2/ M!m @ X2X Bb +HH2/ i?2 `2/m+2/ m@i? Q`/2` SHK T`Q##BHBiv i (x1 , . . . , xm ) ∈ E m X aBKBH`Hv iQ i?2 +b2 m = 1- r2 +M +?QQb2 7Q` HH Γ  p2`bBQM Q7 i?2 T`Q##BHBiv ! @HKQbi HH (x1 , . . . , xm ) ∈ E m Px! 1 ,...,xm (Γ) bm+? i?i 7Q` Mm Px! 1 ,...,xm (·) Bb  T`Q##BHBiv K2bm`2 QM (Mp (E), Mp (E)) M/ 7Q` }t2/ Γ

x → Px! 1 ,...,xm (Γ) Bb B(E)⊗m @K2bm`#H2 .

h?2 `2/m+2/ J2+F2 7Q`KmH Q7 h?2Q`2K 3X8XR MQr `2/b    g(x1 , . . . , xm , μ − εx1 − · · · − εxm ) μ(dx) P (dμ) Mp (E) E    = E

Mp (E)

g(x1 , . . . , xm , μ) Px! 1 ,...,xm (dμ)

! Mm (dx1 × · · · × dxm ) ,

7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb g : E × Mp (E) → RX h?2 `2bmHi 2ti2M/BM; h?2Q`2K 3X8Xj Bb, h?2Q`2K 3X8XRy h?2 `2/m+2/ SHK K2bm`2 Px! 1 ,...,xm M/ i?2 MQM@`2/m+2/ SHK K2bm`2 Px1 ,...,xm `2 HBMF2/ #v i?2 7QHHQrBM; `2HiBQM, 7Q` HH Γ ∈ Mp (E)Px1 ,...,xm (Γ) = Px! 1 ,...,xm ({μ ; μ + εx1 + · · · + εxm ∈ Γ})

! Mm @X2X

U3Xk8V

aHBpMvFǶb i?2Q`2K Uh?2Q`2K 3X8X9V 2ti2M/b iQ, h?2Q`2K 3X8XRR 6Q` i?2 T`Q##BHBiv P iQ KF2 Q7 i?2 m@i? Q`/2` TQBMi T`Q+2bb N  bBKTH2 SQBbbQM T`Q+2bb rBi? MQM@iQKB+ BMi2MbBiv K2bm`2 ν- Bi Bb M2+2bb`v M/ bm{+B2Mi i?i 7Q` HH Γ ∈ Mp (Rm )Px! 1 ,...,xm (N ∈ Γ) = P (N ∈ Γ)

dν@X2X

_2K`F 3X8XRk h?2Q`2K 3X8XRR 7Q`KHBx2b i?2 `2bmHi Q7 1t2`+Bb2 jX3XkX

U3XkeV

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1tpBbB#H2 AMi2MbBiv,  :HBKTb2 i i?2 :2M2`H *b2 h?Bb iQTB+ rb H`2/v i`2i2/ BM *?Ti2` e 7Q` }MBi2 TQBMi T`Q+2bb2bX h?2 ;2M2`H +b2 Bb KQ`2 BMpQHp2/ M/ r2 b?HH QMHv T`QpB/2 ?2m`BbiB+b- bbmKBM; i?i i?2 `2/m+2/ J2+F2 K2bm`2 C ! Bb #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ m ⊗ P M/  μ)X h?2M /2MQiBM; i?2 +Q``2bTQM/BM; _/QMĜLBFQ/ɷK /2`BpiBp2 #v λ(x,      μ)P (dμ) dx g(x, μ − εx ) μ(dx) P (dμ) = g(x, μ)λ(x, Rm

Rm

Mp (Rm )

Q`- BM i?2 Qi?2` MQiiBQM    g(x, N − εx ) = E x∈N

Mp (Rm )

   N )g(x, N ) dx . E λ(x, Rm

AM T`iB+mH`- rBi? g(x, μ) = 1B (x) UB ∈ Rm V    N ) dx , E λ(x, ν(B) = B

r?B+? BKTHB2b i?i ν ?b  /2MbBiv UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2V ;Bp2M #v     N) . λ(x) := E λ(x, h?2 `2/m+2/ J2+F2 7Q`KmH i?2`27Q`2 `2/b      dx , g(x, N − εx ) = E!x [g(x, N )] λ(x) E Rm

x∈N

M/ i?2`27Q`2 7Q` HKQbi HH UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2V x ∈ Rm    N) λ(x, , U3XkdV E!x [g(x, N )] = E g(x, N )  λ(x) r?B+? bvb i?i 7Q` HKQbi HH x ∈ Rm λ(x, N ) :=

 N) λ(x,  λ(x)

U+HH2/ i?2 2tpBbB#H2 BMi2MbBiv i TQBMi x Q7 i?2 TQBMi T`Q+2bb N V Bb i?2 _/QMĜ LBFQ/ɷK /2`BpiBp2 Q7 Px! rBi? `2bT2+i iQ P X h?2 bK2 ?2m`BbiB+ `;mK2Mi b i?2 QM2 7Q` }MBi2 TQBMi T`Q+2bb2b b?Qrb i?i λ(x, N ) dx = P (N (dx) > 0 | N (· ∩ dx)) , λ(x) i?2 +QM/BiBQMH T`Q##BHBiv i?i i?2`2 Bb  TQBMi Q7 N BM M BM}MBi2bBKH M2B;?#Qm`@ ?QQ/ dx Q7 x- ;Bp2M i?2 HQ+iBQM Q7 HH TQBMib Q7 N QmibB/2 i?Bb M2B;?#Qm`?QQ/X _2K`F 3X8XRj _2+HH i?2 HBMF rBi? i?2 K`iBM;H2 TT`Q+? Q#b2`p2/ BM *?T@ i2` eX a22 HbQ 1t2`+Bb2b 3XeX3 M/ 3XeXNX

3XeX 1s1_*Aa1a

3Xe

jjR

1t2`+Bb2b

1t2`+Bb2 3XeXRX MQi?2` 7Q`K Q7 i?2 BMp2`bBQM 7Q`KmH AM i?2 mMB/BK2MbBQMH +b2- b?Qr ?Qr iQ Tbb 7`QK i?2 BMp2`bBQM 7Q`KmH Q7 1t@ KTH2 3XRXe iQ i?2 BMp2`bBQM 7Q`KmH BM h?2Q`2K dXjX9X 1t2`+Bb2 3XeXkX p2`;2 pQHmK2 Q7 i?2 oQ`QMQŖ +2HH pb BMi2MbBivX G2i N #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb rBi? BMi2MbBiv 0 < λ < ∞X a?Qr i?i mM/2` i?2 SHK T`Q##BHBiv- i?2 2tT2+i2/ pQHmK2 Q7 i?2 oQ`QMQś +2HH V0 Bb 2[mH iQ i?2 BMp2`b2 BMi2MbBivX 1t2`+Bb2 3XeXjX 62HH2`ǰb T`/Qt G2i N #2  bBKTH2 biiBQM`v TQBMi T`Q+2bb rBi? BMi2MbBiv 0 < λ < ∞X G2i W (0, N ) #2 i?2 oQ`QMQB +2HH +2Mi2`2/ i  TQBMi Q7 N +Qp2`BM; 0 Ui?Bb +2HH Bb mMB[m2- #v G2KK 3XRX9VX a?Qr i?i 7Q` Mv MQM@M2;iBp2 7mM+iBQM g Q7 W0 (N )E [g(W (0, N ))] = λE0N [g(V0 (N ))pQH(V0 (N ))] M/ BM T`iB+mH` E pQH(W (0, N ))−1 = E0N [pQH(V0 (N ))]−1 . a?Qr i?i E [pQH(W (0, N ))] ≥ E0N [pQH(V0 (N ))] . Uh?Bb Bb i?2 bTiBH p2`bBQM Q7 62HH2`Ƕb T`/Qt- Q` #mb T`/Qt XV 1t2`+Bb2 3XeX9X SHK /Bbi`B#miBQM Q7 i?2 BMi2MbBiv Q7  KBt2/ SQBbbQM T`Q+2bb G2i N #2  ?QKQ;2M2Qmb *Qt T`Q+2bb Q7 BMi2;`#H2 +QM/BiBQMH BMi2MbBiv ΛX S`Qp2 i?i E Λ1{Λ≤λ} Px (Λ ≤ λ) = . E [Λ] 1t2`+Bb2 3XeX8X .2MQi2 #v Nx1 ,...,xn M/ Nx! 1 ,...,xn TQBMi T`Q+2bb2b QM Rnm rBi? i?2 /Bbi`B#miBQMb Px1 ,...,xn M/ Px! 1 ,...,xn `2bT2+iBp2HvX bbmK2 MNn+m iQ #2 σ@}MBi2X 6Q` A ∈ B(E)n M/ B ∈ B(E)m   MNn+m (A × B) = MNx!m,...,xn (dx1 , . . . , dxm )MNn (dx1 , . . . , dxn ) . A

B

1

! (A × B)X >BMi, *QKTmi2 Cn+m

1t2`+Bb2 3XeXeX LQM@R@BbQHi2/ TQBMib Q7  SQBbbQM T`Q+2bb AM 1tKTH2 3X8Xe- b?Qr i?i i?2 iQiH MmK#2` Q7 MQM@R@BbQHi2/ TQBMib Bb HKQbi@ bm`2Hv BM}MBi2X U"2r`2,  MQM@M2;iBp2 }MBi2 `M/QK p`B#H2 Kv ?p2 M BM}MBi2 K2MXV

jjk

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1t2`+Bb2 3XeXdX >B;?2bi /2MbBiv Q7 R@BbQHi2/ TQBMib Q7  SQBbbQM T`Q@ +2bb AM 1tKTH2 3X8Xe- }M/ i?2 BMi2MbBiv λ Q7 i?2 ?TT i?i KtBKBx2b i?2 BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb Q7 R@BbQHi2/ TQBMibX *QKT`2 i?Bb rBi? i?2 BMi2MbBiv Q7 i?2 TQBMi T`Q+2bb Q7 R@BbQHi2/ TQBMib BM i?2 bim`i2/ Jiû`M KQ/2H Q7 1tKTH2 3X9XReX 1t2`+Bb2 3XeX3X SHK S`Q##BHBiv M/ biQ+?biB+ BMi2MbBiv G2i N #2  HQ+HHv }MBi2 bBKTH2 TQBMi T`Q+2bb QM i?2 HBM2 rBi? i?2 Ft @BMi2MbBiv {λ(t)}t∈R X G2i A ∈ Fa X S`Qp2 i?i 7Q` G2#2b;m2 HKQbi HH t ≥ aPt (F ) =

E [1F λ(t)] , λ(t)

r?2`2 λ(t) := E [λ(t)]X 1t2`+Bb2 3XeXNX J`iBM;H2b BM/2t2/ #v b2ib a?Qr i?i mM/2` TT`QT`Bi2 +QM/BiBQMb Ub22 h?2Q`2K eXjX8V   g(x, μ − εx )μ(dx) − E!x [g(x, μ)] ν(dx) MA (g) := A

Bb bm+? i?i

A

E [MA (g) | FA ] = 0 ,

r?2`2 7Q` Mv b2i C ⊆ B(E)- FC := σ(N ∩ C)X aT2+BHBx2 iQ i?2 +b2 r?2`2 N Bb M ?TT Q7 BMi2MbBiv λX

*?Ti2` N h?2 SQr2` aT2+i`H J2bm`2 h?2 MQiBQM Q7 TQr2` bT2+i`H K2bm`2 7Q` rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb2bR Bb  Mim`H 2ti2MbBQM Q7 i?2 MHQ;Qmb MQiBQM 7Q` rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb2b- i?2 "Q+?M2` bT2+i`H K2bm`2X AM i?Bb +?Ti2`- T`iB+mH` ii2MiBQM rBHH #2 TB/ iQ i?2 TQr2` bT2+i` Q7 +QKTH2t bB;MHb #b2/ QM TQBMi T`Q+2bb2b bm+? b i?Qb2 `BbBM; BM +QKKmMB+iBQMb BM i?2 T`2b2M+2 Q7 `2~2+iBQMb 7Q` BMbiM+2Xk h?2 KBM `2bmHi rBi? `2bT2+i iQ i?Bb ;QH Bb  bBKTH2 mMBp2`bH 7Q`KmH 7`QK r?B+?  MmK#2` Q7 bT2+B}+ TQr2` bT2+i` +M #2 Q#iBM2/ i HBiiH2 2tT2Mb2X

NXR

h?2 *Qp`BM+2 J2bm`2

aBM+2 i?2 TQr2` bT2+i`H i?2Q`v Q7 TQBMi T`Q+2bb2b BKBii2b i?2 +HbbB+H i?2Q`v +QM+2`MBM; rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb2b- r2 b?HH iF2 bQK2 iBK2 iQ `2pB2r i?2 #bB+ `2bmHib Q7 i?2 Hii2` i?2Q`vX qB/2@a2Mb2 aiiBQM`v aiQ+?biB+ S`Q+2bb2b  72r `2bmHib +QM+2`MBM; i?2 bT2+i`H MHvbBb Q7 rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb2b rBHH #2 `2+HH2/ BM pB2r Q7 +QKT`BbQM rBi? i?2 MHQ;Qmb QM2b 7Q` rB/2@ b2Mb2 biiBQM`v TQBMi T`Q+2bb2bX h?2B` T`QQ7b `2 QKBii2/ bBM+2 i?2v +M #2 B;MQ`2/ 7Q` i?2 /2p2HQTK2Mi Q7 i?2 i?2Q`v Q7 i?Bb +?Ti2` U2t+2Ti 7Q` h?2Q`2K RyX8XR BM i?2 M2ti +?Ti2`VX  rB/2@b2Mb2 biiBQM`v UrbbV biQ+?biB+ T`Q+2bb {X (t)}t∈Rm Bb- #v /2}MBiBQM-  b2+QM/@Q`/2` biQ+?biB+ T`Q+2bb Ui?i Bb- E |X(t)|2 < ∞ 7Q` HH t ∈ Rm V bm+? i?i E [X(t)] = μ UBM/2T2M/2Mi Q7 tV M/ +Qp (X(t + τ ), X(t)) = CX (τ ) UBM/2T2M/2Mi Q7 tVX h?2 7mM+iBQM CX Bb +HH2/ i?2 +Qp`BM+2 7mM+iBQMX  h?2Q`2K NXRXR 6Q` HH ϕ ∈ L1C (Rm )- i?2 BMi2;`H Rm ϕ(t)X(t) dt Bb r2HH /2}M2/ M/ BM L1C (Rm ) ∩ L2C (Rm )X HbQ- 7Q` HH 7mM+iBQMb ϕ, ψ ∈ L1C (Rm )   +Qp ϕ(t)X(t) dt, ψ(s)X(s) ds Rm Rm    = ϕ(t)ψ ∗ (t + s) dt CX (s) ds . UNXRV Rm

Rm

R

("`iH2ii- RNej)X k (_B/QH}- RNNN)X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_9

jjj

jj9

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 Uj V _2+HH i?2 /2}MBiBQM Q7 i?2 6Qm`B2` i`Mb7Q`K ϕ Q7  7mM+iBQM ϕ ∈ L1C (Rm ),  ϕ(t)e−2iπν,t h(t) dt . ϕ(ν) := Rm

h?2Q`2K NXRXk lM/2` i?2 //BiBQMH bbmKTiBQM Q7 +QMiBMmBiv BM i?2 [m/`iB+ K2M U2[mBpH2Mi iQ +QMiBMmBiv Q7 i?2 +Qp`BM+2 7mM+iBQMV i?2`2 2tBbib U"Q+?M2`Ƕb i?2Q`2KV  mMB[m2 }MBi2 K2bm`2 μ QM Rm - +HH2/ i?2 TQr2` bT2+i`H K2bm`2 Q7 i?Bb rbb biQ+?biB+ T`Q+2bb- bm+? i?i  CX (τ ) = e2iπν,τ  μX (dν) (τ ∈ Rm ) . Rm

HbQ- 7Q` HH ϕ ∈ L1C (Rm ) rBi? 6Qm`B2` i`Mb7Q`K ϕ 2 

 2 ϕ(t)X(t) dt |ϕ(ν)| μ(dν) . =E E Rm

Rm

UNXkV

h?2Q`2K NXRXj G2i {X (t)}t∈Rm #2  rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb +QM@ iBMmQmb BM i?2 [m/`iB+ K2M rBi? TQr2` bT2+i`H K2bm`2 μX X G2i h(t) Ut ∈ Rm V #2  /2i2`KBMBbiB+ 7mM+iBQM BMi2;`#H2 rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 QM R M/ /KBiiBM; i?2 6Qm`B2` i`Mb7Q`K hX h?2M- i?2 biQ+?biB+ T`Q+2bb  Y (t) := h(t − s)X(s) ds (t ∈ Rm ) Rm

Bb  r2HH@/2}M2/ rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb +QMiBMmQmb BM i?2 [m/`iB+ K2M M/ rBi? TQr2` bT2+i`H K2bm`2 & &2 & & μY (dν) = &h(ν)& μX (dν) .

h?2Q`2K NXRX9 G2i {X(t)}t∈R #2  +2Mi2`2/ rbb biQ+?biB+ T`Q+2bb- +QMiBM@ mQmb BM [m/`iB+ K2M- M/ H2i μ #2 Bib TQr2` bT2+i`H K2bm`2X h?2`2 2t@ Bbib  mMB[m2 UKQ`2 T`2+BbBQM #2HQr i?2 i?2Q`2KV +2Mi2`2/ biQ+?biB+ T`Q+2bb {x(ν)}ν∈R rBi? mM+Q``2Hi2/ BM+`2K2Mib M/ rBi? bi`m+im`H K2bm`2 μ Ui?i BbE [|x(ν1 ) − x(ν2 )|2 ] = μ((ν1 , ν2 ])V- bm+? i?i P @XbX X(t) = e2iπνt dx(ν) (t ∈ R) , UNXjV R

r?2`2 i?2 BMi2;`H Q7 i?2 `B;?i@?M/ bB/2 Bb  .QQ# BMi2;`HX h?2 /2+QKTQbBiBQM UNXjV Bb i?2 *`Kû`ĜE?BM+?BM /2+QKTQbBiBQMX q2 MQr im`M iQ i?2 bT2+i`H i?2Q`v Q7 TQBMi T`Q+2bb2bX j 6Q`  T`QQ7 Q7 i?2 #Qp2 `2bmHi M/ Q7 i?2 M2ti QM2b- b22 7Q` BMbiM+2 ("`ûKm/- kyR9- `2bTX kyky)- *?TX j- `2bTX RkX

NXRX h>1 *Po_AL*1 J1al_1

jj8

aTBF2 h`BMb  TQBMi T`Q+2bb N QM R rBi? b2[m2M+2 Q7 2p2Mi iBK2b {Tn }n∈Z +M #2 `2T`2b2Mi2/ bvK#QHB+HHv #v i?2 dz`M/QK .B`+ +QK#Ǵ  X(t) := δ(t − Tn ), UNX9V n∈Z

i?2 bmK i?2`2BM 2ti2M/BM;- ++Q`/BM; iQ i?2 mbmH +QMp2MiBQM- iQ HH 2p2Mi iBK2b i }MBi2 /BbiM+2X h?2 `B;?i@?M/ bB/2 Bb +2`iBMHv MQi  #QM }/2 biQ+?biB+ T`Q+2bb bBM+2 i?2 /2Hi 7mM+iBQM Bb MQi  7mM+iBQM BM i?2 Q`/BM`v b2Mb2- #mi  /Bbi`B#miBQMX AM T`iB+mH`- QM2 +MMQi /2}M2 7Q` i?2 `M/QK .B`+ +QK# bbQ+Bi2/ rBi?  biiBQM`v TQBMi T`Q+2bb  TQr2` bT2+i`H K2bm`2 b BM i?2 +b2 Q7 rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb2bX h?2 *Qp`BM+2 J2bm`2 Q7  SQBMi S`Q+2bb _2+HH  72r /2}MBiBQMb M/ 7+ib 7`QK a2+iBQM RXkX  TQBMi T`Q+2bb N QM Rm Bb +HH2/  b2+QM/@Q`/2` TQBMi T`Q+2bb B7 7Q` HH +QKT+i b2ib C ⊂ Rm E N (C)2 < ∞ . h?Bb BKTHB2b BM T`iB+mH` i?i i?2 BMi2MbBiv K2bm`2 ν Bb HQ+HHv }MBi2X h?2 b2+QM/ KQK2Mi K2bm`2 M2 Bb  HQ+HHv }MBi2 K2bm`2 QM Rm × Rm , B(Rm ) ⊗ B(Rm ) i?i Bb r2HH M/ mMB[m2Hv /2}M2/ #v Bib pHm2b QM i?2 K2bm`#H2 `2+iM;H2b, M2 (A × B) = E [N (A) N (B)]

(A, B ∈ B(Rm )) .

6Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb g : Rm × Rm → R     g(t, s) N (dt)N (ds) = g(t, s) M2 (dt × ds) , E Rm

Rm

Rm

Rm

 7Q`KmH r?B+? 2ti2M/b iQ HH K2bm`#H2 7mM+iBQMb g : Rm × Rm → C bm+? i?i   |g(t, s)| M2 (dt × ds) < ∞ . Rm

Rm

AM T`iB+mH`- B7 ϕ, ψ ∈ L2N (M2 )  ∗   = ϕ(t) N (dt) ψ(t)N (dt) E Rm

Rm

 Rm

Rm

ϕ(t)ψ ∗ (s)M2 (dt × ds) . UNX8V

.2}MBiBQM NXRX8 .2}M2 L2N (M2 ) iQ #2 i?2 +QHH2+iBQM Q7 K2bm`#H2 7mM+iBQMb ϕ : R → C bm+? i?i   |ϕ(t)ϕ(s)| M2 (dt × ds) < ∞ Rm

Q`- 2[mBpH2MiHv-

Rm

E N (|ϕ|)2 < ∞ .

UNXeV

jje

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

h?2Q`2K NXRXe L2N (M2 ) Bb  p2+iQ` bT+2 i?i +QMiBMb HH #QmM/2/ 7mM+iBQMb rBi? +QKT+i bmTTQ`iX AM 7+i- r2 ?p2 i?2 7QHHQrBM; BM+HmbBQM, L2N (M2 ) ⊆ L1C (ν) ∩ L2C (ν) .

UNXdV

S`QQ7X AM2[mHBiv UNXeV BKTHB2b BM T`iB+mH` i?i E [N (|ϕ|)] < ∞ M/ i?2`27Q`2 U*KT#2HHǶb 7Q`KmHV ϕ ∈ L1C (ν)X aBM+2  2   2 |ϕ(Xn )| ≤ |ϕ(Xn )| , n∈N 2 B7  ϕ ∈ LN2(M2 )- i?2M E |ϕ(t)| ν(dt) < ∞X Rm

n∈N





Rm

|ϕ(t)|2 N (dt) < ∞- i?i Bb- #v *KT#2HHǶb 7Q`KmH

.2}MBiBQM NXRXd  rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb Bb- #v /2}MBiBQM-  b2+QM/@ Q`/2` TQBMi T`Q+2bb N bm+? i?i 7Q` HH b2ib C ∈ B(Rm ) M/ HH t ∈ Rm E [N (C + t)] = E [N (C)] , M/ bm+? i?i 7Q` HH #QmM/2/ A, B ∈ B(Rm ) M/ HH t ∈ Rm E [N (A + t)N (B + t)] = E [N (A)N (B)] . h?2 Hbi `2[mB`2K2Mi BKTHB2b BM T`iB+mH` i?i 7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb ϕ, ψ : Rm → R- i?2 [mMiBiv    

E ϕ(τ + t) N (dt) ψ(τ + t) N (dt) R

R

Bb BM/2T2M/2Mi Q7 τ ∈ RX U1t2`+Bb2 NX8XjXV G2i N #2  rB/2@b2Mb2 biiBQM`v bBKTH2 M/ HQ+HHv }MBi2 TQBMi T`Q+2bb QM Rm X AM T`iB+mH`- Bib BMi2MbBiv K2bm`2 ν Bb BMp`BMi mM/2` i`MbHiBQM M/ i?2`27Q`2 Uh?2Q`2K XRX3V ν (C) = λ m (C) U m Bb i?2 G2#2b;m2 K2bm`2 QM Rm V 7Q` bQK2 λ ∈ R+ - +HH2/ i?2 BMi2MbBivX AM T`iB+mH` LpC (ν) = LpC (Rm ) M/ i?2`27Q`2- #v h?2Q`2K NXRXeL2N (M2 ) ⊆ L1C (Rm ) ∩ L2C (Rm ) .

UNX3V

"v biiBQM`Biv ;BM- 7Q` HH b2ib A, B ∈ B(Rm ) M/ HH t ∈ Rm M2 ((A + t) × (B + t)) = M2 (A × B) . Ai 7QHHQrb 7`QK i?Bb M/ h?2Q`2K XRXR9- i?i 7Q` HH ϕ, ψ ∈ L2N (M2 )     ϕ (t) ψ ∗ (s) M2 (dt × ds) = ϕ (t) ψ ∗ (s + t) dt σ (ds) Rm

Rm

7Q` bQK2 HQ+HHv }MBi2 K2bm`2 σX

Rm

Rm

UNXNV

NXRX h>1 *Po_AL*1 J1al_1

jjd

h?2Q`2K NXRX3 6Q` HH ϕ, ψ ∈ L2N (M2 ) +Qp

  ϕ (t) N (dt) , ψ (s) N (ds) m Rm   R = ϕ (t) ψ ∗ (t + s) dt ΓN (ds) Rm

Rm

UNXRyV

r?2`2 i?2 HQ+HHv }MBi2 K2bm`2 ΓN := σ − λ2 m

UNXRRV

Bb +HH2/ i?2 +Qp`BM+2 K2bm`2 Q7 i?2 biiBQM`v b2+QM/@Q`/2` TQBMi T`Q+2bb N X

S`QQ7X ai`iBM; 7`QK UNXNV Bi bm{+2b iQ Q#b2`p2 i?i 7Q` ϕ, ψ ∈ L1C (Rm )     ∗ E [N (ϕ)] E [N (ψ)] = λ ϕ (t) dt λ ψ (s) ds m Rm   R  = λ2 ϕ (t) ψ ∗ (t + s) dt ds . ∗

Rm

Rm

 LQi2 i?2 7Q`KH `2b2K#HM+2 Q7 1[Mb UNXRV M/ UNXRyVX 1tKTH2 NXRXN, h?2 *Qp`BM+2 J2bm`2 Q7 i?2 >QKQ;2M2Qmb SQBbbQM S`Q+2bbX G2i N #2  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM i?2 HBM2 rBi? BMi2MbBiv λ > 0X "v i?2 +Qp`BM+2 7Q`KmH UjX9V +Qp (N (ϕ), N (ψ)) = λ

ϕ (t) ψ ∗ (t) dt .

R

i 0- r2 ?p2 7Q` i?2 `B;?i@?M/ bB/2 Q7 i?2 Hbi .2MQiBM; #v ε0 i?2 .B`+ K2bm`2  /BbTHv i?2 2tT`2bbBQM λ R R ϕ (t) ψ ∗ (t + s) dt ε0 (ds) M/ i?2`27Q`2- +QKT`BM; rBi? i?2 `B;?i@?M/ bB/2 Q7 UNXRVΓN = λε0 .

1tKTH2 NXRXRy, h?2 *Qp`BM+2 J2bm`2 Q7 i?2 _2M2rH S`Q+2bbX G2i N #2  biiBQM`v `2M2rH TQBMi T`Q+2bb rBi? `2M2rH 7mM+iBQM R Ui?2 bK2 MQiiBQM rBHH #2 mb2/ 7Q` i?2 bbQ+Bi2/ `2M2rH K2bm`2VX 6Q` Mv K2bm`#H2 MQM@M2;iBp2 7mM+iBQMb ϕ, ψ : R+ → R-

jj3

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 

  

ϕ(t) N (dt) ψ(t) N (dt) R R     =E ϕ(Tn ) ψ(Tn + (Tn+m − Tn ))

E

m≥n

n∈Z





+E

=







E ϕ(Tn )

ψ(Tn )



+



 ϕ(Tn + (Tn+m − Tn ))

m>1

n∈Z



ψ(Tn + (Tn+m − Tn ))

m≥1

n∈Z









E ψ(Tn )



 ϕ(Tn + (Tn+m − Tn ))

.

m>1

n∈Z

*HH A M/ B i?2 bmKb QM i?2 `B;?i@?M/ bB/2X aBM+2 {(Tn+m − Tn )}m≥1 Bb M mM@ /2Hv2/ `2M2rH T`Q+2bb BM/2T2M/2Mi Q7 Tn M/ rBi? i?2 bK2 BMi2`pH /Bbi`B#miBQM b i?2 Q`B;BMH `2M2rH T`Q+2bb- TTHvBM; *KT#2HHǶb 7Q`KmH irB+2 ;Bp2b      A= E ϕ(Tn )E ψ(Tn + (Tn+m − Tn )) | Tn n∈Z

=



E ϕ(Tn )



ψ(t) R(dt + Tn ) R

n∈Z



=E

m≥1





ϕ(Tn )

n∈Z



ψ(t) R(dt + Tn )



R

ϕ(s)

=E R

aBKBH`Hv

R ϕ(s)

=E







ψ(t) R(dt + s) λ ds 

ψ(s + t) R(dt) λ ds .

R

[s,+∞)







ψ(s)

B=E R

R



ϕ(s + t) R(dt) λ ds . (s,+∞)

6BMHHv   

  ϕ(t) N (dt) ψ(t) N (dt) =E E R







ϕ(s) Rm



ψ(s + t) R(dt) λ ds , R

7`QK r?B+? Bi 7QHHQrb i?i σ = λ R M/ i?2 +Qp`BM+2 K2bm`2 Bb BM i?Bb +b2, ΓN (dt) = λ(R(dt) − λ dt) .

UNXRkV

Ua22 HbQ 1tKTH2 NXkXN 7Q` M Hi2`MiBp2 T`QQ7XV h?2 `2M2rH K2bm`2 Q7 i?2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ Bb i?2 .B`+ K2bm`2 ε0 THmb λ iBK2b i?2 G2#2b;m2 K2bm`2X h?2`27Q`2 ΓN (dt) = λ ε0 (dt)- b H`2/v T`Qp2/ BM 1tKTH2 NXRXNX

NXRX h>1 *Po_AL*1 J1al_1

jjN

_BTH2vǶb 6mM+iBQM _2+HH i?2 /2}MBiBQM Q7 i?2 7+iQ`BH KQK2Mi K2bm`2 M2! Q7  b2+QM/@Q`/2` TQBMi T`Q+2bb QM Rm ,    ! 1A (Xj )1B (Xk )1j=k (A, B ∈ B(Rm )) . M2 (A × B) := E j

k

6Q` Mv MQM@M2;iBp2 K2bm`#H2 7mM+iBQM f : Rm × Rm → R- Bi ?QH/b i?i      f (Xj , Xk )1j=k = f (t, s) M2! (dt × ds) . E j

Rm

k

Rm

A7 N Bb KQ`2Qp2` rB/2@b2Mb2 biiBQM`v rBi? p2`;2 BMi2MbBiv λ- `;mK2Mib bBKBH` iQ i?Qb2 H2/BM; iQ i?2 /2}MBiBQM Q7 i?2 +Qp`BM+2 K2bm`2 b?Qr i?2 2tBbi2M+2 Q7  σ@}MBi2 K2bm`2 K QM Rm bm+? i?i       f (Xj , Xk )1j=k = λ f (t, t + s) dt K(ds) . E j

Rm

k

Rm

TTHvBM; i?Bb 7Q`KmH iQ f (t, s) := 1A (t)1B (s − t) UA, B ∈ B(Rm )V ;Bp2b    1A (Xj )1B (Xk − Xj )1j=k = λ1A (t)1B (u) dx K(du) = λ m (A)K(B) , E j

k

i?i BbK(B) =

E[

n

1A (Xn )N ((B + Xn )\Xn )] . E [N (A)]

UNXRjV

.2}MBiBQM NXRXRR U9 V h?2 _BTH2v 7mM+iBQM Bb i?2 7mM+iBQM K : R+ → R+ /2}M2/ #v 1 K(r) := K(B(0, r) . λ aT2+BHBxBM; UNXRjV iQ B = B(0; r)- r2 Q#iBM i?2 2tT`2bbBQM

E [ n 1A (Xn )N ((B(0; r) + Xn )\Xn )] K(r) = E [N (A)]

E [ n 1A (Xn (N (B(0; r) + Xn ) − 1)] . = E [N (A)] h?2 [mMiBiv λK(r) `2T`2b2Mib 7Q` Mv TQBMi Xn i?2 p2`;2 MmK#2` Q7 TQBMib Xj bm+? i?i 0 < d(Xj , Xk ) ≤ rX 6`QK J2+F2Ƕb 7Q`KmH, K(r) =

E0N [N (B(0; r) − 1] . λ

_BTH2vǶb 7mM+iBQM Bb M BKTQ`iMi biiBbiB+H iQQH M/ Bib 2biBKi2b `2 mb27mH BM /2i2`KBMBM; i?2 Mim`2 Q7  TQBMi T`Q+2bbX h?2 [mMiBiv 9

(_BTH2v- RNde)X

j9y

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

K(r) :=

j

k

1W (Xj )1W (Xk )1d(Xj ,Xk )≤r 1k=j

,

λW r?2`2 W Bb M Q#b2`piBQM rBM/Qr M/ λW :=

N (W ) ,

m (W )

Bb mb2/ b M 2biBKiQ` Q7 K(r) i?i +M #2 mb2/ iQ i2bi M ?TT ?vTQi?2bBb- b i?2 M2ti 2tKTH2 bm;;2bibX8 1tKTH2 NXRXRk, _BTH2v 6mM+iBQM Q7  ?TTX A7 N Bb M ?TT- i?2M K(r) = νm rm , r?2`2 νm Bb i?2 m@pQHmK2 Q7 i?2 m@/BK2MbBQMH bT?2`2 Q7 `/Bmb 1 U1t2`+Bb2 NX8XkXV

NXk h?2 "`iH2ii aT2+i`H J2bm`2 h?2 2ti2MbBQM Q7 i?2 MQiBQM Q7 TQr2` bT2+i`H K2bm`2 7Q` biiBQM`v TQBMi T`Q@ +2bb2b i?i Bb bmBi#H2 7Q` biiBQM`v TQBMi T`Q+2bb2b Bb i?2 bQ@+HH2/ "`iH2ii bT2+i`H K2bm`2X .2}MBiBQM NXkXR G2i N #2  bBKTH2 rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λX  K2bm`2 μN QM Rm Bb +HH2/ i?2 TQr2` bT2+i`H K2bm`2 Q7 N B7 Bi Bb i?2 mMB[m2 HQ+HHv }MBi2 K2bm`2 μN bm+? i?i    ϕ (t) N (dt) = |ϕ (ν)|2 μN (dν) UNXR9V o` Rm

Rm

7Q` HH ϕ ∈ BN - r?2`2 BN ⊆ L2N (M 2 ) Bb  p2+iQ` bT+2 Q7 7mM+iBQMb +HH2/ i?2 i2bi 7mM+iBQM bT+2-  7mM+iBQM i?2`2BM #2BM; +HH2/  i2bi 7mM+iBQMX AM T`iB+mH`,

  o` ULUUy-h)V =

R

sin πνT πν

2 μN (dν) .

UNXR8V

h?Bb BM 7+i +?`+i2`Bx2b i?2 TQr2` bT2+i`H K2bm`2Xe _2K`F NXkXk LQi2 i?2 7Q`KH `2b2K#HM+2 Q7 1[Mb UNXkV M/ UNXR9VX "v TQH`BxiBQM Q7 UNXR9V- 7Q` HH ϕ, ψ ∈ BN  +Qp (N (ϕ) , N (ψ)) = ϕ(ν)ψ ∗ (ν)μN (dν). Rm

UNXReV

8 6Q` i?2 biiBbiB+H TTHB+iBQMb Q7 i?2 i?2Q`v Q7 TQBMi T`Q+2bb2b- b22 (*?Bm- aiQvM- E2M/HH M/ J2+F2- kyRj)X e (.H2v- RNdR)X

NXkX h>1 "_hG1hh aS1*h_G J1al_1

j9R

_2K`F NXkXj h?2 2ti2Mi Q7 i?2 i2bi 7mM+iBQM bT+2 BN Bb ;Bp2M BM 2+? bBim@ iBQMXd Ai Kv MQi #2 i?2 H`;2bi QM2- #mi Bi b?QmH/ BM Mv +b2 +QMiBM  +Hbb Q7 7mM+iBQMb `B+? 2MQm;? iQ ;m`Mi22 mMB[m2M2bb Q7 i?2 K2bm`2 μN X "v i?Bb- i?2 7QHHQrBM; Bb K2MiX A7 i?2 HQ+HHv }MBi2 K2bm`2b μ1 M/ μ2 `2 bm+? i?i   |ϕ(ν)|2 μ1 (dν) = |ϕ(ν)|2 μ2 (dν) Rm

Rm

7Q` HH ϕ ∈ BN - i?2M μ1 ≡ μ2 X h?2Q`2K NXkXRy rBHH i2HH mb i?i BN ⊇ B- i?2  b2i Q7 +QMiBMmQmb 7mM+iBQMb bm+? i?i #Qi? f M/ Bib 6Qm`B2` i`Mb7Q`K f `2 O 1/ |t|2 b |t| → ∞X _2K`F NXkX9 LQi2 i?i BN ⊆ L1C (Rm ) bBM+2- b r2 Q#b2`p2/ 2`HB2`- L2N (M 2 ) ⊆ L1C (Rm )X AM T`iB+mH`- i?2 6Qm`B2` i`Mb7Q`K Q7 Mv ϕ ∈ BN Bb r2HH /2}M2/X h?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 "`iH2ii bT2+i`H K2bm`2 Bb T`Qp2/ BM h?2Q`2K NXkXRy #2HQr- r?2`2 Bi rBHH #2 b?QrM i?i BN +QMiBMb  i H2bi i?2 7mM+iBQMb i?i `2- iQ;2i?2` rBi? i?2B` 6Qm`B2` i`Mb7Q`K- O 1/ |t|2 b |t| → ∞X _2K`F NXkX8 PM2 bQK2iBK2b T`272`b iQ bii2 i?2 `2bmHib BM i2`Kb Q7 dzTmHbiBQMbǴ BMbi2/ Q7 dz7`2[m2M+B2bǴX h?2 6Qm`B2` i`Mb7Q`K ϕ Q7 ϕ i?2M #2BM; /2}M2/ #v  ϕ(t)e−iωt dt , ϕ(ω) := R

M 2H2K2Mi`v +?M;2 Q7 p`B#H2b vB2H/b i?2 7Q`KmH 

 o`

ϕ (t) N (dt) R

 = R

|ϕ(ω)|2 fN (ω) dω ,

r?2`2 i?2 dzω@bT2+i`H /2MbBivǴ Bb fN (ω) =

1  ω f . 2π N 2π

h?2 /2}MBiBQM Q7 i?2 "`iH2ii bT2+i`H K2bm`2 BKK2/Bi2Hv vB2H/b i?2 bT2+i`H K2bm`2 Q7 i?2 ?QKQ;2M2Qmb SQBbbQM T`Q+2bb, 1tKTH2 NXkXe, SQBbbQM AKTmHbBp2 q?Bi2 LQBb2X Ai rb b?QrM BM 1tK@ TH2 NXRXRy i?i i?2 +Qp`BM+2 7mM+iBQM Q7  SQBbbQM T`Q+2bb Bb λ iBK2b i?2 .B`+ K2bm`2 i i?2 Q`B;BM M/ i?2`27Q`2 Bib bT2+i`H K2bm`2 Bb λ iBK2b i?2 G2#2b;m2 K2bm`2- i?i Bb- Bi /KBib i?2 TQr2` bT2+i`H /2MbBiv fN (ν) ≡ λ.

d AM i?2 i?2Q`2iB+H HBi2`im`2- i?2 +?QB+2 Bb mbmHHv  b2i Q7 p2`v bKQQi? 7mM+iBQMb- r?B+? Kv #2 iQQ `2bi`B+iBp2 7Q` TTHB+iBQMbX

j9k

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

6Q` i?Bb `2bQM i?2 `M/QK .B`+ +QK# UNX9V +Q``2bTQM/BM; iQ  SQBbbQM T`Q+2bb Bb bQK2iBK2b +HH2/ i?2 SQBbbQM BKTmHbBp2 r?Bi2 MQBb2X3 1tKTH2 NXkXd, a?Qi LQBb2X G2i N #2  rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb QM Rm X 6Q` HH h ∈ BN M/ HH u, v ∈ Rm - H2i ϕ(t) := h(u − t), ψ(t) = h(v − t). "v UNXReV  &   &2 & & h(u − t)N (dt), h(v − t)N (dt) = +Qp &h(ν)& e2iπν(v−u) μN (dν). R

R

R

AM i?2 mMBp`Bi2 +b2 Um = 1V- i?2 TQBMi T`Q+2bb Bb `2T`2b2Mi2/ #v Bib b2[m2M+2 Q7 2p2Mi iBK2b {Tn }n∈Z - M/  X(t) = h(t − Tn ) n∈Z

Bb i?2 biQ+?biB+ T`Q+2bb Q#iBM2/ #v TbbBM; i?2 .B`+ +QK# n δ(t − Tn ) i?`Qm;? i?2 }Hi2`X h?2 TQr2` bT2+i`H K2bm`2 Q7 i?2 QmiTmi {X(t)}t∈R Bb & &2 & & UNXRdV μX (dν) = &h(ν)& μN (dν) , r?B+? BM/22/ +Q``2bTQM/b iQ i?2 }Hi2`BM; 7Q`KmH Q7 h?2Q`2K NXRXj 7Q` #QM }/2

rbb bB;MHb B7 r2 bbBKBHi2 i?2 .B`+ +QK# δ(t − Tn ) iQ  #QM }/2 rbb bB;MH Ur?B+? Q7 +Qm`b2 Bi Bb MQiV rBi? bT2+i`H K2bm`2 μN X h?2 bT2+i`H K2bm`2 μN Bb ?Qr2p2` MQi  }MBi2 K2bm`2 b Bi rQmH/ #2 7Q` Q`/BM`v rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb2bX

1tBbi2M+2 M/ lMB[m2M2bb h?2 T`QQ7 Q7 2tBbi2M+2 Q7 i?2 "`iH2ii bT2+i`H K2bm`2 rBHH #2 /QM2 BM i?2 mMB@ p`Bi2 +b2 7Q` i?2 bF2 Q7 HB;?i2` MQiiBQM- #mi Bi `2KBMb i?2 bK2 BM i?2 ;2M2`H +b2X G2i N #2  bBKTH2 HQ+HHv }MBi2 rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb QM R M/ H2i σ #2 i?2 HQ+HHv }MBi2 K2bm`2 bm+? i?i 7Q` HH 7mM+iBQMb ϕ, ψ : R → R BM L2N (M2 )   ϕ(t)ψ(s + t) dt

E [N (ϕ)N (ψ)] = R

σ(ds) .

R

G2KK NXkX3 6Q` HH +QKT+i b2ib K ⊂ Rsup σ (K + t) < ∞ .

UNXR3V

t∈R

JQ`2Qp2`- B7 f : R → R Bb  MQM@M2;iBp2 7mM+iBQM

i?i Bb MQM@BM+`2bBM; QM [1, ∞)-MQM@/2+`2bBM; QM (−∞, 0] M/ bm+? i?i n∈Z f (n) < ∞- r2 ?p2 i?i supt∈R R f (t + s) σ (ds) < ∞X 3 AM S?vbB+b- QM2 mbmHHv +HHb dzr?Bi2 MQBb2Ǵ  `M/QK 7mM+iBQM rBi?  ~i bT2+i`mK Qp2`  H`;2 `M;2 Q7 7`2[m2M+B2bX

NXkX h>1 "_hG1hh aS1*h_G J1al_1

j9j

S`QQ7X a2H2+i ϕ : R → R-  MQM@M2;iBp2 7mM+iBQM rBi? +QKT+i bmTTQ`i bm+? i?i ϕ ∗ ϕˇ ≥ 1K Ur?2`2 ϕ(t) ˇ = ϕ(−t)VX ˇ h?2M- τt /2MQiBM; i?2 b?B7i QT2`iQ` s → s − t ˇ (s − t) σ (ds) σ (K + t) ≤ (ϕ ∗ ϕ) R = ((ϕ ◦ τt ) ∗ ϕ) ˇ (s) σ (ds) = E [N (ϕ ◦ τt ) N (ϕ)] . R

h?2`27Q`2- #v a+?r`xǶb BM2[mHBiv σ (K + t) ≤ E N (ϕ)2 ,  }MBi2 [mMiBiv bBM+2 ϕ Bb #QmM/2/ rBi? +QKT+i bmTTQ`i M/ N Bb  b2+QM/@Q`/2` TQBMi T`Q+2bbX LQr- 7Q` f b BM i?2 bii2K2Mi Q7 i?2 H2KK   f (n) σ ([n, n + 1) − t) + f (n) σ ([n − 1, n) − t) sup f (t + s) σ (ds) ≤ t∈R

R

n≥0





n≤−1

 f (n)



sup σ ([l, l + 1) − t)

≤C

l∈Z,t∈R

n∈Z



f (n) ,

n∈Z

r?2`2 C < ∞- #v TTHB+iBQM Q7 UNXR3V iQ i?2 +QKT+i b2i [0, 1]X



1tKTH2 NXkXN, *Qp`BM+2 J2bm`2 M/ SHK S`Q##BHBivX q2 MQr ;Bp2 MQi?2` T`QQ7- #b2/ QM J2+F2Ƕb 7Q`KmH- Q7 h?2Q`2K XRXR9 M/ G2KK NXkX3 7Q` i?2 +Qp`BM+2 K2bm`2 Q7  b2+QM/@Q`/2` TQBMi T`Q+2bb BM i?2 bi`B+iHv biiBQM`v +b2X G2i (Ω, F, P) #2  T`Q##BHBiv bT+2 2M/Qr2/ rBi?  K2bm`#H2 ~Qr {θt }t∈Rm - M/ bmTTQb2 i?i P Bb θt @BMp`BMi 7Q` HH t ∈ Rm X G2i N #2  bBKTH2 TQBMi T`Q+2bb /2}M2/ QM i?Bb T`Q##BHBiv bT+2- M/ θt @+QKTiB#H2X Ai Bb BM T`iB+mH` biiBQM`v- rBi? BMi2MbBiv λ bbmK2/ }MBi2 M/ TQbBiBp2X *QMbB/2` i?2 K2bm`2 σ QM (Rm , B(Rm )) /2}M2/ #v σ (C) = λE0N [N (C)]

UNXRNV

7Q` HH C ∈ B(Rm )- r?2`2 E0N /2MQi2b BMi2;`iBQM rBi? `2bT2+i iQ i?2 SHK T`Q#@ #BHBiv PN0 X h?2 K2bm`2 σ Bb  HQ+HHv }MBi2 K2bm`2c 2p2M KQ`2, sup σ (K + t) < ∞

UNXkyV

t∈Rm

7Q` HH +QKT+i K ⊂ Rm X HbQ 7Q` f, g : Rm → R- K2bm`#H2 M/ MQM@M2;iBp2 E [N (f ) N (g)] = (f ∗ gˇ) dσ . UNXkRV Rm

U_2+HH i?2 MQiiBQM fˇ(t) = f (−t)XV

j99

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

S`QQ7X 6Q` MQM@M2;iBp2 K2bm`#H2 f : Rm → R g dσ = λE0N [N (g)] , Rm

M/ i?2`27Q`2

 g dN E [N (f ) N (g)] =E N (f ) Rm   = g (t) N (f ) (ω) N (ω, dt) P (dω) m Ω  R =λ g (t) N (f ) (θ−t ω) dt PN0 (dω) Ω Rm     =λ g (t) f (t + s) N (ω, ds) dt PN0 (dω) m Rm  Ω R  =λ f (t + s) g (t) dt N (ω, ds) PN0 (dω) , Ω

Rm

Rm

r?2`2 i?2 i?B`/ 2[mHBiv mb2b J2+F2Ƕb 7Q`KmHX 6BMHHv 

 0 (f ∗ gˇ) dN = E [N (f ) N (g)] = λEN Rm

Rm

(f ∗ gˇ) dσ.

q2 ?p2 i?2`27Q`2 T`Qp2/ UNXkRVX AM Q`/2` iQ T`Qp2 UNXkyV- b2H2+i ϕ : Rm → R-  MQM@M2;iBp2 7mM+iBQM rBi? +QKT+i bmTTQ`i bm+? i?i ϕ ∗ ϕˇ ≥ 1K X h?2M   σ (K + t) ≤ (ϕ ∗ ϕ) ˇ (s − t) σ (ds) = ((ϕ ◦ τt ) ∗ ϕ) ˇ (s) σ (ds) Rm

Rm

=E [N (ϕ ◦ τt ) N (ϕ)] = E [(N (ϕ) ◦ θt ) (N (ϕ))] . h?2`27Q`2- #v a+?r`xǶb BM2[mHBiv- σ (K + t) ≤ E N (ϕ)2 -  }MBi2 [mMiBiv bBM+2 ϕ Bb #QmM/2/ rBi? +QKT+i bmTTQ`i M/ N Bb  b2+QM/@Q`/2` TQBMi T`Q+2bbX 

h?2Q`2K NXkXRy G2i N #2 b #Qp2X h?2`2 2tBbib  mMB[m2 MQM@M2;iBp2 HQ+HHv }MBi2 K2bm`2 σ QM (R, B(R)) bm+? i?i- B7 f Bb +QMiBMmQmb M/ #Qi? f M/ Bib   6Qm`B2` i`Mb7Q`K f `2 O 1/ |t|2 b |t| → ∞- i?2M   f (ν) σ (dν) = f (t) σ (dt) UNXkkV R

R

M/- B7 g biBb}2b i?2 bK2 +QM/BiBQMb b f  E [N (f ) N (g)] = λ f (ν) gˇ (ν) σ (dν) . R

S`QQ7XN G2i 7Q` a > 0 N

(L2p2m- RNdd)- S`QTX AAXk9- TX j83X

UNXkjV

NXkX h>1 "_hG1hh aS1*h_G J1al_1 ha (t) :=

j98

1 , 1 + 4π 2 a2 t2

 7mM+iBQM rBi? 6Qm`B2` i`Mb7Q`K ha (ν) = P#b2`p2 i?i

1 −a|ν| e . 2a

2 ha/2 ∗ ha/2 = ha a

UNXk9V

bBM+2 BM i?2 6Qm`B2` /QKBM- (ha/2 )2 = a2 ha X HbQ- MQi2 i?i Bi Bb 2[mBpH2Mi iQ bv   i?i f Bb O 1/ |t|2 b |t| → ∞ Q` i?i f Bb #QmM/2/ #v  KmHiBTH2 Q7 h1 X "v G2KK NXkX3- 7Q` HH a > 0- σa (dt) = ha (t) σ (dt) /2}M2b  }MBi2 K2bm`2 QM (R, B(R))X Aib 6Qm`B2` i`Mb7Q`K  e−2iπνt σa (dt) σa (ν) = R

Bb i?2`27Q`2 #QmM/2/ M/ +QMiBMmQmbX lbBM; UNXk9V- r2 ?p2 & &2   & & a −2iπνt E && ha/2 (t) e N (dt)&& = ha (t) e−2iπνt σ (dt) = σa (ν) 2 R R M/ i?2`27Q`2 σa (ν) ≥ 0 7Q` HH a > 0 M/ HH ν ∈ RX h?2`27Q`2- i?2 K2bm`2 σa (ν) h1 (ν) dν Bb  MQM@M2;iBp2 }MBi2 K2bm`2 QM (R, B(R))X Aib +?`+i2`BbiB+ 7mM+iBQM UrBi? t b `;mK2MiV Bb    e−2iπνt σa (ν) h1 (ν) dν = e−2iπνt ha (s) e−2iπνs σ (ds) h1 (ν) dν R R R     = e−2iπν(t+s) h1 (ν) dν ha (s) σ (ds) = h1 (t + s) ha (s) σ (ds) UNXk8V



R

R

R

U6m#BMBǶb i?2Q`2K Bb TTHB+#H2 ?2`2 #2+mb2 (ν, s) →1 (ν) ha (s) Bb ×σ@BMi2;`#H2VX h?2`27Q`2   −2iπνt σa (ν) h1 (ν) dν = lim h1 (t + s) ha (s) σ (ds) lim e a↓0 R a↓0 R  = h1 (t + s) σ (ds) < ∞ R

U#v /QKBMi2/ +QMp2`;2M+2 bBM+2 h1 (t + s) ha (s) ≤ h1 (t + s)-  σ@BMi2;`#H2 7mM+@ iBQM #v G2KK NXkX3VX h?2 HBKBi Bb +QMiBMmQmb BM t U#v /QKBMi2/ +QMp2`;2M+2 ;BMVX h?2`27Q`2- #v i?2 2ti2MbBQM iQ }MBi2 K2bm`2b Q7 SmH GûpvǶb i?2Q`2KRy i?2 K2bm`2 σa (ν) h1 (ν) dν +QMp2`;2b r2FHv b a ↓ 0 iQ  }MBi2 K2bm`2 r?B+? r2 Kv Hrvb r`Bi2 b h1 (ν) σ (dν) bBM+2 h1 (ν) > 0- 7Q` HH ν ∈ RX q2F +QMp2`;2M+2 K2Mb i?i 7Q` Mv #QmM/2/ M/ +QMiBMmQmb 7mM+iBQM ϕ : R → R Ry

a22 7Q` BMbiM+2 ("`ûKm/- kyR9)- h?2Q`2K kXjX8X

j9e

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 

 ϕ (ν) h1 (ν) σa (ν) dν =

lim a↓0

R

ϕ (ν) h1 (ν) σ (ν) dν . R

AM T`iB+mH`- B7 f Bb +QMiBMmQmb M/ #QmM/2/ #v  KmHiBTH2 Q7 h1   lim f (ν) σa (ν) dν = f (ν) σ (ν) dν . a↓0

R

R

PM i?2 Qi?2` ?M/- #v  +QKTmiiBQM bBKBH` iQ i?i BM UNXk8V  f (ν) σa (ν) dν = f (t) ha (t) σ (dt) , R

R

M/ i?2`27Q`2    f (t) σ (dt) , lim f (ν) σa (ν) dν = lim f (t) ha (t) σ (dt) = a↓0

R

a↓0

R

R

 r?2`2 f Bb #QmM/2/ #v  KmHiBTH2 Q7 h1 Ui?Bb BKTHvBM; i?i R |f |(t)σ (dt) < ∞VX h?Bb T`Qp2b UNXkkVX   A7 f M/ g `2 O 1/ |t|2 b r2HH b i?2B` 6Qm`B2` i`Mb7Q`Kb- bQ Bb f ∗ gˇ- M/ i?2`27Q`2   (f ∗ gˇ) dσ = f gˇ dσ . E [N (f ) N (g)] = Rm

Rm

Ai `2KBMb iQ T`Qp2 mMB[m2M2bbX AM pB2r Q7 UNXkkV- i?Bb 7QHHQrb 7`QK i?2 7+i i?i irQ HQ+HHv }MBi2 K2bm`2b ;`22BM; QM HH 7mM+iBQMb i?i `2- iQ;2i?2` rBi? i?2B` 6Qm`B2` i`Mb7Q`Kb- Q7 i?2 Q`/2` Q7 |t|−2 i BM}MBiv- `2 B/2MiB+HX  h?2 7QHHQrBM; Bb  `2bii2K2Mi M/ M Q#pBQmb 2ti2MbBQM iQ i?2 +QKTH2t +b2 Q7 h?2Q`2K NXkXRyX *Q`QHH`v NXkXRR .2MQi2 #v B i?2 +QHH2+iBQM Q7 +QMiBMmQmb 7mM+iBQMb ϕ : Rm → C i?i `2 O 1/ |t|2 b r2HH b i?2B` 6Qm`B2` i`Mb7Q`KX G2i N #2  bBKTH2 rB/2@ b2Mb2 biiBQM`v TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λX h?2M- i?2`2 2tBbib  mMB[m2 σ@}MBi2 K2bm`2 μN U= σ − λ2 ε0 V bm+? i?i 7Q` HH ϕ : Rm → C BM B +Qp (N (ϕ) , N (ψ)) = ϕ(ν)ψ ∗ (ν)μN (dν) . Rm

h?2Q`2K NXkXRy M/ Bib +Q`QHH`v `2 ;2M2`H `2bmHib +QM+2`MBM; i?2 2tBbi2M+2 M/ mMB+Biv Q7 i?2 "`iH2ii bT2+i`H K2bm`2X h?2 M2ti 2tKTH2b ;Bp2 2tTHB+Bi 7Q`@ KmHb 7Q` i?2 bT2+i`H K2bm`2 M/  /2b+`BTiBQM Q7  bT+2 Q7 i2bi 7mM+iBQMb BM i?`22 BKTQ`iMi bT2+BH +b2bX h?Bb +QHH2+iBQM Q7 2tTHB+Bi 2tKTH2b rBHH #2 +QMbB/2`#Hv 2MH`;2/ BM a2+iBQM NX9X 1tKTH2 NXkXRk, h?2 "`iH2ii aT2+i`mK Q7 i?2 aiiBQM`v _2;mH` :`B/X *QMbB/2` i?2 TQBMi T`Q+2bb QM R2 r?Qb2 TQBMib 7Q`K  `2;mH` (T1 , T2 )@;`B/ QM R2 rBi? `M/QK Q`B;BM- i?i Bb  (T1 > 0 , T2 > 0) , N = (n1 T1 + U1 , n2 T1 + U2 ) , (n1 , n2 ) ∈ Z2

NXkX h>1 "_hG1hh aS1*h_G J1al_1

j9d

r?2`2 U1 - U2 `2 BM/2T2M/2Mi mMB7Q`K `M/QK p`B#H2b QM [0, T1 ] M/ [0, T2 ] `2@ bT2+iBp2HvX h?2 TQBMi T`Q+2bb Bb b2+QM/@Q`/2` biiBQM`v rBi? p2`;2 BMi2MbBiv λ = 1/(T1 T2 )X Aib "`iH2ii bT2+i`H K2bm`2 Bb μN =

1



ε( T 1 , T 2 ) 1 2 T12 T22 (n1 ,n2 )=(0,0) n

M/ QM2 +M iF2 7Q` i?2 i2bi 7mM+iBQM bT+2   BN := ϕ ∈ L1C (R2 ) ∩ L2C (R2 ) M/ n1 ,n2 ∈Z

n

UNXkeV

,

 &  & & & n n 1 2 &Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1



ϕ (t) N (dt) R2  = E [ϕ (U1 + n1 T1 , U2 + n2 T2 )]

E

n1 ,n2 ∈Z

=

1 T1 T2



T1



T2

ϕ (u1 + n1 T1 , u2 + n2 T2 ) du = 0

0

1 T1 T2

 ϕ (t) dt = R2

1 ϕ (0, 0) . T1 T2

h?2`27Q`2 

 ϕ (t) N (dt)

Var R2

 &&  n1 n2 &&2 1 & − 1 |ϕ (0, 0)|2 &ϕ = 2 2 , T1 T2 n ,n ∈Z & T1 T2 & T12 T22 1 2 &  &2  & n1 n2 && 1 & = 2 2 &ϕ T1 T2 & T1 T2 (n1 ,n2 )=(0,0)   = |ϕ (ν1 , ν2 )|2 μN (dν1 × dν2 ) , R

R



r?2`2 μN Bb ;Bp2M #v UNXkeVX

1tKTH2 NXkXRj, h?2 "`iH2ii aT2+i`mK Q7  aiiBQM`v *Qt S`Q+2bbX G2i N #2  *Qt TQBMi T`Q+2bb QM Rm rBi? biQ+?biB+ BMi2MbBiv {λ(t)}t∈Rm X _2K2K#2` i?i i?Bb K2Mb i?2 7QHHQrBM;X 6B`biHv- {λ(t)}t∈Rm Bb  MQM@M2;iBp2 HQ@ +HHv BMi2;`#H2 T`Q+2bb- M/ b2+QM/Hv- +QM/BiBQMHHv QM i?Bb T`Q+2bb- N Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t)X amTTQb2 i?i {λ(t)}t∈Rm Bb  rbb T`Q+2bb rBi? K2M λ M/ TQr2` bT2+i`H K2bm`2 μλ X h?2M i?2 "`iH2ii bT2+i`mK Q7 N Bb μN (dν) = μλ (dν) + λdν ,

UNXkNV

M/ r2 +M iF2 BN = L1C (Rm ) ∩ L2C (Rm )X 1p2M KQ`2- BM i?Bb +b2 BN = L2N (M2 ) λ S`QQ7X .2MQi2 #v F∞ i?2 σ@}2H/ ;2M2`i2/ #v i?2 biQ+?biB+ T`Q+2bb {λ(t)}t∈R X LQi2 i?i B7 ϕ ∈ L2C (Rm ) 

 E |ϕ(t)|2 λ(t)dt = λ |ϕ(t)|2 dt < ∞ Rm

Rm

M/ i?2`27Q`2- HKQbi bm`2Hv Rm

|ϕ(t)|2 λ(t)dt < ∞ .

aBKBH`Hv- bBM+2 ϕ ∈ L1C (Rm )- HKQbi bm`2Hv  |ϕ(t)| λ(t)dt < ∞ . Rm

"v i?2 7Q`KmHb Q#iBM2/ BM h?2Q`2K jXkXR-

NXkX h>1 "_hG1hh aS1*h_G J1al_1 



2

E Rm

λ |F∞

|ϕ(t)| N (dt)

j9N 

 |ϕ(t)| λ(t)dt +

2

2

= Rm

Rm

|ϕ(t)| λ(t)dt

. UNXjyV

q2 i?2`27Q`2 ?p2  E Rm

2 

|ϕ(t)| N (dt)



 |ϕ(t)| dt + E

2 

2

=λ Rm

Rm

|ϕ(t)| λ(t)dt

 

2 ≥λ |ϕ(t)| dt + E |ϕ(t)| λ(t)dt Rm Rm   2  2 =λ |ϕ(t)| dt + λ |ϕ(t)| dt . 

2

Rm

Rm

h?2`27Q`2 B7 ϕ ∈ L2N (M2 )- i?2M ϕ ∈ L1C (Rm ) ∩ L2C (Rm )X HbQ- 7`QK 

2 

E Rm

|ϕ(t)| N (dt)

2 



 |ϕ(t)| dt + E 2

=λ Rm

M/ i?2 7+i i?i B7 ϕ ∈ L1C (Rm )- i?2M E



Rm

|ϕ(t)| λ(t)dt

Rm L1C (Rm )

|ϕ(t)| λ(t)dt

2 

,

< ∞ Ui?Bb Bb #2+mb2

∩ i?2M ϕ ∈ L2N (M2 )X {λ(t)}t∈R Bb rbbV- r2 b22 i?i B7 ϕ ∈ h?2`27Q`2 BM i?2 +b2 Q7 *Qt T`Q+2bb2b rBi?  rbb +QM/BiBQMH BMi2MbBiv- ϕ ∈ L1C (Rm ) ∩ L2C (Rm ) Bb 2[mBpH2Mi iQ ϕ ∈ L2N (M2 )X L2C (Rm )-

"v i?2 +QM/BiBQMH p`BM+2 7Q`KmH o`

Rm

 ϕ(t)N (dt) & &  

 

 & λ & λ = E o` + o` E ϕ(t)N (dt)&& F∞ ϕ(t)N (dt)&& F∞ m Rm 

 R =E ϕ(t)2 λ(t)dt + o` ϕ(t)λ(t)dt Rm Rm    =λ ϕ(t)2 dt + o` ϕ(t)λ(t)dt . Rm

Rm

"v /2}MBiBQM Q7 i?2 TQr2` bT2+i`H K2bm`2- 7Q` ϕ ∈ L1C (Rm ) o`

 ϕ(t)λ(t)dt

 =

Rm

Rm

|ϕ(ν)|2 μλ (dν) ,

M/ #v i?2 SHM+?2`2HĜS`b2pH 7Q`KmH 

 2

ϕ (t)dt = Rm

Rm

|ϕ(ν)|2 dν .

j8y

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 

h?2`27Q`2 o`

 ϕ(t)N (dt)

 =

Rm

Rm

|ϕ(ν)|2 (μλ (dν) + λdν) , 

r?B+? T`Qp2b UNXkNVX

1tKTH2 NXkXR9, h?2 "`iH2ii aT2+i`mK Q7 i?2 aiiBQM`v _2M2rH SQBMi S`Q+2bbXRj AM Q`/2` iQ +QKTmi2 i?2 "`iH2ii bT2+i`mK Q7  biiBQM`v `2M2rH TQBMi T`Q+2bb rBi? BMi2MbBiv λ M/  MQM@HiiB+2 `2M2rH /Bbi`B#miBQM F /2}M2  F (2iπν) := e−2iπνt dF (t) . R+

LQi2 i?i- bBM+2 F Bb MQM@HiiB+2- F (ν) = 1- 2t+2Ti 7Q` ν = 0X h?2 +Qp`BM+2 K2bm`2 Bb ;Bp2M #v i?2 7Q`KmH Γ(dx) = λE0 [N (dx)] − λ2 (dx) , r?2`2 E0 /2MQi2b 2tT2+iiBQM rBi? `2bT2+i iQ i?2 T`Q##BHBiv KFBM; Q7 N M mM/2Hv2/ `2M2rH T`Q+2bbX h?2 K2bm`2 E0 [N (dx)] Bb i?2 bmK Q7  .B`+ K2bm`2 i 0- ε(dx)- M/ Q7  bvKK2i`B+ K2bm`2 U (dx)- ;Bp2M- 7Q` dx ⊂ (0, ∞)- #v  U (dx) = F ∗n (dx) . n≥1

h?2 K2bm`2 U rBHH MQr #2 bmTTQb2/ iQ /KBi  /2MbBiv u rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2 M/ r2 bbmK2 i?i  ∞ |u(t) − λ|dt < ∞ . UNXjRV 0

.2}MBM;

 g(ν) :=



e−2iπνt (u(t) − λ) dt

0

M/ iFBM; BMiQ ++QmMi i?2 bvKK2i`v Q7 u e−2iπνt (u(t) − λ)dt = g(ν) + g ∗ (ν) . R

Ai rBHH #2 T`Qp2/ #2HQr i?i g(ν) =

F (2iπν) 1 − F (2iπν)

+

1 . 2iπν

UNXjkV

*QK#BMBM; i?2 #Qp2 `2bmHib b?Qrb i?i i?2 "`iH2ii bT2+i`mK Q7 N /KBib i?2 /2MbBiv   F (2iπν) fN (ν) = λ(1 + Re 1 − F (2iπν) Rj

(.H2v M/ o2`2@CQM2b- RN33)- 1tKTH2 RRXk U#V- TX 9RkX

NXjX GPL:@_L:1 .1S1L.1L*1

j8R

6Q` i?2 T`QQ7 Q7 UNXjkV- r`Bi2 7Q` θ > 0  ∞  ∞ e−(θ+2iπν)t (u(t) − λ)dt = e−(θ+2iπν)t F ∗n (dt) − 0

n≥1

=



0

e−(θ+2iπν)t λdt

0

F (θ + 2iπν)n −

n≥1

=



F (θ + 2iπν) 1 − F (θ + 2iπν)



λ θ + 2iπν λ . θ + 2iπν

6Q` ν = 0- H2iiBM; θ i2M/ iQ 0 BM i?2 }`bi i2`K Q7 i?2 #Qp2 2[mHBiv- r2 Q#iBM #v ∞ /QKBMi2/ +QMp2`;2M+2 0 e−2iπνt (u(t) − λ)dtX G2iiBM; θ i2M/ iQ 0 BM F (θ + 2iπν)r2 Q#iBM F (2iπν)- ;BM #v /QKBMi2/ +QMp2`;2M+2X >2M+2 i?2 MMQmM+2/ `2bmHiX lM/2` i?2 +QM/BiBQMb Q7 i?2 #Qp2 2tKTH2- QM2 +M iF2 7Q` i?2 i2bi 7mM+iBQMb bT+2 BN = L1C (R) ∩ L2C (R)X AM/22/- 7Q`  7mM+iBQM ϕ ∈ L2N (M2 ) UBM T`iB+mH`ϕ ∈ L1C (R) ∩ L2C (R)V     ∗ ϕ (t) N (dt) = ϕ (t) ϕ (t + s) dt ΓN (ds) , o` Rm

r?2`2

Rm

Rm

ΓN (dt) = λR(dt) − λ2 dt = λ(u(t) − λ) dt + λε(dt) .

UNXjjV

∩ h?2`27Q`2- 7Q` ϕ ∈   ϕ(t) N (dt) o`   R    2 −2iπνs 2 −2iπνs = |ϕ(ν)| e dν (u(s) − λ) ds + λ |ϕ(ν)| e dν ε(ds) R R   R  R = |ϕ(ν)|2 e−iπνs (u(s) − λ) ds dν + λ |ϕ(ν)|2 dν Rm Rm Rm  = |ϕ(ν)|2 f (ν) dν . L1C (R)

L2C (R)-

Rm

_2K`F NXkXR8 LQi2 i?i BM HH i?`22 Q7 i?2 #Qp2 2tKTH2b- i?2 +QHH2+iBQMb BN +QMiBM B M/ i?2`27Q`2 `2 ;2MmBM2 i2bi 7mM+iBQM bT+2b UH`;2 2MQm;? iQ ;m`Mi22 mMB+Biv Q7 i?2 bT2+i`H K2bm`2b i?2`2BMc b22 _2K`F NXkXjVX

NXj

GQM;@`M;2 .2T2M/2M+2

h?2 MQiBQM Q7 HQM;@`M;2 /2T2M/2M+2 ?b BKTQ`iMi BKTHB+iBQMb BM i?2 i?2Q`v Q7 biQ+?biB+ T`Q+2bb2b- BM+Hm/BM; TQBMi T`Q+2bb2bX Ai Bb +QMp2MB2Mi iQ BMi`Q/m+2 Bi BM i2`Kb Q7 i?2 +Qp`BM+2 7mM+iBQM Q7 rB/2@b2Mb2 biQ+?biB+ T`Q+2bb2bX h?2 dz+HbbB+HǴ KQ/2Hb Q7 rbb biQ+?biB+ T`Q+2bb2b ?p2  +Qp`BM+2 7mM+iBQM r?B+? Bb ;2QK2i`B@ +HHv #QmM/2/- i?i Bb,

j8k

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 C(|τ |) ≤ Ke−α|τ | ,

r?2`2 K > 0 M/ α > 0X  HQM; K2KQ`v rbb biQ+?biB+ T`Q+2bb Bb QM2 7Q` r?B+? 2tTQM2MiBH /2+v Bb `2TH+2/ #v TQHvMQKBH /2+vX JQ`2 T`2+Bb2HvC(|τ |) ∼ K|τ |2d−1

b τ → ∞,

UNXj9V

r?2`2 K > 0 M/ d < 12 X PM2 bQK2iBK2b KF2b i?2 /BbiBM+iBQM #2ir22M BMi2`K2@ /Bi2 K2KQ`v M/ HQM; K2KQ`v ++Q`/BM; iQ r?2i?2` d < 0- BM r?B+? +b2  C(τ ) dτ < ∞ , R

Q` 0 < d < 12 - BM r?B+? +b2

 R

C(τ ) dτ = ∞ .

h?2 i2`KBMQHQ;v Bb MQi v2i biM/`/Bx2/R9 M/ bQK2iBK2b QM2 T`272`b HQM;@`M;2 /2T2M/2M+2 iQ HQM;@K2KQ`vX >2`2-  HQM;@`M;2 /2T2M/2Mi biQ+?biB+ T`Q+2bb rBHH #2 QM2 r?Qb2 +Qp`BM+2 7mM+iBQM Bb MQi BMi2;`#H2X AM T`iB+mH`  HQM;@K2KQ`v biQ+?biB+ T`Q+2bb Bb HQM;@`M;2 /2T2M/2MiX GQM;@`M;2 /2T2M/2M+2 +M HbQ #2 TT`Q+?2/ 7`QK i?2 bT2+i`H TQBMi Q7 pB2rX AM/22/- bmTTQb2 i?i  rbb biQ+?biB+ T`Q+2bb /KBib  TQr2` bT2+i`H /2MbBiv f r?Qb2 #2?pBQm` i i?2 Q`B;BM Bb i?2 7QHHQrBM;, lim f (ν)ν β = r > 0 ,

UNXj8V

ν→0

7Q` bQK2 β ∈ (0, 1)X h?2M r`BiBM;  +∞  C(τ )τ 1−β = f (ν)τ 1−β e+2iπντ dν = −∞

M/ Q#b2`pBM; i?i

+∞ −∞

1 ν  τ 1−β f dν τ τ

1 ν  r lim τ 1−β f = β τ ↑∞ τ τ ν

r2 ?p2 i?i UrBi? KBH/ //BiBQMH +QM/BiBQMb QM f ;m`Mi22BM; i?i G2#2b;m2Ƕb /QKBMi2/ +QMp2`;2M+2 i?2Q`2K Bb TTHB+#H2V  ∞ +2iπν e 1−β lim C(τ )τ =r dν . τ ↑∞ νβ 0 h?2`27Q`2- i BM}MBiv- i?2 +Qp`BM+2 7mM+iBQM #2?p2b b i?2 T`Q+2bb Bb HQM;@`M;2 /2T2M/2MiX

1 τ 1−β

r?2`2 1 − β ∈ (0, 1),

 7Q`KmHiBQM BM i2`Kb Q7 i?2 TQr2` bT2+i`H /2MbBiv b BM UNXj8V Bb HbQ /Ti2/ iQ rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb2bX JQ/2Hb 7Q` r?B+? i?Bb T`QT2`iv ?QH/b `2 2bv iQ Q#iBMX b  Kii2` Q7 7+i- b b?QrM BM 1tKTH2 NXkXRj-  *Qt T`Q+2bb r?Qb2 +QM/BiBQMH BMi2MbBiv T`Q+2bb Bb  rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb R9

6Q`  KQ`2 /2iBH2/ BMi`Q/m+iBQM iQ HQM;@`M;2 /2T2M/2M+2- b22 (aKQ`Q/MBibFv- kyye)X

NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1

j8j

rBi? TQr2` bT2+i`H /2MbBiv fλ /KBib- ++Q`/BM; iQ 7Q`KmH UNXkNV- i?2 "`iH2ii bT2+i`H /2MbBiv fN (ν) = fλ (ν) + λ ,

UNXjeV

M/ Bi Bb i?2M +H2` i?i Bib HQM;@`M;2 /2T2M/2M+2 T`QT2`iB2b /B`2+iHv 7QHHQr 7`QK i?Qb2 Q7 i?2 BMi2MbBiv T`Q+2bbX h?2`27Q`2 QM2 +M }M/ b KMv HQM;@`M;2 /2T2M@ /2Mi biiBQM`v TQBMi T`Q+2bb2b b i?2`2 `2 MQM@M2;iBp2 HQM;@`M;2 /2T2M/2Mi biiBQM`v biQ+?biB+ T`Q+2bb2b- M/ i?Bb 7+i Bb r2HH bmTTQ`i2/ #v BMimBiBQMX h?2 TQBMi T`Q+2bb2b +QMbB/2`2/ BM i?2 T`2pBQmb T`;`T? `2 dz/Qm#Hv biQ+?b@ iB+Ǵ- M/ QM2 Kv rQM/2` #Qmi i?2 2tBbi2M+2 Q7  dzb2H7@2t+BiBM;Ǵ TQBMi T`Q+2bb 2t?B#BiBM; HQM;@`M;2 /2T2M/2M+2X h?2 Mbr2` Bb 2bv, iF2 i?2 bK2 *Qt T`Q+2bbX dz"mi Bi Bb MQi b2H7@2t+BiBM;Ǵ Kv QM2 `;m2X h?Bb TQBMib QM+2 ;BM iQ i?2 T`Q#H2K `BbBM; r?2M QM2 [mHB}2b  TQBMi T`Q+2bb b dzb2H7@2t+BiBM;Ǵ r?2M QM2 b?QmH/ BM@ bi2/ bT2F Q7 Bib b2H7@2t+BiBM; dz/2b+`BTiBQMǴ Ub22 i?2 /Bb+mbbBQM Q7 a2+iBQM 8XkVX Mv *Qt T`Q+2bb ?b  b2H7@2t+BiBM; /2b+`BTiBQM r?B+? Bb i?2 T`QD2+iBQM Q7 i?2 /Qm#Hv biQ+?biB+ BMi2MbBiv QM i?2 Mim`H }Hi`iBQMX _Qm;?Hv bT2FBM;- Bib dzb2H7@2t+BiBM;Ǵ BMi2MbBiv Bb {E λ(t) | FtN }t∈R X *M r2 }M/  HQM;@`M;2 /2T2M/2Mi TQBMi T`Q+2bb Q7 r?B+? i?2`2 Bb MQ Q#pBQmb /Qm#Hv biQ+?biB+ /2b+`BTiBQM #mi QMHv  Mim`H BMi`BMbB+ /2b+`BTiBQM\ h?Bb Bb  p;m2 [m2biBQM- iQ r?B+? i?2`2 Bb  TQbBiBp2 Mbr2`- mbBM; i?2 Hi2`MiBp2 /2}MBiBQM Q7 HQM;@`M;2 /2T2M/2M+2- MK2Hv i?i +QM/BiBQM lim t↑∞

Var N ([0, t]] =∞ t

UNXjdV

#2 biBb}2/X AM 7+i-R8  `2M2rH T`Q+2bb biBb}2b +QM/BiBQM UNXjdV B7 M/ QMHv B7 Bib BMi2`pH /Bbi`B#miBQM ?b BM}MBi2 b2+QM/ KQK2MibX h?2 HBMF rBi? i?2 #2?pBQm` BM i?2 pB+BMBiv Q7 i?2 MmHH 7`2[m2M+v Q7 i?2 /2}MBiBQM UNXjdV Bb K/2 pB .H2vǶb /2}MBiBQM UNXR8V Q7 i?2 "`iH2ii bT2+i`H K2bm`2X AM a2+iBQM RkX8- r2 b?HH b22  KQ`2 bT2+i+mH` 2tKTH2- i?i Q7  >rF2b #`M+?BM; TQBMi T`Q+2bb rBi?Qmi M+2biQ`b- 72im`BM;  T?b2 i`MbBiBQM T?2@ MQK2MQMX

NX9

h`Mb7Q`KiBQMb Q7 i?2 aT2+i`H J2bm`2

h?Bb b2+iBQM 72im`2b  7Q`KmH i?i HHQrb QM2 iQ Tbb 7`QK i?2 bT2+i`H K2bm`2 Q7  #bB+ rB/2@b2Mb2 biiBQM`v TQBMi T`Q+2bb iQ i?Qb2 Q7 TQBMi T`Q+2bb2b Q#iBM2/ #v p`BQmb i`Mb7Q`KiBQMb Q7 i?2 BMBiBH TQBMi T`Q+2bbX Ai Bb  FBM/ Q7 dzarBbb `Kv FMB72Ǵ- r?2`2 2+? dz#H/2Ǵ +QM+2`Mb  bT2+B}+ i`Mb7Q`KiBQM, i?BMMBM;- i`MbHi@ BM;- DBii2`BM; M/ +Hmbi2`BM;X R8

(h2m;2Hb- RNe3)- (.H2v- RNNN)X

j89

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

 lMBp2`bH *Qp`BM+2 6Q`KmH G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM Rm rBi? TQBMi b2[m2M+2 {Xn }n∈N X G2i {Zn }n∈N #2  7KBHv Q7 BB/ `M/QK p`B#H2b rBi? pHm2b BM i?2 K2bm`#H2 bT+2 (K, K) rBi? +QKKQM T`Q##BHBiv /Bbi`B#miBQM Q M/ BM/2T2M/2Mi Q7 N X HbQ bbmK2 i?i N Bb  b2+QM/@Q`/2` biiBQM`v TQBMi T`Q+2bb rBi? "`iH2ii bT2+i`H K2bm`2 μN M/ i2bi 7mM+iBQM bT+2 BN X G2i Z #2  `M/QK 2H2K2Mi rBi? /Bbi`B#miBQM QX _2+HH i?2 MQiiBQM LpC ( × Q) 7Q` i?2 b2i Q7 K2bm`#H2 7mM+iBQMb ϕ : Rm × K → C bm+? i?i    |ϕ(t, z)|p Q(dz) dt = E [|ϕ(t, Z)|p ] dt < ∞ . Rm

Rm

K

AM T`iB+mH`- ϕ(t, Z) ∈ K2bm`2VX

LpC (P )

7Q` HKQbi HH t ∈ R UrBi? `2bT2+i iQ i?2 G2#2b;m2

G2i ϕ : Rm × K → R #2  K2bm`#H2 7mM+iBQM bm+? i?i ϕ ∈ L1C ( × Q) .

UNXj3V

AM T`iB+mH`- ϕ(t, Z) ∈ L1C (P ) 7Q` HKQbi HH t ∈ R UrBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2V M/ i?2 [mMiBiv ϕ(t) := E [ϕ(t, Z)] Bb r2HH /2}M2/ 7Q` HKQbi HH tX Ai HbQ 7QHHQrb 7`QK bbmKTiBQM UNXj3V i?i ϕ ∈ L1C (Rm ) M/ 7Q` Q@HKQbi HH z ∈ K- ϕ(·, z) ∈ L1C (Rm )X .2MQi2 i?2 6Qm`B2` i`Mb7Q`Kb Q7 i?2b2 irQ 7mM+iBQMb #v ϕ M/ ϕ(·, z) `2bT2+iBp2HvX amTTQb2 i?i ϕ ∈ L2C ( × Q). UNXjNV  aBM+2 +QM/BiBQM UNXjNV Bb i?2 bK2 b Rm |E [ϕ(t, Z)]|2 dt < ∞- r2 b22 i?i ϕ ∈ L2C (Rm ) M/ i?i 7Q` Q@HKQbi HH z ∈ K- ϕ(·, z) ∈ L2C (Rm )X LQi2 i?i ϕ(ν) = E [ϕ(ν, Z)] := ϕ(ν). 6BMHHv- bmTTQb2 i?i

ϕ ∈ BN .

UNX9yV

h?2Q`2K NX9XR URe V G2i N M/ {Zn }n∈N #2 b #Qp2 M/ H2i ϕ, ψ : Rm × K → R biBb7v +QM/BiBQMb UNXj3V- UNX9yV M/ UNXjNVX h?2M  cov





ϕ(Tn , Zn ) , 

n∈N

=

 ψ(Tn , Zn ) 

n∈N ∗

ϕ(ν) ψ (ν)μN (dν) + λ Rm

  cov ϕ(ν, Z), ψ ∗ (ν, Z) dν, UNX9RV

Rm

r?2`2 λ Bb i?2 BMi2MbBiv Q7 N M/ Z Bb  K@pHm2/ `M/QK p`B#H2 rBi? /Bbi`B#miBQM Q- i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 Zn ǶbX Re

("`ûKm/ M/ JbbQmHBû- kyyk)X

NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1

j88

S`QQ7X AM Q`/2` iQ ;Bp2

KQ`2 `2/#BHBiv iQ i?2 +H+mHiBQMb #2HQr- i?2 7QHHQrBM; MQiiBQM rBHH #2 mb2/, x∈N ϕ(x, Z(x)) := n∈N ϕ(Xn , Zn )X 6Q`KHHv,  E



 ϕ(t, Z(t))

t∈N



=E



 ψ(t, Z(t))

t∈N



 



=E

 ϕ(t)





=E 



ϕ(t)ψ (t ) + E

t,t ∈N,t=t









ψ (t )



−E

 t∈N



 ∗

ϕ(t, Z(t))ψ (t, Z(t))

t∈N





t ∈N

t∈N



ϕ(t, Z(t))ψ(t , Z(t )) + E

t,t ∈N,t=t







 ∗

ϕ(t, Z)ψ (t, Z) 







ϕ(t)ψ (t) +E

t∈N

 ∗

ϕ(t, Z)ψ (t, Z) .

t∈N

.2MQi2 #v (a)−(b)+(c) i?2 Hbi HBM2 /BbTHv2/X h?2 #Qp2 7Q`KH +QKTmiiBQMb `2 DmbiB}2/ #2+mb2 HH i?`22 i2`Kb `2- r?2M ϕ M/ ψ `2 `2TH+2/ #v i?2B` #bQHmi2 pHm2b- }MBi2X h?Bb 7QHHQrb 7`QK a+?r`xǶb BM2[mHBiv M/ i?2 7+ib i?i U7Q` (a) 2 m M/ (b)V ϕ M/ ψ `2 BM L2N (M2 ) M/ BM LC (R )c M/ 7Q` (c) #2+mb2 Q7 +QM/BiBQM UNXjNVX aBM+2 E t∈N ϕ(t, Z(t)) = E t∈N ϕ(t)  cov



ϕ(t, Z(t)),

t∈N



cov



 ψ(t, Z(t))

t∈N

 t∈N

ϕ(t),





ψ(t)

= 



−E

t∈N

 ∗

ϕ(t)ψ (t) + E

 

t∈N

 ∗

ϕ(t, Z)ψ (t, Z) .

t∈N

.2MQi2 #v A- B M/ C i?2 i?`22 i2`Kb QM i?2 `B;?i@?M/ bB/2 Q7 i?2 #Qp2 2[miBQMr?B+? i?2M `2/b A−B +CX "v /2}MBiBQM Q7 i?2 "`iH2ii bT2+i`mK M/ ?vTQi?2bBb UNX9yV ∗

A=

ϕ(ν)ψ (ν)μN (dν) . Rm

"v /2}MBiBQM Q7 i?2 BMi2MbBiv λ  ∗ B=λ ϕ(t)ψ(t) dt, C = λ Rm

E [ϕ(t, Z)ψ(t, Z)∗ ] dt . Rm

"v i?2 SHM+?2`2HĜS`b2pH B/2MiBiv  ϕ(ν)ψ(ν)∗ dν = λ ϕ(ν)ψ(ν)∗ dν B=λ m Rm R    =λ E [ϕ(ν, Z)] E ψ(ν, Z)∗ dν , Rm



M/ C = λE

ϕ(ν, Z)ψ(ν, Z)∗ dν ,

Rm

M/ i?2 `2bmHi UNX9RV 7QHHQrbX



j8e

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

1tKTH2 NX9Xk, a?Qi LQBb2X *QMbB/2` i?2 #Qp2 K`F2/ TQBMi T`Q+2bb UrBi? BM/2T2M/2Mi BB/ K`FbV M/ H2i h : Rm × K → R biBb7v +QM/BiBQMb UNX9yV M/ UNXjNVX G2i  X(t) := h(t − Xn , Zn ). n∈N

U_2K2K#2` i?i i?2 biQ+?biB+ T`Q+2bb {X(t)}t∈Rm Bb +HH2/  b?Qi MQBb2XV h?2M  h(t)dt,

E [X(t)] = λ Rm

M/

 +Qp (X(u), X(v)) =

r?2`2

e2iπν,u−v μX (dν) , Rm

&  &2   & & μX (dν) = &E h(ν, Z) & μN (dν) + λo` h(ν, Z) dν .

UNX9kV

h?Bb 7QHHQrb 7`QK i?2 7mM/K2MiH BbQK2i`v 7Q`KmH TTHB2/ iQ ϕ(t, z) = h(u−t, z) M/ ψ(t, z) = h(v − t, z)X 1tKTH2 NX9Xj, h?BMMBM;X G2i N #2  rbb biiBQM`v TQBMi T`Q+2bb QM Rm rBi? bT2+i`H K2bm`2 μN M/ i2bi 7mM+iBQM bT+2 BN X G2i {Zn }n∈Z #2 M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b BM/2T2M/2Mi Q7 N - iFBM; i?2B` pHm2b BM {0, 1} M/ bm+? i?i P (Z1 = 1) = αX .2}M2 i?2 TQBMi T`Q+2bb Nα #v Nα (C) =



(C ∈ B(Rm ) .

Zn 1{Xn ∈C} ,

n≥1

h?mb- Nα Bb Q#iBM2/ #v i?BMMBM; N -  TQBMi Q7 N #2BM; `2iBM2/ rBi? T`Q##BHBiv αX h?2 i?BMM2/ TQBMi T`Q+2bb Nα /KBib i?2 bT2+i`H K2bm`2 μNα := α2 μN + λα(1 − α) m

UNX9jV

M/ r2 Kv iF2 BNα := ϕ ∈ L1C (Rm ) ∩ L2C (Rm ) ∩ BN 7Q` i?2 i2bi 7mM+iBQM bT+2X hQ T`Qp2 i?Bb r2 Kmbi b?Qr i?i 7Q` Mv 7mM+iBQM ϕ˜ ∈ BNα 

 o` LQr

ϕ(x) ˜ Nα (dx) = Rm

Rm

 ϕ(x) ˜ Nα (dx) = Rm

& & & & ˜ & μNα (dν) . &ϕ(ν)



Zn ϕ(X ˜ n) ,

n≥1

M/ i?2`27Q`2- i?2 `2bmHi 7QHHQrb 7`QK h?2Q`2K NX9XR- #v TTHvBM; 7Q`KmH UNX9RV rBi? ϕ(x, z) = ψ(x, z) = z ϕ(x)  rBi? ϕ  ∈ BN U1t2`+Bb2 NX8XNVX

NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1

j8d

1tKTH2 NX9X9, CBii2`BM;X *QMbB/2` i?2 K`F2/ TQBMi T`Q+2bb Q7 h?2Q`2K  Bb /2}M2/ #v Bib b2[m2M+2 Q7 TQBMib NX9XR- rBi? K = Rm X  TQBMi T`Q+2bb N {Xn + Zn }n∈N . h?2M- rBi? λ i?2 BMi2MbBiv Q7 N M/ μN i?2 "`iH2ii bT2+i`mK Q7 N   μN (dν) = |ψZ (ν)|2 μN (dν) + λ 1 − |ψZ (ν)|2 dν , r?2`2

ψZ (ν) := E e2iπν,Z

UNX99V UNX98V

Bb i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 i?2 `M/QK /BbTH+2K2Mib /Bbi`B#mi2/ b QX q2 +M iF2   ˜ + Z)] ∈ BN M/ ϕ˜ ∈ L1C (Rm ) ∩ L2C (Rm ) . UNX9eV BN˜ = ϕ˜ ; E [ϕ(t 6Q` i?2 T`QQ7- /2}M2 ϕ(t, z) = ϕ(t+z)X  *QM/BiBQMb UNXj3V M/ UNXjNV 7Q` i?2 7mM+iBQM ϕ `2 2[mBpH2Mi iQ +QM/BiBQMb ϕ˜ ∈ L1C (Rm ) M/ ϕ˜ ∈ L2C (Rm ) `2bT2+iBp2Hv- bBM+2 7Q` Mv p ≥ 0 

 E Rm

|ϕ(t, Z)|p dt =

Rm

|ϕ(t)| ˜ p dt .

*QM/BiBQM UNX9yV 7Q` i?2 7mM+iBQM ϕ Bb biBb}2/ #v i?2 / ?Q+ /2}MBiBQM Q7 BN˜ X q2 Kv i?2`27Q`2 TTHv h?2Q`2K NX9XRX q2 ?p2 ϕ(ν, z) = e2iπν,z ϕ(ν),  ϕ(ν) = ϕ(ν) = ψZ (ν)ϕ(ν),  &2  &&  & ∗ +Qp (ϕ(ν, Z), ϕ(ν, Z) ) = 1 − |ψZ (ν)|2 &ϕ(ν)  & . HbQ-

 o`

Rm

 (dt) ϕ(t)  N



 = o`

M/ i?2`27Q`2- TTHvBM; 7Q`KmH UNX9RV    (dt) = ϕ(t)  N o` Rm



 ϕ(X ˜ n + Zn )

,

n∈N

Rm

& &2 & &  & μN (dν) , &ϕ(ν)

r?2`2 μN Bb b BM UNX99VX 1tKTH2 NX9X8, h?2 CBii2`2/ :`B/X q2 +QMbB/2` i?2 +b2 r?2`2 N Bb i?2 ;`B/ Q7 1tKTH2 NXkXRkX q2 +M iF2    && n1 n2 && 1 2 2 2 & & BN˜ = ϕ˜ ; , ) < ∞ M/ ϕ˜ ∈ LC (R ) ∩ LC (R ) . ˜ &ϕ( T1 T2 & n1 ,n2 ∈Z

S`QQ7X AM/22/- Q#b2`pBM; i?i

j83

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1 & & & & &  & & & ˜ + Z)](ν)& = &ϕ(ν) ˜ &, &E [ϕ(·

r2 b22 i?i +QM/BiBQM E [ϕ(t ˜ + Z)] ∈ BN Bb 2[mBpH2Mi iQ i?2 +QM/BiBQM  && n1 n2 && &ϕ( & & ˜ T1 , T2 )& < ∞ . n ,n ∈Z 1

2



1tKTH2 NX9Xe, h?2 CBii2`2/ *Qt S`Q+2bbX G2i N #2 i?2 *Qt T`Q+2bb Q7 1tKTH2 NXkXRjX q2 +M iF2   BN˜ = ϕ˜ ; ϕ˜ ∈ L1C (Rm ) ∩ L2C (Rm ) . AM/22/- +QM/BiBQM E [ϕ(t ˜ + Z)] ∈ BN - i?i Bb- BM i?Bb T`iB+mH` +b2- E [ϕ(t ˜ + Z)] ∈ L1C (Rm ) ∩ L2C (Rm )- Bb 2t+iHv ϕ˜ ∈ L1C (Rm ) ∩ L2C (Rm )X 1tKTH2 NX9Xd, h?2 *Hmbi2` SQBMi T`Q+2bbX G2i N #2  rB/2@b2Mb2 biiBQM@ `v TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λ > 0 `2T`2b2Mi2/ #v Bib TQBMi b2[m2M+2 {Xn }n≥1 - rBi? TQr2` bT2+i`H K2bm`2 μN M/ i2bi 7mM+iBQM bT+2 BN X G2i {Zn }n≥1 #2 M BB/ +QHH2+iBQM Q7 TQBMi T`Q+2bb2b QM Rm - BM/2T2M/2Mi Q7 N X G2i Z #2  TQBMi T`Q+2bb QM Rm rBi? i?2 bK2 /Bbi`B#miBQM b i?2 +QKKQM /Bbi`B#miBQM Q7 i?2 Zn ǶbX .2}M2 

ψZ (ν) := E e2iπν,t Z (dt) . Rm

h?2 7mM+iBQM ψZ Bb r2HH /2}M2/ mM/2` i?2 bbmKTiBQM E [Z (Rm )] < ∞. UAM T`iB+mH`- Z Bb HKQbi bm`2Hv  }MBi2 TQBMi T`Q+2bbXV q2 MQr /2}M2 irQ TQBMi  M/ N -QM Rm #v T`Q+2bb N   (C) =N (C) + N Zn (C − Xn ) , N (C) =



n≥1

Zn (C − Xn ) ,

n≥1

rBi? i?2 Q#D2+iBp2 Q7 +QKTmiBM; i?2B` "`iH2ii bT2+i`H K2bm`2b- bi`iBM; rBi? i?2 }`bi QM2X 6Q`KHHv        (dt) =o` o` ϕ (Xn ) + ϕ(t)N ϕ (Xn + s) Zn (ds) Rm

 =o`

n≥1

 n≥1



Rm

ϕ (Xn , Zn ) ,

NX9X h_La6P_JhAPLa P6 h>1 aS1*h_G J1al_1

j8N



r?2`2 ϕ (x, z) = ϕ (x) +

ϕ (x + s) z (ds) . Rm



q2 ?p2



E [ϕ (x, Z)] = ϕ (x) + E

ϕ (x + s) Z (ds) Rm





 ϕ (ν, z) = ϕ (ν) + 

Rm



Rm

= ϕ (ν) + R

ϕ (t + s) z (ds) e−2iπν,t dt  ϕ (t + s) e−2iπν,t dt z (ds)

Rm

m

ϕ (ν) e2iπν,s z (ds) = ϕ (ν) + m  R  2iπν,s = ϕ (ν) 1 + e z (ds) . Rm

LQi2 i?i i?2 2t+?M;2 Q7 Q`/2` Q7 BMi2;`iBQM Bb MQi  T`Q#H2K B7 z Bb  }MBi2 TQBMi T`Q+2bb- BM T`iB+mH` B7 z Bb `2TH+2/ #v Bib `M/QK p2`bBQM ZX HbQ ϕ (ν) = ϕ (ν) (1 + ψZ (ν)) . TTHvBM; 7Q`KHHv h?2Q`2K NX9XR- r2 Q#iBM  o`



 ϕ (Xn , Zn )

 = Rm

n≥1

|ϕ (ν)|2 |1 + ψZ (ν)|2 μN (dν) 



+λ Rm

P#b2`p2 i?i o`



|ϕ (ν)| o` 2



 e2iπν,s Z (ds)

1+ Rm



iQ Q#iBM o`

 n∈N

 ϕ (Xn )



 1+

e

2iπν,s

Z (ds) dν .

Rm



 e2iπν,s Z (ds)

= o`

Rm

 = Rm

r?2`2

|ϕ (ν)|2 μN (dν) , 

μN (dν) = |1 + ψZ (ν)|2 μN (dν) + λo`

 e2iπν,s Z (ds) dν

Rm

 X aBKBH` +QKTmiiBQMb H2/ iQ Bb i?2 "`iH2ii bT2+i`H K2bm`2 Q7 N   2 2iπν,s e Z (ds) dν. μN (dν) = |ψZ (ν)| μN (dν) + λo` Rm

UNX9dV

UNX93V

jey

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

JmHiBp`Bi2 SQBMi S`Q+2bb2b ii2MiBQM rBHH #2 `2bi`B+i2/ iQ #Bp`Bi2 TQBMi T`Q+2bb2b- i?2 2ti2MbBQM iQ M `#B@ i``v MmK#2` Q7 /BK2MbBQMb #2BM; BKK2/Bi2X  #Bp`Bi2 TQBMi T`Q+2bb QM Rm Bb Dmbi  TB` (N1 , N2 ) Q7 TQBMi T`Q+2bb2b QM Rm X Ai Bb +HH2/  b2+QM/@Q`/2` #Bp`Bi2 TQBMi T`Q+2bb B7 #Qi? N1 M/ N2 `2 b2+QM/@Q`/2` TQBMi T`Q+2bb2bX Ai Bb +HH2/ rbb B7 #Qi? N1 M/ N2 `2 rbb M/ B7 KQ`2Qp2` i?2v `2 DQBMiHv rbb- i?i Bb B7 7Q` HH #QmM/2/ b2ib A, B ∈ B(Rm ) M/ HH t ∈ Rm E [N1 (A + t)N2 (B + t)] = E [N1 (A)N2 (B)] . AM i?Bb bBimiBQM- r2 b?HH bv i?i N1 M/ N2 /KBi i?2 +`Qbb@bT2+i`H K2bm`2 μN1 ,N2 B7 i?2 Hii2` Bb  σ@}MBi2 bB;M2/ K2bm`2 bm+? i?i 7Q` HH ϕ1 ∈ BN1 - ϕ2 ∈ BN2  cov (N1 (ϕ1 ), N2 (ϕ2 )) = ϕ 21 (ν)2 ϕ2 (ν)∗ μN1 ,N2 (dν) . Rm

1tKTH2 NX9X3, h?2 "Bp`Bi2 rbb *Qt S`Q+2bbX G2i N1 M/ N2 #2 rbb *Qt T`Q+2bb2b QM R r?Qb2 `2bT2+iBp2 biQ+?biB+ BMi2MbBiB2b {λ1 (t)}t∈R M/ {λ2 (t)}t∈R `2 DQBMiHv biiBQM`v rbb biQ+?biB+ T`Q+2bb2b rBi? `2bT2+iBp2 bT2+i`H K2bm`2 μλ1 M/ μλ2 - M/ +`Qbb@bT2+i`H K2bm`2 μλ1 ,λ2 X 6Q` ϕ1 , ϕ2 ∈ L1C (R) ∩ L2C (R) biM/`/ +QKTmiiBQMb b?Qr i?i E [N1 (ϕ1 )N2 (ϕ2 )∗ ] = E E N1 (ϕ1 )N2 (ϕ2 )∗ |F λ1 ,λ2 = E E N1 (ϕ1 )|F λ1 ,λ2 E N1 (ϕ1 )∗ |F λ1 ,λ2

  ϕ1 (t)λ1 (t) dt × ϕ2 (t)∗ λ2 (t) dt =E R

M/

 E [N1 (ϕ1 )] E [N2 (ϕ2 )∗ ] = E R

R



ϕ1 (t)λ1 (t) dt × E ϕ2 (t)∗ λ2 (t) dt . R

h?2`27Q`2

   cov (N1 (ϕ1 ), N2 (ϕ2 )) = cov ϕ1 (t)λ1 (t) dt, ϕ2 (t)λ2 (t) dt R R  ∗ = ϕ 21 (ν)2 ϕ2 (ν) μλ1 ,λ2 (dν) . Rm

h?2`27Q`2 μN1 ,N2 = μλ1 ,λ2 .

h?2 mMBp2`bH +Qp`BM+2 7Q`KmH +M #2 mb2/ iQ +QKTmi2 +`Qbb@bT2+i`H K2@ bm`2b #2ir22M  TQBMi T`Q+2bb M/ Bib i`Mb7Q`K #v i?2 #Qp2 QT2`iBQMb Q7 i?BM@ MBM;- +Hmbi2`BM;- i`MbHiBQM Q` DBii2`BM;- b r2HH b #2ir22M i?2 i`Mb7Q`Kb i?2K@ b2Hp2bX 6Q` BMbiM+2,

NX8X 1s1_*Aa1a

jeR

1tKTH2 NX9XN, *`Qbb@bT2+i`H J2bm`2 Q7  SQBMi S`Q+2bb M/ Bib CBii2`2/ o2`bBQMX h?2 Q#i2MiBQM Q7 i?2 +`Qbb@bT2+i`H K2bm`2 Q7  rbb TQBMi T`Q+2bb N := N1 rBi? Bib DBii2`2/ p2`bBQM N2 Q7 *Q`QHH`v NX9X9 `2[mB`2b mb iQ +QKTmi2     cov ϕ(Xn ), ψ(Xn + Zn ) . n∈Z

n∈Z

6Q`KmH UNX9RV `2/m+2b BM i?Bb +b2 iQ      ϕ(Xn ) , ψ(Xn + Zn ) = cov n≥1

Rm

n≥1

"mi

  ϕ(ν)E ψ(ν + Z)∗ μN (dν).



ψ(t + Z) e−2iπνt dt

ψ(ν + Z) = m R

=

ψ(t) e−2iπν(t−Z) dt = ψ(ν)E e+2iπνZ ,

Rm

r?2`2 i?2 2tT2+iiBQM Bb rBi? `2bT2+i iQ Z  `M/QK p`B#H2 rBi? i?2 +QKKQM T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 K`FbX 6BMHHv      ϕ(Xn ), ψ(Xn + Zn ) = ϕ(ν)ψ(ν)∗ E e−2iπνZ μN (dν) , cov n∈Z

n∈Z

M/ i?2`27Q`2

NX8

Rm

μN1 ,N2 (dν) = E e−2iπνZ μN (dν) .

1t2`+Bb2b

1t2`+Bb2 NX8XRX a2+QM/ JQK2Mi J2bm`2 Q7  SQBbbQM S`Q+2bb G2i N #2  SQBbbQM T`Q+2bb QM Rm rBi? BMi2MbBiv K2bm`2 νX UBV a?Qr i?i BM Q`/2` 7Q` N iQ #2  b2+QM/@Q`/2` TQBMi T`Q+2bb- Bi Bb M2+2bb`v M/ bm{+B2Mi i?i i?2 BMi2MbBiv K2bm`2 #2 HQ+HHv }MBi2X a?Qr i?i BM i?Bb +b2i?2 b2+QM/ KQK2Mi K2bm`2 M2 Bb ;Bp2M #v M2 (A × B) = ν(A ∩ B) − ν(A)ν(B) . UBBV a?Qr i?i 7Q` HH ϕ, ψ ∈ L1Rm (ν) ∩ L2Rm (ν)   E [N (ϕ)N (ψ)∗ ] = ϕ (t) ψ ∗ (t) ν(dt) + ϕ (t) ν(dt) ψ ∗ (t) ν(dt) . R

R

R

jek

*>Sh1_ NX h>1 SPq1_ aS1*h_G J1al_1

1t2`+Bb2 NX8XkX h?2 _BTH2v 7mM+iBQM :Bp2 i?2 _BTH2v 7mM+iBQM Q7 i?2 L2vKMĜa+Qii +Hmbi2` TQBMi T`Q+2bbX 1t2`+Bb2 NX8XjX aiiBQM`Biv BM h2`Kb Q7 AMi2;`Hb G2i N #2  b2+QM/@Q`/2` TQBMi T`Q+2bb QM Rm X a?Qr i?i 7Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+iBQMb ϕ, ψ : Rm → R- i?2 [mMiBiv    

E ϕ(τ + t) N (dt) ψ(τ + t) N (dt) R

R

Bb BM/2T2M/2Mi Q7 τ ∈ RX 1t2`+Bb2 NX8X9X h2bi 6mM+iBQMb 7Q` i?2 :`B/ AM 1tKTH2 NXkXRk- b?Qr i?i i?2 i2bi 7mM+iBQM bT+2   &  &  n1 n2 && 1 2 2 2 & & BN := ϕ ; ϕ ∈ LC (R ) ∩ LC (R ) M/ &ϕ T1 , T2 & < ∞ n1 ,n2 ∈Z

BM/22/ biBb}2b i?2 mMB[m2M2bb `2[mB`2K2Mi- i?i Bb, B7 i?2 HQ+HHv }MBi2 K2bm`2b μ1 M/ μ2 `2 bm+? i?i   2 |ϕ(ν)| μ1 (dν) = |ϕ(ν)|2 μ2 (dν) Rm

Rm

7Q` HH ϕ ∈ BN - i?2M μ1 ≡ μ2 X 1t2`+Bb2 NX8X8X lMB[m2M2bb BM h?2Q`2K NXkXRy S`QpB/2 i?2 /2iBHb 7Q` i?2 mMB[m2M2bb bii2K2Mi BM h?2Q`2K NXkXRyX 1t2`+Bb2 NX8XeX "`iH2ii aT2+i`mK Q7  _2M2rH S`Q+2bb- A a?Qr i?i B7 i?2 ?x`/ `i2 Q7 i?2 `2M2rH /Bbi`B#miBQM UbbmK2/ iQ ?p2  /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2V Bb #QmM/2/ #Qp2 M/ #2HQr #v }MBi2 M/ bi`B+iHv TQbBiBp2 +QMbiMib- +QM/BiBQM UNXjRV Bb biBb}2/X 1t2`+Bb2 NX8XdX "`iH2ii aT2+i`mK Q7  _2M2rH S`Q+2bb- AA *QMbB/2`  `2M2rH T`Q+2bb r?Qb2 `2M2rH /Bbi`B#miBQM ?b  /2MbBiv rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2X amTTQb2 i?i i?2 mM/2Hv2/ p2`bBQM M/ i?2 biiBQM`v p2`bBQM `2 +QMbi`m+i2/ pB :`B;2HBQMBbǶ BK#2//BM; i?2Q`2K Uh?2Q`2K 8XdXjV 7`QK i?2 bK2 SQBbbQM K2bm`2 QM R2 M/ i?i i?2v +QmTH2 i i?2 `M/QK iBK2 T X a?Qr i?i B7 i?2 ?x`/ `i2 Q7 i?2 `2M2rH /Bbi`B#miBQM Bb #QmM/2/ 7`QK #Qp2+QM/BiBQM UNXjRV Bb biBb}2/ B7 i?2 2tT2+i2/ +QmTHBM; iBK2 Bb }MBi2X 1t2`+Bb2 NX8X3X _2M2rH S`Q+2bb2b rBi? /7` .Bbi`B#miBQM h?Bb 2t2`+Bb2 +QMiBMm2b 1tKTH2 8XdX9X a?Qr i?i BM i?Bb +b2  ∞  ∞ |u(t) − λ| dt = (u(t) − λ) dt 0

0

NX8X 1s1_*Aa1a

jej

M/ i?i i?Bb [mMiBiv Bb bKHH2` i?M E [N ((0, T1 ))]X *QM+Hm/2X 1t2`+Bb2 NX8XNX h?BMMBM; S`QpB/2 i?2 /2iBHb BM i?2 T`QQ7 Q7 1tKTH2 NX9XjX 1t2`+Bb2 NX8XRyX *Hmbi2`BM; i?2 :`B/ *QMbi`m+i  TQBMi T`Q+2bb N QM R b 7QHHQrbX G2i T > 0 M/ H2i U #2  `M/QK p`B#H2 mMB7Q`KHv /Bbi`B#mi2/ QM [0, T ]X G2i L #2  }t2/ TQbBiBp2 BMi2;2`X G2i {Vn, }n∈Z,1≤≤L #2 M BB/ b2[m2M+2 Q7 `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, T ]- BM/2T2M/2Mi Q7 U X h?2 +QHH2+iBQM Q7 TQBMib Q7 N Bb {nT +Vn, }n∈Z,1≤≤L X a?Qr i?i N Bb biiBQM`v- M/ ;Bp2 Bib "`iH2ii bT2+i`H K2bm`2 rBi? M bbQ+Bi2/ i2bi 7mM+iBQM bT+2X 1t2`+Bb2 NX8XRRX *Hmbi2`BM; i?2 SQBbbQM S`Q+2bb  M/ N BM i?2 _272` iQ 1tKTH2 NX9XdX *QKTmi2 i?2 TQr2` bT2+i`H K2bm`2b Q7 N bT2+BH +b2 r?2`2 N Bb  SQBbbQM T`Q+2bb QM R rBi? BMi2MbBiv λ M/ i?2 b2[m2M+2 Q7 TQBMib Q7 i?2 ;2M2`B+ +Hmbi2` Z Bb a- 2a- Ę- Θa- r?2`2 a Bb  TQbBiBp2 MmK#2` M/ Θ Bb  SQBbbQM `M/QK p`B#H2 rBi? K2M θ > 0X 1t2`+Bb2 NX8XRkX *`Qbb@bT2+i` *QKTmi2 i?2 +`Qbb@bT2+i`H K2bm`2b #2ir22M  TQBMi T`Q+2bb M/ Bib i`Mb7Q`K #v i?2 QT2`iBQMb Q7 i?BMMBM;- +Hmbi2`BM;- i`MbHiBQM Q` DBii2`BM;- b r2HH b #2ir22M i?2 i`Mb7Q`Kb i?2Kb2Hp2bX

*?Ti2` Ry h?2 AM7Q`KiBQM *QMi2Mi Q7 SQBMi S`Q+2bb2b h?2 }`bi irQ b2+iBQMb Q7 i?Bb +?Ti2` /2KQMbi`i2 i?2 TQr2` Q7 i?2 K`iBM;H2 TT`Q+? iQ TQBMi T`Q+2bb2b #v K2Mb Q7 bBKTH2 2tKTH2b r?B+? Kv b2`p2 b M BMi`Q/m+iBQM iQ i?2 KQ`2 2H#Q`i2 i?2Q`v M/ TTHB+iBQMbXR h?2 Hbi b2+iBQM i`2ib irQ Bbbm2b i?i +QK2 mT r?2M bKTHBM;  rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb i `M/QK iBK2b 7Q`KBM;  TQBMi T`Q+2bb, }`biHv- ?Qr Km+? Q7 i?2 Q`B;BMH TQr2` bT2+i`H K2bm`2 +M #2 `2+Qp2`2/ 7`QK i?2 `M/QK bKTH2b\ M/ b2+QM/Hv?Qr 7` 7`QK i?2 Q`B;BMH Bb i?2 T`Q+2bb Q#iBM2/ #v HBM2` }Hi2`BM; TTHB2/ iQ i?2 bKTH2/ bB;MH\ h?2 +HbbB+H a?MMQMĜLv[mBbi `2+QMbi`m+iBQM i?2Q`2K Q7 #M/@ HBKBi2/ bB;MHb Bb  T`iB+mH` +b2 Q7 i?2 ;2M2`H i?2Q`vX

RyXR

6BHi2`BM;

h?2 i2`K }Hi2`BM; `272`b iQ i?2 T`Q#H2K Q7 2biBKiBM; i?2 pHm2 i Mv iBK2 Q7 bQK2 biQ+?biB+ T`Q+2bb ;Bp2M bQK2 T`iBH BM7Q`KiBQM ;i?2`2/ i i?Bb iBK2X JQ`2 T`2@ +Bb2Hv- H2i {X(t)}t≥0 #2 M Ft @T`Q;`2bbBp2Hv K2bm`#H2 biQ+?biB+ T`Q+2bb- +HH2/ i?2 bii2 T`Q+2bb- rBi? pHm2b BM bQK2 K2bm`#H2 bT+2 (E, E)X G2i {Ot }t≥0 #2  }Hi`iBQM- +HH2/ i?2 Q#b2`p2/ ?BbiQ`v- bm+? i?i Ot ⊆ Ft Ut ≥ 0VX h?2 σ@}2H/ Ot `2T`2b2Mib i?2 BM7Q`KiBQM pBH#H2 i iBK2 t iQ Q#iBM M 2biBKi2 Q7 i?2 T`Q#@ #BHBiv /Bbi`B#miBQM Q7 i?2 bii2 T`Q+2bb i iBK2 tX aT2+B}+HHv QM2 b22Fb iQ +H+mHi2 7Q` Mv K2bm`#H2 7mM+iBQM ϕ : (E, E) → (R, B(R)) i?2 [mMiBiv E [Z(t) | Ot ]r?2`2 Z(t) := ϕ(X(t)) Bb bbmK2/ BMi2;`#H2X AM i?2 bQ@+HH2/ BMMQpiBQMb T@ T`Q+? iQ i?2 }Hi2`BM; T`Q#H2K- i?2 biQ+?biB+ T`Q+2bb {Z(t)}t≥0 iF2b i?2 7Q`K Q7  dzb2KB@K`iBM;H2Ǵ M/ i?2 K`iBM;H2 `2T`2b2MiiBQM i?2Q`2K THvb  TBpQiH `QH2X (N,Z)

AM i?Bb +?Ti2`- i?2 Q#b2`p2/ ?BbiQ`v Bb Q7 i?2 7Q`K Ot = Ft Bb  K`F2/ TQBMi T`Q+2bbX R

- r?2`2 (N, Z)

a22 7Q` BMbiM+2 ("Q2H- o`Bv M/ qQM;- RNd8)- ("`ûKm/- RN3R)X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_10

je8

jee

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

a2KB@K`iBM;H2 aii2 S`Q+2bb h?2 bii2 T`Q+2bb {Z(t)}t≥0 Bb bbmK2/ iQ #2  `2H@pHm2/ biQ+?biB+ T`Q+2bb Q7 i?2 7Q`K  t Z(t) = Z(0) + f (s) ds + m(t) , URyXRV 0

r?2`2 UBV {f (t)}t≥0 Bb M Ft @T`Q;`2bbBp2 biQ+?biB+ T`Q+2bb bm+? i?i E ∞ 7Q` HH t ≥ 0- M/



t 0

 |f (s)| ds
0- 7`QK i?2 pHm2 a iQ i?2 pHm2 b- Q7  TQBMi T`Q+2bb N QM i?2 TQbBiBp2 ?H7@HBM2X JQ`2 T`2+Bb2Hv- N Bb  *Qt TQBMi T`Q+2bb rBi? `2bT2+i iQ i?2 σ@}2H/ G := σ(τ ) r?Qb2 Ft @BMi2MbBiv ěr?2`2 Ft := FtN ∨ σ(τ )ě Bb λ(t) := a + (b − a)Z(t) , r?2`2 Z(t) := 1{τ ≤t} . .2MQiBM; #v F i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 τ - M/ bbmKBM; i?i Bi ?b  /2MbBiv f rBi? `2bT2+i iQ i?2 G2#2b;m2 K2bm`2- Bib 7BHm`2 `i2 Bb f (t) 1 − F (t)

h(t) := M/ i?2`27Q`2 i?2 T`Q+2bb

 Z(t) −

t∧τ

h(s)ds 0

Bb M Ft @K`iBM;H2X 1[mBpH2MiHv  t h(s)(1 − Z(s)) ds + m(t) , Z(t) = 0

r?2`2 m(t) = Ft @K`iBM;H2X

1tKTH2 RyXRXk, J`FQp *?BM a2KB@K`iBM;H2 1[miBQMX G2i {X(t)}t≥0 #2  bi#H2 M/ +QMb2`piBp2 ?K+ rBi? /2MmK2`#H2 bii2 bT+2 E M/ BM}MBi2b@ BKH ;2M2`iQ` AX 6Q` 2+? TB` (i, j) Q7 /BbiBM+i bii2b- H2i Ni,j #2 i?2 TQBMi k AM 7+i-  T`iB+mH` 7Q`K Q7 Bi- #mi r2 b?HH MQi M22/ iQ +QMbB/2` KQ`2 ;2M2`H b2KB@ K`iBM;H2b BM i?Bb +?Ti2`X

RyXRX 6AGh1_AL:

jed

T`Q+2bb +QmMiBM; i?2 i`MbBiBQMb 7`QK i iQ j- r?Qb2 T`2/B+i#H2 biQ+?biB+ FtX @ BMi2MbBiv Bb Ub22 1tKTH2 8XRXRjV λij (t) := qij 1{X(t−)=i} X Ai rb b?QrM BM am#@ b2+iBQM kX8 M/ 1t2`+Bb2 kXdXR8 i?i- /2}MBM; Zi (t) := 1{X(t)=i} M/ Mij (t) := t Nij ((0, t]) − 0 Zi (s)qij ds UM FtX @K`iBM;H2V

t

AZ(s) ds + M (t) ,

Z(t) = Z(0) +

URyXkV

0

r?2`2 Z(t) = {Zi (t)}i∈E - M (t) = {Mi (t)}i∈E - M/ 7Q` 2+? i ∈ E- {Mi (t)}t≥0 Bb M FtX @K`iBM;H2X 6Q`KmH URyXkV `2/b BM 2tTM/2/ 7Q`K ⎛



t

Zi (t) = Zi (0) + 0





⎞ Zj (s)qji − Zi (s)qi ⎠ ds + Mi (t)

(i ∈ E) .

URyXjV

j∈E\i

S`QD2+iBQM Q7 i?2 a2KB@J`iBM;H2 aii2 QM i?2 P#b2`piBQM >BbiQ`v PM2 b22Fb iQ +QKTmi2 7Q` 2+? iBK2 t i?2 2biBKi2 Z(t) := E [Z(t) | Ot ]X h?2 }`bi bi2T Q7 i?2 BMMQpiBQMb K2i?Q/ +QMbBbib BM T`QD2+iBM; 1[MX URyXRV QM i?2 Q#b2`p2/ ?BbiQ`vX G2KK RyXRXj lM/2` i?2 #Qp2 bbmKTiBQMb Z(t) = Z(0) +

t

f (s) ds + m(t) ,

URyX9V

0

r?2`2, UV {f (t)}t≥0 Bb M Ot @T`2/B+i#H2 biQ+?biB+ T`Q+2bb mMB[m2Hv /2}M2/ #v i?2 `2[mB`2K2Mi i?i 7Q` HH t ≥ 0 M/ HH #QmM/2/ Ot @T`2/B+i#H2 biQ+?biB+ T`Q+2bb2b {C(t)}t≥0  t

 t

E C(s)f (s) ds = E C(s)f (s) ds . 0

0

U#V {m(t)}t≥0 Bb  +2Mi2`2/ Ot @K`iBM;H2X S`QQ7X h?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 {f (t)}t≥0 biBb7vBM; UV 7QHHQrb 7`QK i?2 bK2 _/QMĜLBFQ/ɷK `;mK2Mib b BM i?2 T`QQ7 Q7 i?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 T`2/B+i#H2 BMi2MbBivX h?2 /2iBHb `2 H27i b 1t2`+Bb2 RyXeXRX 6`QK URyXRV 

t

E [Z(t) | Ot ] = E [Z(0) | Ot ] + E

f (s) ds | Ot + E [m(t) | Ot ] . 0

h?2 T`Q+2bb



je3

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a m(t) := (E [Z(0) | Ot ] − E [Z(0)])   t

 t  + E f (s) ds | Ot − f (s) ds 0

0

+ E [m(t) | Ot ] := A + B + C Bb  0@K2M Ot @K`iBM;H2X h?Bb Bb +H2`Hv i`m2 7Q` i?2 T`ib A M/ CX hQ T`Qp2 i?Bb 7Q` B r2 ?p2 iQ b?Qr i?i 7Q` HH s U0 ≤ s ≤ tV M/ HH A ∈ Os

 s



 t   t

f (u) du | Ot − E f (u) du | Os = E 1A E 1A E f (u) du . () 0

0

s

h?2 H27i@?M/ bB/2 Q7 UV +M #2 `2r`Bii2M b 

 t

  t  s f (u) du − f (u) du = E 1A f (u) du = E E 1A 0

0

s



f (u)C(u) du

0

r?2`2 C(u, ω) := 1A (ω)1(s,t] (u) /2}M2b  MQM@M2;iBp2 Ou @T`2/B+i#H2 biQ+?biB+ T`Q+2bbX b 7Q` i?2 `B;?i@?M/ bB/2 Q7 UV- Bi Bb 2[mH iQ  t

 ∞

E 1A f (u) du = E f (u)C(u) du . s

0

h?2 2[mHBiv UV i?2M 7QHHQrb 7`QK i?2 /2}MBiBQM Q7 {f (t)}t≥0 X



Ai Bb ?2M+27Q`i? bbmK2/ i?i i?2 Q#b2`p2/ ?BbiQ`v Bb Q7 i?2 7Q`K (N,Z)

Ot = G ∨ F t

,

r?2`2 (N, Z) Bb  bBKTH2 HQ+HHv }MBi2 K`F2/ TQBMi T`Q+2bb QM R+ rBi? K`Fb BM K M/ bbQ+Bi2/ HB7i2/ T`Q+2bb NZ QM R+ × KX h?2`27Q`2 i iBK2 t QM2 Bb iQ #b2 i?2 2biBKi2 Q7 i?2 bii2 Z(t) QM i?2 Q#b2`piBQM Q7 (N, Z) #27Q`2 t- THmb Mv //BiBQMH BM7Q`KiBQM +QMiBM2/ BM GX Ai Bb KQ`2Qp2` bbmK2/ i?i NZ /KBib i?2 (P, Ot )@HQ+H +?`+i2`BbiB+b (λ(t), Φ(t, dz))X h?2 AMMQpiBQMb :BM h?2 b2+QM/ bi2T BM i?2 BMMQpiBQMb K2i?Q/ /2T2M/b +`m+BHHv QM i?2 K`iBM;H2 `2T`2b2MiiBQM i?2Q`2K Uh?2Q`2K 8XjX9V- r?B+? ;m`Mi22b i?2 2tBbi2M+2 Q7 M BM/2t2/ biQ+?biB+ T`Q+2bb K ∈ P(O· ) ⊗ K- +HH2/ i?2 BMMQpiBQMb ;BM- bm+? i?i  t |K(s, z)|λ(s) Φ(s, dz) ds < ∞ (t ≥ 0) 0

K



M/



2Z (ds × dz) K(s, z)M

m(t) = (0,t]

(t ≥ 0) ,

K

2Z (ds × dz) := NZ (ds × dz) − λ(s)Φ(s, dz) dsX h?2`27Q`2r?2`2 M

RyXRX 6AGh1_AL:

jeN 





t

2Z (ds × dz) . K(s, z)M

f (s) ds +

Z(t) = Z(0) + 0

(0,t]

URyX8V

K

Ai `2KBMb iQ /2i2`KBM2 i?2 BMMQpiBQMb ;BM K BM i2`Kb i?i Kv #2 K/2 2tTHB+Bi Hi2` QMX h?2 ;2M2`H T`Q+2/m`2 Bb i?2 7QHHQrBM;X aBM+2 Z(t) Bb i?2 +QM/BiBQMH 2tT2+iiBQM Q7 Z(t) ;Bp2M Ot - M2+2bb`BHv  E [Z(t)U (t)] = E Z(t)U (t) URyXeV 7Q` HH #QmM/2/ Ot @K`iBM;H2b {U (t)}t≥0 U1t2`+Bb2 RyXeXkVX am+? K`iBM;H2b ?p2 i?2 7Q`K   U (t) = H(s, z)M Z (ds × dz) , (0,t]

K

t r?2`2 H Bb P(O. ) × K@K2bm`#H2 M/ KQ`2Qp2` 0 K |H(s, z)|λ(s) Φ(s, dz) ds < ∞X 1[mHBiv URyXeV rBHH T`QpB/2 M 2tT`2bbBQM Q7 i?2 BMMQpiBQMb ;BMX h?2 /2iBHb rBHH #2 ;Bp2M 7Q` M Q#b2`piBQM i?i Bb  KmHiBp`Bi2 TQBMi T`Q+2bbb /2b+`B#2/ BM 1tKTH2 8XjXe- iQ r?B+? r2 `272` 7Q` i?2 MQiiBQMX h?mb Ot = FtN M/ i?2`27Q`2  t   2i (ds) , Ki (s)M f (s) ds + Z(t) = Z(0) + 0

(0,t]

i∈E

K

2i (ds) := Ni (ds) − λi (s) ds M/ r?2`2 {λi (t)}t≥0 Bb i?2 Ot @T`2/B+i#H2 r?2`2 M biQ+?biB+ BMi2MbBiv Q7 Ni X h?2Q`2K RyXRX9 G2i {Ψij (t)}t≥0 #2 Ot @T`2/B+i#H2 T`Q+2bb2b bm+? i?i- 7Q` HH i ∈ E M/ HH MQM@M2;iBp2 #QmM/2/ Ot @T`2/B+i#H2 T`Q+2bb2b {C(t)}t≥0 ,

 t

 t C(s)Z(s)λi (s) ds = E C(s)Ψi1 (s)λi (s) ds , E 0



t



0



C(s)Ψi2 (s)λi (s) ds

C(s)Z(s)λi (s) ds = E

E 0



t

,

0

M/- rBi? Δg(s) := g(s) − g(s−)   t

 i E C(s)Δm(s)ΔNi (s) = E C(s)Ψ3 (s)λi (s) ds . 0

0Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

_2K`F RyXRX8 PM2 Hrvb ?b Ψi2 (t) = Z(t−) . HbQ- B7 {m(t)}t≥0 /Q2b MQi ?p2 /Bb+QMiBMmBiB2b i i?2 TQBMib QM N - i?2M Ψi3 ≡ 0X S`QQ7X q2 /Q i?2 mMBp`Bi2 +b2 7Q` i?2 bF2 Q7 MQiiBQMH bBKTHB+BivX h?2 T`QQ7 #2HQr Bb BM7Q`KH BM i?i i 2+? bi2T r2 M2Bi?2` +?2+F B7 i?2 [mMiBiB2b r?Qb2 2tT2+iiBQM Bb iF2M `2 BMi2;`#H2- MQ` B7 i?2 bKQQi?BM; i?2Q`2K Bb TTHB+#H2X Ai +M #2 K/2 `B;Q`Qmb #v `2TH+BM; t #v t ∧ Sn - r?2`2 {Sn }n≥1 Bb  b2[m2M+2 Q7 TT`QT`Bi2 biQTTBM; iBK2b i2M/BM; iQ ∞- M/ i?2M H2iiBM; n ↑ ∞X AM ai2T R- QM2 +QmH/ iF2 7Q` BMbiM+2 i?2 Ft @biQTTBM; iBK2b  t (|H(s) + 1|λ(s) + f (s))ds ≥ n} , Sn := inf{t ≥ 0 ; N (t) + Z(t−) + 0

M/ BM ai2T k M/ ai2T j- i?2 FtN @biQTTBM; iBK2b  t Sn := inf{t ≥ 0 ; N (t) + (1 + |K(s)|)λ(s) ds ≥ n} . 0

h?2 p2`B}+iBQMb `2 H27i 7Q` i?2 `2/2`X ai2T RX q2 }`bi +QKTmi2 i?2 H27i@?M/ bB/2 Q7 URyXeV- rBi?  2(ds) , U (t) = H(s)M (0,t]

t r?2`2 i?2 FtN @T`2/B+i#H2 T`Q+2bb {H(t)}t≥0 Bb bm+? i?i 0 |H(s)|λ(s) ds < ∞X 6Q` i?Bb- }`bi /2+QKTQb2 i?2 bii2 b  t  Z(t) = Zc (t) + Zd (t) := mc (t) + f (s) ds + md (t) . 0

"v i?2 aiB2HiD2bĜG2#2b;m2 `mH2 Q7 BMi2;`iBQM #v T`ib  Zd (s−) dU (s) + U (s) dZd (s) Zd (t)U (t) = (0,t] (0,t]   2(s) + Zd (s−)H(s) dM U (s) (dmd (s) + f (s) ds) = 

(0,t]

(0,t] t



  Zd (s−)H(s) λ(s) − λ(s) ds

Zd (s−)H(s) dM (s) +

= (0,t]



U (s)f (s) ds +

+ 0

+

0



t



U (s−) dmd (s) (0,t]

H(s)Δmd (s)ΔN (s) .

0Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

G2KK RyXRXRy 6Q` HH t ≥ 0 EQ L(t) | FtN EP Z(t) | FtN = EQ Z(t)L(t) | FtN , Q`- 2[mBpH2MiHvEP



Z(t) | FtN



EQ Z(t)L(t) | FtN = , EQ [L(t) | FtN ]

Q@XbX

P @XbX

S`QQ7X h?Bb Bb Dmbi  `2T?`bBM; Q7 G2KK jX8XeX



1tKTH2 RyXRXRR, 1biBKiBM; i?2 _M/QK AMi2MbBiv Q7  >QKQ;2@ M2Qmb *Qt S`Q+2bbX h?2 #Qp2 H2KK HHQrb mb iQ `2TH+2  }Hi2`BM; T`Q#H2K rBi? `2bT2+i iQ P #v QM2 rBi? `2bT2+i iQ Q- r?B+? Kv #2  bBKTHB}+iBQM r?2M Q ?b  bBKTH2 bi`m+im`2X 6Q` BMbiM+2- B7 Q Bb  T`Q##BHBiv i?i KF2b i?2 TQBMi T`Q+2bb N SQBbbQM rBi? BMi2MbBiv 1- M/ B7 Λ Bb M BMi2;`#H2 p`B#H2 BM/2T2M/2MimM/2` Q- Q7 N - i?2 K2bm`2 P /2}M2/ #v dPt = ΛN (t) exp{(1 − Λ)t} dQt KF2b N  /Qm#Hv biQ+?biB+ T`Q+2bb rBi? BMi2MbBiv ΛX h?2M  ∞ N (t)+1 −λt λ e dF (λ) N E Λ | Ft = 0 ∞ N (t) −λt . λ e dF (λ) 0 U1t2`+Bb2 RyXeX8XV

RyXk a2T`iBQM Q7 .2i2+iBQM M/ 6BHi2`BM;  iBK2 p`vBM; bB;MH {θ(t)}t∈[0,T ] Bb i`MbKBii2/ pB  TQBMi T`Q+2bb +?MM2HX JQ`2 T`2+Bb2Hv-  dzb2M/2`Ǵ /2bB;Mb  +Q/2 7Q` i?Bb bB;MH- i?i Bb-  biQ+?biB+ T`Q+2bb {μ(t)}t∈[0,T ] +QMiBMBM; BM7Q`KiBQM #Qmi i?2 bB;MH- M/  dz`2+2Bp2`Ǵ Q#b2`p2b  TQBMi T`Q+2bb N rBi? M Ft @biQ+?biB+ BMi2MbBiv Q7 i?2 7Q`K λ(t) = λ + μ(t) , r?2`2 Ft := Ftθ ∨ FtN X h?Bb KQ/2H HHQrb 7Q` dz722/#+FǴ- BM i?2 b2Mb2 i?i μ(t) Kv /2T2M/- #2bB/2b i?2 Tbi i iBK2 t Q7 i?2 bB;MH- QM i?2 Tbi i iBK2 t Q7 i?2 Q#b2`@ piBQM N X h?2 +Q/BM; T`Q+2bb {μ(t)}t∈[0,T ] Bb KQ`2Qp2` bbmK2/ Ft @T`2/B+i#H2X AM  +QKKmMB+iBQMb +QMi2ti- i?2 +Q/BM; /Q2b MQi 2`b2 BM7Q`KiBQM #Qmi i?2 bB;MH M/ /Q2b MQi +``v BM7Q`KiBQM MQi +QMiBM2/ BM N Q` i?2 bB;MH- i?i Bb- 7Q` HH t ∈ R+ Ftθ ∨ FtN ≡ Ftμ ∨ FtN . (†) h?2 mM/2`HvBM; T`Q##BHBiv +Q``2bTQM/BM; iQ  +Q/2 μ rBHH #2 /2MQi2/ #v Pμ X Ai Bb +QMbi`m+i2/ b 7QHHQrbX q2 bi`i 7`QK  `272`2M+2 T`Q##BHBiv P0 mM/2` r?B+? N Bb 

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SQBbbQM T`Q+2bb Q7 `i2 λ M/ r?2`2 i?2 bB;MH {θ(t)}t∈[0,T ] M/ N `2 BM/2T2M/2MiX h?2 +Q/BM; Bb `2[mB`2/ iQ biBb7v i?2 dzT2F +QMbi`BMiǴ 0 ≤ μ(t) ≤ c . h?2 T`Q##BHBiv Pμ Bb +QMbi`m+i2/ 7`QK P0 b 7QHHQrb BM bm+?  rv i?i Pμ  P0 X JQ`2 2tTHB+BiHv     T dPμ μ(s) N (ds) − |FT := L(T ) := exp log 1 + μ(s) ds . () dP0 λ [0,T ] 0 Ai rb b?QrM BM 1tKTH2 8X8Xj i?i i?Bb 7Q`KmH /2}M2b  T`QT2` _/QMĜLBFQ/ɷK /2`BpiBp2 M/ i?i mM/2` i?2 T`Q##BHBiv Pμ bQ /2}M2/- N /KBib QM i?2 BMi2`pH [0, T ] i?2 Ft @BMi2MbBiv {λ(t)}t∈[0,T ] X JQ`2Qp2`- i?2 /Bbi`B#miBQMb Q7 i?2 bB;MH mM/2` Pμ Q` P0 `2 i?2 bK2X AM/22/- 7Q` Mv 2p2Mi A ∈ FTθ Pμ (A) = Eμ [1A ] = E0 [1A L(T )]

 L(s−)(λ(s) − λ)(N (ds) − λ(s) ds)) = E0 1A (1 + [0,T ]

 1A L(s−)(λ(s) − λ)(N (ds) − λ(s) ds) = P0 (A) , = E0 [1A ] + E0 [0,T ]

r?2`2 i?2 Hbi BM2[mHBiv KF2b mb2 Q7 i?2 7+i i?i {1A L(t−)(λ(t)−λ)}t∈[0,T ] Bb M FTθ ∨ FtN @T`2/B+i#H2 T`Q+2bb M/ i?i mM/2` P0 - N /KBib i?2 FTθ ∨ FtN @BMi2MbBiv λX _2K`F RyXkXR h?2 #QmM/2/M2bb +QM/BiBQMb QM i?2 (P, Ft )@BMi2MbBiv Q7 N ;m`@ Mi22 i?i P (L(T ) > 0) = 1 M/ i?2`27Q`2 P0 M/ Pμ `2 KmimHHv #bQHmi2Hv +QMiBMmQmbX h?2 JmimH AM7Q`KiBQM 6Q`KmH "v /2}MBiBQM- i?2 KmimH BM7Q`KiBQM #2ir22M N QM [0, T ] M/ {θ(t)}t∈[0,T ] Bb i?2 [mMiBiv   dPμ |FTN ∨FTθ IT (θ, N ) := Eμ log , dPμ |FTθ dPμ |FTN rBi? i?2 7QHHQrBM; BMi2`T`2iiBQM dPμ |FTN ∨FTθ

dPμ |FTθ dPμ |FTN aBM+2

dPμ | θ dP0 FT

:=

dPμ dPμ dPμ | FT  |FTθ | N. dP0 dP0 dP0 FT

= 1 IT (θ, N ) := Eμ log

r?2`2 i?2 Hbi 2[mHBiv mb2b U†VX h?2 #bB+ `2bmHi Bb,

dPμ |FTN ∨FTθ dPμ |FTN



 = Eμ log

dPμ |FTN ∨FTμ dPμ |FTN

 ,

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h?2Q`2K RyXkXk lM/2` i?2 #Qp2 +QM/BiBQMb- i?2 KmimH BM7Q`KiBQM #2ir22M N QM [0, T ] M/ {θ(t)}t∈[0,T ] Bb ;Bp2M #v i?2 7Q`KmH  T

(ϕ(μ(s)) − ϕ(μ(s))) ds , IT (θ, N ) = Eμ 0

r?2`2

ϕ(x) := (λ + x) log(λ + x) − λ log λ .

S`QQ7X Ai /2T2M/b QM i?2 7QHHQrBM; H2KK- +HH2/ i?2 i?2Q`2K Q7 b2T`iBQM Q7 /2i2+iBQM M/ }Hi2`BM;X9 G2KK RyXkXj dPμ | N = L(T ) := exp dP0 FT



    T μ(s) N (ds) − log 1 + μ(s) ds , λ 0 [0,T ]

()

r?2`2 λ(t) = λ + μ(t) Bb i?2 T`2/B+i#H2

FtN @BMi2MbBiv

Q7 N X

S`QQ7X h?2 +QM/BiBQMb Q7 #QmM/2/M2bb 7Q` {λ(t)}t∈[0,T ] `2 HbQ biBb}2/ 7Q` {λ(t)}t∈[0,T ] mM/2` Pμ - M/ i?2`27Q`2 mM/2` P0 UPμ M/ P0 `2 KmimHHv #bQHmi2Hv +QMiBMmQmbc b22 _2K`F RyXkXRVX h?2`27Q`2 i?2 T`Q##BHBiv K2bm`2 QdQ |F := L(T ) dP0 T Bb bm+? i?i N ?b QM [0, T ] i?2 (Q, FTN )@BMi2MbBiv {λ(t)}t∈[0,T ] X h?2`27Q`2 Q M/ Pμ ;`22 QM FTN - r?B+? b?Qrb i?i dPμ | N = L(T ) . dP0 FT  h?2 T`QQ7 Q7 h?2Q`2K RyXkXk rBHH MQr #2 +QKTH2i2/X _2+HH i?i

dPμ dPμ |FT − log |FTN . IT θ, N ) = Eμ log dP0 dP0 LQr

 

 T dPμ μ(s) N (ds) − | FT = E μ log 1 + μ(s) ds Eμ log dP0 λ [0,T ] 0   

 T μ(s) λ(s) ds − log 1 + μ(s) ds = Eμ λ [0,T ] 0 9

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M/ bBKBH`Hv

  

 T dPμ μ(s) Eμ log λ(s) ds − |FTN = Eμ log 1 + μ(s) ds . dP0 λ 0 [0,T ] h?2 `2bmHi 7QHHQrb 7i2` Q#b2`pBM; i?i  T 

μ(s) ds = Eμ Eμ 0

T

μ(s) ds .

0

 *T+Biv Q7 i?2 SQBbbQM *?MM2H h?2 +T+Biv Q7 i?2 SQBbbQM +?MM2H Bb- #v /2}MBiBQM- i?2 [mMiBiv C :=

1 sup sup IT (θ, N ) , T θ μ

r?2`2 i?2 supθ #2`b QM HH bB;MH T`Q+2bb2b {θ(t)}t∈[0,T ] - M/ i?2 supμ #2`b QM HH +Q/BM;b biBb7vBM; i?2 +QMbi`BMibX Ai `2T`2b2Mib- BM `Qm;? i2`Kb- i?2 KtBKmK `i2 Q7 i`MbKBbbBQM Q7 BM7Q`KiBQM rBi?Qmi 2``Q` i?`Qm;? i?2 +?MM2H +QMbB/2`2/ Ua?MMQMǶb +T+Biv i?2Q`2KVX AM i?2 +b2 r?2`2 QMHv M KTHBim/2 +QMbi`BMi Bb BKTQb2/- r2 ?p2 i?2 7QHHQrBM; `2bmHiX8 h?2Q`2K RyXkX9 C=λ

 

  1 e 1+λ/c c λ 1+ log 1 + . − 1+ e λ c λ

()

S`QQ7X "v C2Mb2MǶb BM2[mHBivEμ [ϕ(μ(s))] ≤ ϕ(Eμ [μ(s)]) = ϕ(Eμ [μ(s)]) , M/ i?2`27Q`2



T

IT θ, N ) ≤

(Eμ [ϕ(μ(s))] − ϕ(Eμ [μ(s)]) ds .

URyXRRV

0

G2i A #2 i?2 +QHH2+iBQM Q7 T`Q##BHBiv K2bm`2b Q QM [0, c]X h?2 [mMiBiv Eμ [ϕ(μ(s))]− ϕ(Eμ [μ(s)] Bb Q7 i?2 7Q`K    ϕ(x) Q(dx) − ϕ x Q(dx) [0,c]

M/ i?2`27Q`2

[0,c]



 IT θ, N ) ≤ sup Q∈A

8

(E#MQp- RNd3)X



ϕ(x) Q(dx) − ϕ [0,c]

x Q(dx) [0,c]

.

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.2MQiBM; #v U i?2 `B;?i@?M/ bB/2 Q7 i?2 T`2pBQmb BM2[mHBiv- r2 ?p2 i?i U := max U (v) , 0≤v≤c



r?2`2 U (v) :=



Q∈A;



 ϕ(x) Q(dx) − ϕ(v)

max [0,c]

x Q(dx)=v

.

[0,c]

  x v ϕ(x) − ϕ(c) Q(dx) = ϕ(x) Q(dx) − ϕ(c) , c c [0,c] [0,c]  x r2 Kmbi KtBKBx2 h(v) − [0,c] h(x) Q(dx)- r?2`2 h(x) := c ϕ(c) − ϕ(x)X h?2 7mM+@ iBQM h Bb +QMiBMmQmb- +QM+p2- TQbBiBp2 2t+2Ti i x = 0 M/ x = c r?2`2 Bi Bb MmHHX h?2 T`Q##BHBiv Q QM [0, c] +?B2pBM; i?2 KtBKmK Q7 U (v) bm#D2+i iQ i?2  +QMbi`BMi [0,c] x Q(dx) = v iF2b QMHv i?2 irQ pHm2b 0 M/ c- M/

aBM+2



Q({0}) = 1 −

v v , Q({c}) = . c c



h?2`27Q`2 U = max

0≤v≤c

i?i BbU =λ

ϕ(c) − ϕ(v) v c

 ,

 

  1 e 1+λ/c c λ 1+ log 1 + . − 1+ e λ c λ

h?2 pHm2 Q7 v +?B2pBM; i?2 KtBKmK Bb 

e 1+λ/c 1 vm = λ 1+ −1 . e λ aQ 7`- r2 ?p2 T`Qp2/ i?i C Bb H2bb i?i i?2 `B;?i@?M/ bB/2 Q7 UVX AM Q`/2` iQ b?Qr 2[mHBiv- Bi bm{+2b iQ }M/  b2[m2M+2 Q7 bB;MHb {θ(n) (t)}t≥0 Un ≥ 1V M/  +Q/2 μ `2bT2+iBM; i?2 +QMbi`BMib M/ bm+? i?i lim IT (θ(n) , N ) = C .

n↑∞

h?Bb Bb +?B2p2/ #v  HBM2` +Q/2 μ(t) = cθ(t) M/  b2[m2M+2 Q7 dzi2H2;`T? bB;MHbǴX S`2+Bb2Hv- {θ(n) (t)}t≥0 Bb  +QMiBMmQmb iBK2 ?K+ rBi? bii2 bT+2 E = {0, 1} M/ BM}MBi2bBKH +?`+i2`BbiB+b q0,1 = n, ,

q1,0 =

r?2`2

n(1 − p) , p

vm . c 1+? bB;MH Bb bbmK2/ biiBQM`v- i?i Bb- i?2 BMBiBH /Bbi`B#miBQM Bb i?2 biiBQM`v /Bbi`B#miBQM, p=

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P (θ(n) (0) = 1) = E θ(n) (0) = p . 6`QK 1t2`+Bb2 RyXeX9- i?2 2biBKi2 θ(t) := E θ(t) | FtN Q7 i?2 i2H2;`T? bB;MH 2pQHp2b ++Q`/BM; iQ UQKBiiBM; i?2 bmT2`b+`BTiV  n t θ(t) = p + (p − θ(s)) ds p 0  cθ(s−)(λ − θ(s−)) (N (ds) − (λ + cθ(s)) ds) . URyXRkV + 1 + cθ(s−) (0,t] aBM+2 ϕ(cθ(t)) = θ(t)ϕ(c) + (1 − θ(t)ϕ(0) = θ(t)ϕ(c) M/ Eμ [θ(t)] = p = ?p2 i?i vm Eμ [ϕ(cθ(t))] = ϕ(c) . c Ai i?2`27Q`2 `2KBMb iQ b?Qr i?i   lim Eμ ϕ(cθ(n) (t)) = ϕ(vm ) .

vm c

r2

n↑∞

h?2 /2`BpiBp2 Q7 ϕ #2BM; #QmM/2/ QM [0, c] #v  }MBi2 +QMbiMi- bv M |ϕ(cθ(t) − ϕ(vm )| ≤ M |cθ(t) − vm | = M c|θ(t) − p| . "v a+?r`xǶb BM2[mHBiv   12  Eμ |ϕ(cθ(t)) − ϕ(vm )| ≤ M cEμ |θ(t) − p|2 .   Ai i?2`27Q`2 `2KBMb iQ T`Qp2 i?i limn↑∞ Eμ |θ(n) (t) − p|2 = 0X "v i?2 T`Q/m+i `mH2 Q7 aiB2HiD2bĜG2#2b;m2 +H+mHmb TTHB2/ iQ URyXRkV  t c2 [θ(s)(1 − θ(s))]2 2 2 2 θ(s)(p − θ(s)) + θ(t) = p + ds + m(t)  , np 0 1 + θ(s) N r?2`2 {m(t)}  t∈[0,T ] Bb M Ft @K`iBM;H2X h?2`27Q`2     Eμ |θ(n) (t) − p|2 = Eμ θ(n) (t)2 − p2  t

2n Eμ = θ(s)(p − θ(s)) ds p  0  t 2 c [θ(s)(1 − θ(s))]2 + Eμ ds . λ + θ(s) 0   qBi? z (n) (t) := Eμ θ(n) (t)2 − p2 - r2 ?p2 i?i

z (n) (t) = − r?2`2

2n p





t

z (n) (s) ds + 0

t

g (n) (s) ds , 0

j39

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(n)

(s) := Eμ

c2 [θ(n) (s)(1 − θ(n) (s))]2



λ + θ(n) (s)

Bb #QmM/2/X h?2`27Q`2- #v :`QMrHHǶb H2KK- limn↑∞ z (n) (t) = 0X



q2 MQr BMi`Q/m+2 M //BiBQMH +QMbi`BMi- i?2 ǵ2M2`;v +QMbi`BMiǴ QM i?2 +Q/BM;MK2Hv  T μ(s) ds ≤ E0 . 0

.2}MBM; h(x) :=

x ϕ(c) − ϕ(x) , c

r2 ?p2 7`QK URyXRRVIT θ, N ) ≤

1 T



T

h(Eμ [μ(s)] ds . 0

T A7 E0 ≥ vm - bBM+2 h ?b  KtBKmK i x = vm - i?2 KtBKmK Q7 0 h(Eμ [μ(s)] ds Bb iiBM2/ 7Q` Eμ [μ(s)] = vm M/ i?2 2M2`;v +QMbi`BMi Bb biBb}2/X AM i?2 Qi?2` +b2 E0 < vm - i?2 KtBKmK Eμ [μ(s)] = vm HHQr2/ #v i?2 2M2`;v +QMbi`BMi Bb E0 M/ h Bb KtBKBx2/ QM [0, E0 ] i x = E0 X h?2`27Q`2 IT (θ, N ) ≤ T h(E0 ).  b2[m2M+2 Q7 bB;MHb M/ +Q/2b +?B2pBM; i?2 #QmM/ Bb i?2 bK2 b i?2 QM2 mb2/ #27Q`2- rBi? p = Ec0 UBMbi2/ Q7 p = vcm VX AM bmKK`v, h?2Q`2K RyXkX8 Ue V A7 E0 ≥ vm - i?2 T2F +QMbi`BMi /QKBMi2b M/ i?2 +T+Biv Bb i?2 bK2 b 7QmM/ #27Q`2X A7 E0 < vm - i?2 2M2`;v +QMbi`BMi Kmbi #2 iF2M BMiQ ++QmMi M/ i?2 +T+Biv Bb i?2M ;Bp2M #v E0 C = ϕ(c) − ϕ(E0 ) . c

RyXj >B/BM; AM7Q`KiBQM BM  SQBMi S`Q+2bb _2+HH `2bmHi U_kV Q7 a2+iBQM 8X9X A7 N Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ(t) τ (t)  /2}M2/ #v N  ((0, t]) = /2}MBM; τ (t) #v 0 λ(s) ds = t- i?2 TQBMi T`Q+2bb N N ((0, τ (t)]) Bb  biM/`/ ?TTX h?Bb `2bmHi rBHH #2 2ti2M/2/ iQ i?2 i`Mb7Q`KiBQM Q7  TQBMi T`Q+2bb rBi? ;Bp2M biQ+?biB+ BMi2MbBiv BMiQ  biM/`/ ?TTX G2i N #2  bBKTH2 HQ+HHv #QmM/2/ TQBMi T`Q+2bb QM R+ - rBi? i?2 Ft @T`2/B+i#H2 BMi2MbBiv {λ(t)}t≥0 - M/ bmTTQb2 i?i N (0, ∞) = ∞- P @XbX Q`- 2[mBpH2MiHv Uh?2@ ∞ Q`2K 8XRXReV- 0 λ(s) ds = ∞- P @XbX .2}M2 7Q` 2+? t ≥ 0- i?2 MQM@M2;iBp2 `M/QK p`B#H2 τ (t) #v e

(.pBb- RN3y)X

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τ (t)

URyXRjV

λ(s) ds = t . 0

∞ 6Q` 2+? t ≥ 0- τ (t) Bb r2HH@/2}M2/ bBM+2 0 λ(s) ds = ∞ Uh?2Q`2K 8XRXReVX 6Q` 2+? t ∈ R+ - τ (t) Bb  Ft @biQTTBM; iBK2X AM/22/- 7Q` Mv a ∈ R  a λ(s) ds ≤ t ∈ Fa . {τ (t) ≤ a} = 0

 QM R+ #v .2}M2 i?2 bBKTH2 HQ+HHv #QmM/2/ TQBMi T`Q+2bb N  (0, t] := N (0, τ (t)]. N

URyXR9V

  (0, a] := N (0, τ (a)] Bb F N @ LQi2 i?i FtN ⊆ Fτ (t) - bBM+2 7Q` HH a ∈ R+ - N τ (a) K2bm`#H2- M/ i?2`27Q`2 Fτ (a) @K2bm`#H2X

 ?b i?2 Fτ (t) @BMi2MbBiv R UM/ i?2`27Q`2 i?2 FtN @BMi2MbBiv RVX h?2Q`2K RyXjXR N S`QQ7X G2i [a, b] ∈ RX q2 Kmbi b?Qr i?i    (a, b] = E [1A (b − a)] E 1A N "mi i?2 H27i@?M/ bB/2 Bb Dmbi   τ (b)







τ (a)





N (dt) = E 1A

E 1A

(A ∈ Fτ (a) ) .

1(τ (a),τ (b)] (t)N (dt) . 0

aBM+2 i?2 T`Q+2bb 1A 1(τ (a),τ (b)] Bb Ft @T`2/B+i#H2 U#2BM; Ft @/Ti2/ M/ H27i@+QMiBMmQmbVi?2 `B;?i@?M/ bB/2 Q7 i?2 #Qp2 2[mHBiv Bb- #v i?2 bKQQi?BM; 7Q`KmH   



E 1A

τ (b)

λ(t) dt = E [1A (b − a)] .

1(τ (a),τ (b)] (t)λ(t) dt = E 1A 0

τ (a)

  Bb  ?QKQ;2M2Qmb _2K`F RyXjXk "v qiM#2Ƕb i?2Q`2K Uh?2Q`2K kX9XRV- N  (a, b] Bb BM/2T2M/2Mi SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X AM //BiBQM- 7Q` HH [a, b] ∈ R- N Q7 Fτ (a) X _2K`F RyXjXj h?2 `2bmHi Q7 h?2Q`2K RyXjXR Kv #2 mb2/ iQ i2bi i?i  ;Bp2M TQBMi T`Q+2bb /KBib  ;Bp2M ?vTQi?2iB+H BMi2MbBiv, T2`7Q`K i?2 +Q``2bTQM/BM; +?M;2 Q7 iBK2 M/ b22 B7 i?2 `2bmHi Bb  biM/`/ SQBbbQM T`Q+2bb- #v mbBM; Mv pBH#H2 biiBbiB+H i2bi iQ bb2bb i?i  ;Bp2M }MBi2 b2[m2M+2 Q7 `M/QK p`B#H2b Bb BB/ M/ 2tTQM2MiBHHv /Bbi`B#mi2/ rBi? K2M 1X h?2 +Q``2bTQM/BM; K2i?Q/ Bb FMQrM b TQBMi T`Q+2bb `2bB/mH MHvbBbXd d

(P;i- RN33)X

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*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

h?2 7QHHQrBM; [m2biBQM `Bb2b Mim`HHv,3 ?Qr Km+? BM7Q`KiBQM Bb HQbi BM  +?M;2 Q7 iBK2 b+H2\ *QMbB/2` 7Q` BMbiM+2 i?2 bBimiBQM Q7 h?2Q`2K RyXjXRX aBM+2 i?2 +?M;2 Q7 iBK2 i`Mb7Q`Kb i?2 Q`B;BMH TQBMi T`Q+2bb BMiQ  ?QKQ;2M2Qmb SQBb@ bQM T`Q+2bb- ?p2 r2 2`b2/ HH i?2 BM7Q`KiBQM T`2pBQmbHv +QMiBM2/ BM i?2 biQ+?b@ iB+ BMi2MbBiv\ h?2 Mbr2` Bb, Bi /2T2M/bX 6Q` BMbiM+2 bmTTQb2 i?i N Bb  *Qt TQBMi T`Q+2bb rBi? BMi2MbBiv Λ-  MQM@M2;iBp2 `2H@pHm2/ `M/QK p`B#H2X AM Qi?2` rQ`/b- N /KBib i?2 biQ+?biB+ Ft @BMi2MbBiv λ(t) ≡ Λ- r?2`2 Ft := FtN ∨ σ(Λ)X A7  /2}M2/ #v URyXR9V r2 T2`7Q`K i?2 iBK2 +?M;2 τ (t) = Λt - i?2 `2bmHiBM; T`Q+2bb N  (d) − N  (c) Bb BM/2T2M@ Bb  biM/`/ SQBbbQM T`Q+2bbX JQ`2Qp2`- 7Q` HH 0 ≤ c ≤ d- N /2Mi Q7 Fc = FcN ∨ σ(Λ) M/ BM T`iB+mH` Q7 ΛX AM i?Bb b2Mb2- i?2 iBK2 +?M;2 ?b 2`b2/ HH BM7Q`KiBQM +QM+2`MBM; Λ- r?2`2b Λ +QmH/ #2 `2+Qp2`2/ 7`QK N bBM+2#v i?2 bi`QM; Hr Q7 H`;2 MmK#2`bΛ = lim t↑∞

N (t) . t

()

AM i?2 +b2 Q7 M BMi`BMbB+ +?M;2 Q7 iBK2 `2 /`KiB+HHv /Bz2`2MiX h?2 i?BM;b biQ+?biB+ FtN @BMi2MbBiv Q7 N - λ(t) = E Λ | FtN - Bb ;Bp2M #v U1tKTH2 RyXRXRRV,  ∞ N (t)+1 −λt λ e dF (λ) . λ(t) = 0 ∞ N (t) −λt λ e dF (λ) 0 hQ #2 2p2M KQ`2 bT2+B}+- bmTTQb2 i?i P (Λ = a) = P (Λ = b) = 0 < a < b- BM r?B+? +b2λ(t) =

1 2

7Q` bQK2

1 + (b/a)N (t)+1 e(a−b)t . 1 + (b/a)N (t) e(a−b)t

S2`7Q`KBM; i?2 iBK2 +?M;2 

τ(t)

λ(t) dt = t , 0

 /2}M2/ #v N  (t) := N (τ (t))- r?B+? Bb  biM/`/ r2 Q#iBM  TQBMi T`Q+2bb N SQBbbQM T`Q+2bbX >Qr2p2`- i?Bb iBK2- Λ +M #2 2MiB`2Hv `2+Qp2`2/ 7`QK BiX AM 7+i M/ i?2M Λ +M #2 Q#iBM2/ #v b r2 MQr b?Qr- N +M #2 `2+QMbi`m+i2/ 7`QK N  UVX AM 7+i- B7 Tn Bb i?2 n@i? TQBMi Q7 N - i?2M 

Tn+1 Tn

λ(t) dt = Tn+1 − Tn

Q`- KQ`2 2tTHB+BiHvTn+1 − Tn = f (n, Tn+1 ) − f (n, Tn ) , r?2`2

  f (n, t) = at − ln 1 + (b/a)n+1 e(a−b)t .

*H2`Hv i?2M- i?2 b2[m2M+2 {Tn }n≥1 +M #2 `2+Qp2`2/ 7`QK {Tn }n≥1 X 3

("`ûKm/- RNd8 +)X

RyX9X LPAau SPALha

j3d

_2K`F RyXjX9 M BMi2`T`2iiBQM Q7 i?2 #Qp2 `2bmHib BM i2`Kb Q7 +`vTiQ;`T?v Bb i?2 7QHHQrBM;X A7 i?2 BM7Q`KiBQM Bb +QMiBM2/ BM Λ- i?2 BMi`BMbB+ iBK2 +?M;2 vB2H/b  biM/`/ SQBbbQM T`Q+2bb 7`QK r?B+? Λ +M #2 2ti`+i2/ QMHv B7 QM2 FMQrb i?2 dzF2vǴ- i?i Bb- i?2 /Bbi`B#miBQM Q7 ΛX ULQi2 ?Qr2p2` i?i 7`QK  }MBi2 i`D2+iQ`v Q7  QM2 +M QMHv Q#iBM M TT`QtBKiBQM Q7 ΛX AM i?Bb b2Mb2- b2+m`2 i`MbKBbbBQM N rQmH/ #2 i i?2 T`B+2 Q7 bQK2 mM`2HB#BHBivX h?Bb mM`2HB#BHBiv +M #2 +QMi`QHH2/ i i?2 2tT2Mb2 Q7 i`MbKBbbBQM `i2- r?B+? Bb T2`?Tb ++2Ti#H2 B7 QM2 Bb BMi2`2bi2/ QMHv BM biQ`;2 b2+m`BivXV _2K`F RyXjX8 AM i?2 +QMiBMmiBQM Q7 a2+iBQM 8X3- M/ rBi? i?2 MQiiBQM i?2`2Q7 bBKBH` +`vTiQ;`T?B+ 2z2+i Bb Q#b2`p2/ BM i?2 bB;MH THmb MQBb2 KQ/2HX  i`Mb@ t KBii2` rBb?2b iQ b2M/ BM  MQBb2H2bb +?MM2H i?2 BMi2;`i2/ bB;MH { 0 ϕ(s) ds}t≥0 X AMbi2/- ?2 b2M/b  pQHmMi`BHv +Q``mTi2/ p2`bBQM

t

X(t) =

ϕ(s) ds + W (t) ,

()

0

r?2`2 i?2 BMi2;`i2/ MQBb2 Bb BM/2T2M/2Mi Q7 i?2 bB;MHi?2 `2+2Bp2` 7`QK r?B+? +M 2ti`+i M 2biBKi2 {ϕ(t)}t≥0 - r?2`2 ϕ(t) := E ϕ(t) | FtX Ut ≥ 0VX h?Bb Bb TQbbB#H2 bBM+2 i?2 `2+2Bp2` Bb r`2 Q7 i?2 7+i i?i i?2 BMi2;`i2/ MQBb2 Bb  qB2M2` T`Q+2bb Q7 FMQrM p`BM+2 BM/2T2M/2Mi Q7 i?2 bB;MHX LQr- bmTTQb2 i?i i?2 b2M/2` /2+B/2b iQ T2`7Q`K i?2 bK2 +QKTmiiBQMb iQ Q#iBM i?2 #Qp2 2biBKi2 Ur?B+? ?2 +M /Q- bBM+2 ?2 ?BKb2H7 +`2i2/ i?2 MQBb2VX >2 i?2M b2M/b i?2 BMMQpiBQMb T`Q+2bb 2(t)}t≥0 7`QK r?B+? i?2 b2M/2` +M 2ti`+i i?2 Q`B;BMH +Q``mTi2/ bB;MH UV M/ {W Q#iBM i?2 2biBKi2 {ϕ(t)}t≥0 X

RyX9

LQBbv SQBMib

h?2 ;2M2`H bBimiBQM bim/B2/ BM i?Bb b2+iBQM Bb i?2 7QHHQrBM; QM2X h?2`2 Bb  UbBKTH2 M/ HQ+HHv }MBi2V TQBMi T`Q+2bb N i?i Bb Q#b2`p2/ M/ r2 FMQr i?i Bi Bb i?2 bmK Q7 N = N1 + N2 Q7 irQ TQBMi T`Q+2bb2b- QM2 Q7 i?2K- bv N2 - #2BM; +QMbB/2`2/ b  MQBb2X q?i +M r2 /Q iQ /Bb+`BKBMi2 i?2 MQBbv TQBMib 7`QK i?2 BM7Q`KiBp2 TQBMib Q7 N2 \ JQ`2 T`2+Bb2Hv- H2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM i?2 TQbBiBp2 ?H7@HBM2- rBi? i?2 Ft BMi2MbBiv Q7 i?2 7Q`K λ(t) = λ1 (t) + λ2 (t) (t ≥ 0) . N +M #2 i?Qm;?i Q7 b i?2 bmK N1 M/ N2 rBi? i?2 `2bT2+iBp2 Ft BMi2MbBiB2b {λ1 (t)}t≥0 M/ {λ1 (t)}t≥0 X 1tKTH2 RyX9XR, SQBbbQM LQBb2X >2`2 N1 M/ N2 `2 BM/2T2M/2Mi- M/ N2 Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ2 X h?Bb Bb  +HbbB+H bBimiBQM BM QTiB+H i`MbKBbbBQM #v T?QiQMbX h?2 M2ti 2tKTH2 /Q2b MQi BM 7+i BMpQHp2 MQBb2X

j33

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

1tKTH2 RyX9Xk, S`BK`v b?Q+Fb M/ 7i2`b?Q+FbX h?2 b2BbKB+ KQ/2H Q7 1tKTH2 8XRXkd +M #2 2ti2M/2/ iQ BM+Hm/2 7i2`b?Q+Fb,N N (0,t)

λ(t) := eX(0)+ct−

i=1

 Zi

t

+ −∞

>2`2 λ1 (t) = e

N (0,t)

X(0)+ct−

i=1

h(t − s) N (ds) . 

Zi

M/ λ2 (t) =

t

−∞

h(t − s) N (ds)

`2 i?2 `2bT2+iBp2 biQ+?biB+ BMi2MbBiB2b Q7 i?2 T`BK`v b?Q+Fb M/ Q7 i?2 7i2`@ b?Q+FbX :Bp2M  bKTH2 Q7 N QM2 rQmH/ HBF2 iQ b2T`i2 i?2 7i2`b?Q+Fb 7`QK i?2 T`BK`v b?Q+FbX h?2 #2bi i?i +M #2 /QM2 Bb iQ b2H2+i i?2 TQBMib Q7- bv N2 - BM i?2 KMM2` ;Bp2M #v i?2 7QHHQrBM; `2bmHi Ubii2/ BM i?2 ;2M2`H bBimiBQMV, h?2Q`2K RyX9Xj h?2 TQBMi T`Q+2bb2b N1 M/ N2 /2}M2/ #v N1 ((a, b]) :=

∞ 

1U

n=1

λ1 (Tn ) n ≤ λ (T ) 2 n

1

(a,b] (Tn )

M/ N2 = N − N1 - r?2`2 i?2 b2[m2M+2 {Un }n≥1 Bb BB/ M/ mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]- ?p2 i?2 `2bT2+iBp2 Ft @BMi2MbBiB2b {λ1 (t)}t≥0 M/ {λ1 (t)}t≥0 X h?Bb b?Qrb i?i i?2 TQBMib +MMQi #2 b2T`i2/ BM  mMB[m2 KMM2`X :Bp2M N - i?2 T`BK2/ TQBMi T`Q+2bb2b /2T2M/ QMHv QM i?2 `M/QK b2[m2M+2- M/ QM2 +M `2T2i i?2 T`Q+2/m`2 iQ Q#iBM  b2[m2M+2 Q7 BM/2T2M/2Mi U+QM/BiBQMHHv QM N V T`BK2/ TQBMi T`Q+2bb2bX 6Q`  ;Bp2M TQBMi Tk Q7 N - QM2 +M i?2M +QmMi i?2 MmK#2` Q7 iBK2b i?Bb TQBMi ?b #22M bbB;M2/ iQ N1 BM n 2tT2`BK2Mib- M/ B7 i?Bb MmK#2` Bb ;`2i2` i?M 12 n- mHiBKi2Hv bbB;M i?Bb TQBMi iQ N1 X

RyX8 _M/QK aKTHBM; aKTHBM; i i?2 iBK2b {Tn }n∈Z Q7  TQBMi T`Q+2bb N Ui?2 bKTH2`V Q7  +QMiBMmQmb@ iBK2 biQ+?biB+ T`Q+2bb {X(t)}t∈R Ui?2 bKTH2/ T`Q+2bbV vB2H/b  b2[m2M+2 Q7 `M@ /QK bKTH2b Ui?2 bKTH2 b2[m2M+2V {X(Tn )}n∈Z X h?2 Tb2m/Q@T`Q+2bb Y (t) =



X(Tn )δ(t − Tn )

n∈Z

r?2`2 δ (t) Bb i?2 .B`+ Tb2m/Q@7mM+iBQM- Bb +HH2/ i?2 bKTH2 +QK#X N

(a+?Q2M#2`; M/ "QHi- kyyy)X

RyX8X _L.PJ aJSGAL:

j3N X(t) X(Tn )

×

×

×

Tn

×

× ×

t

×

×

×

×

× ×

t

q2 MQr 7Q`KmHi2 i?2 MQiBQM Q7 `M/QK bKTHBM; BM i?2 bTiBH +b2XRy aT2+i`H J2bm`2 Q7 i?2 aKTH2 "`mb? >2`2 i?2 bKTH2/ T`Q+2bb Bb  rbb biQ+?biB+ T`Q+2bb {X(t)}t∈Rm rBi? K2M mX +Qp`BM+2 7mM+iBQM CX - TQr2` bT2+i`H K2bm`2 μX M/ *`Kû`ĜE?BM+?BM bT2+i`H /2+QKTQbBiBQM ZX ,  e2iπν,t ZX (dν) + mX ,

X(t) = Rm

r?2`2 i?2 BMi2;`H i?2`2Q7 Bb  .QQ# BMi2;`HXRR _2+HH i?i 7Q` HH 7mM+iBQMb g ∈ L2C (μX )- i?2 .QQ# BMi2;`H Rm g(ν)ZX (dν) Bb  r2HH@/2}M2/ 2H2K2Mi Q7 L2C (P )X JQ`2Qp2` & &2   & & E && g(ν)ZX (dν)&& = |g (ν)|2 μX (dν) . URyXR8V m m R

R

h?2 bKTH2` Bb  bBKTH2 rbb TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λ ∈ (0, ∞)rBi? TQBMi b2[m2M+2 {Vn }n≥1 X h?2 bKTH2/ T`Q+2bb M/ i?2 bKTH2` `2 bbmK2/ BM/2T2M/2Mi M/ rB/2@b2Mb2 biiBQM`vX h?2 bKTH2 #`mb?  Y (t) = X(Vn )δ(t − Vn ) URyXReV n≥1

Bb B/2MiB}2/ rBi? i?2 bB;M2/ K2bm`2 i aX

n≥1

X(Vn )εVn - r?2`2 εa Bb i?2 .B`+ K2bm`2

.2}M2 i?2 bT2+i`H K2bm`2 Q7 i?2 bKTH2 #`mb? iQ #2  HQ+HHv }MBi2 K2bm`2 μY bm+? i?i- 7Q` Mv ϕ ∈ BY    ϕ (t) X(t)N (dt) = |ϕ (ν)|2 μY (dν) , URyXRdV Var Rm

Rm

r?2`2 BY Bb  H`;2 2MQm;? p2+iQ` bT+2 Q7 7mM+iBQMb- ?2`2 HbQ +HH2/ i?2 i2bi 7mM+iBQMbX "v dzH`;2 2MQm;?Ǵ- r2 K2M i?i i?2`2 +MMQi #2 irQ /Bz2`2Mi HQ+HHv }MBi2 K2bm`2b μY biBb7vBM; URyXRdV 7Q` HH ϕ ∈ BY X P#b2`p2 i?i Ry RR

1`Hv rQ`F BM i?2 +b2 Q7 iBK2 bB;MHb, ("2miH2` M/ G2M2KM- RNee) M/ (Jb`v- RNd3)X a22 7Q` BMbiM+2 ("`ûKm/- kyky)X

jNy

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a 



 ϕ (t) Y (t) dt = Rm

ϕ (t) Rm

=





 X (Vn ) δ (t − Vn ) dt

n≥1



ϕ (Vn ) X (Vn ) =

ϕ (t) X(t)N (dt) , Rm

n≥1

bQ i?i 2[mHBiv URyXRdV #2+QK2b- 7Q`KHHv

 ϕ (t) Y (t) dt

Var Rm

 = Rm

|ϕ (ν)|2 μY (dν) .

Uh?Bb 2tT`2bbBQM `2M/2`b i?2 MHQ;v rBi? i?2 +HbbB+H "Q+?M2` bT2+i`H K2bm`2 KQ`2 TT`2MiXV G2i N #2  rB/2@b2Mb2 biiBQM`v bBKTH2 TQBMi T`Q+2bb QM Rm rBi? BMi2MbBiv λ ∈ (0, ∞)- "`iH2ii bT2+i`mK μN M/ i2bi 7mM+iBQM bT+2 BN X h?2Q`2K RyX8XR URk V amTTQb2 i?i i?2 biQ+?biB+ T`Q+2bb {X(t)}t∈Rm M/ i?2 TQBMi T`Q+2bb N `2 BM/2T2M/2MiX h?2M- i?2 bKTH2/ #`mb? URyXReV /KBib i?2 TQr2` bT2+i`H K2bm`2 μY = μN ∗ μX + λ2 μX + |mX |2 μN .

URyXR3V

A7 BN Bb bi#H2 rBi? `2bT2+i iQ KmHiBTHB+iBQMb #v +QKTH2t 2tTQM2MiBH 7mM+iBQMb- r2 +M iF2 7Q` i?2 i2bi 7mM+iBQM bT+2 BY = BN X _2K`F RyX8Xk h?Bb `2bmHi Bb iQ #2 +QKT`2/ rBi? i?i ;BpBM; i?2 bT2+i`H K2@ bm`2 μY Q7 i?2 T`Q/m+i Q7 irQ BM/2T2M/2Mi rbb biQ+?biB+ T`Q+2bb2b- Y (t) = Z(t)X(t), μY = μZ ∗ μX X h?Bb `2bmHi Bb T`i Q7 i?2 7QHFHQ`2 Q7 i?2 2H2+i`B+H 2M;B@ M22`BM; HBi2`im`2- r?2`2 Bi Bb T`Qp2/ i i?2 2tT2Mb2 Q7 7Q`KH KMBTmHiBQMbX S`QQ7X  7Q`KH 2t+?M;2 Q7 i?2 Q`/2` Q7 BMi2;`iBQM ;Bp2b     ϕ (t) X(t)N (dt) = ϕ (t) e2iπν,t ZX (dν) + mX N (dt) m Rm Rm  R  2iπν,t = ϕ (t) e N (dt) ZX (dν) Rm Rm  ϕ (t) N (dt) . + mX Rm

aBM+2 i?2 BMi2;`Hb rBi? `2bT2+i iQ N M/ ZX `2 Q7  /Bz2`2Mi Mim`2 UQM2 Bb  mbmH BM}MBi2 bmK- i?2 Qi?2` Bb  qB2M2` BMi2;`HV- i?Bb 2t+?M;2 Kmbi #2 7Q`KHHv DmbiB}2/- r?B+? Bb /QM2 i i?2 +HQb2 Q7 i?2 T`QQ7X "v i?2 +QM/BiBQMH p`BM+2 7Q`KmH- r2 ?p2- /2MQiBM; #v F N i?2 σ@}2H/ ;2M2`i2/ #v i?2 TQBMi T`Q+2bb N Rk h?2 `2bmHi Bb bii2/ BM 1tKTH2 RRX j U/V Q7 (.H2v M/ o2`2@CQM2b- RN33)X a22 HbQ ("`ûKm/JbbQmHBû M/ _B/QH}- kyy8)X

RyX8X _L.PJ aJSGAL:

jNR

  Var ϕ (t) X(t)N (dt) Rm &    

  & = E Var ϕ (t) e2iπν,t N (dt) ZX (dν) + mX ϕ (t) N (dt)&& F N m m m &   R  R 

  R & 2iπν,t + Var E ϕ (t) e N (dt) ZX (dν) + mX ϕ (t) N (dt)&& F N . Rm

Rm

Rm

.2MQi2 #v α + β i?2 `B;?i@?M/ bB/2 Q7 i?Bb 2[mHBivX P#b2`p2 i?i- bBM+2 ϕ ∈ L2 (M2 )& &2 & &2 & & & & 2iπν,t & & ≤& & < ∞ , P @XbX ϕ (t) e N (dt) |ϕ (t)| N (dt) URyXRNV & m & & m & R R  lbBM; i?2 7+i i?i- r?2M N Bb }t2/- mX Rm ϕ (t) N (dt) Bb /2i2`KBMBbiB+& 

   & N 2iπν,t & ϕ (t) e N (dt) ZX (dν)& F α = E Var m Rm  & R  &2 & & 2iπν,t & & =E N (dt)& μX (dν) U#v 1[MbX URyXR8V M/ URyXRNVV & m ϕ (t) e Rm R & &2   & & 2iπν,t & = E & ϕ (t) e N (dt)&& μX (dν) m m R R & 

&2     & & 2iπν,t 2iπν,t & = ϕ (t) e N (dt) + &E ϕ (t) e N (dt) && μX (dν) Var m m m R

R





R

& & |ϕ (x − ν)| μN (dx) + && m 2

= Rm

R

ϕ (t) e Rm

2iπν,t

&2  & λdt&& μX (dν)

(?vTQi?2bBb QM BN )   |ϕ (x − ν)|2 μN (dx) μX (dν) + λ2 |ϕ (−ν)|2 μX (dν) = Rm Rm Rm     2 2 = |ϕ (x + ν)| μN (dx) μX (dν) + λ |ϕ (+ν)|2 μX (dν) m m m R R R  2 2 2 = |ϕ (ν)| (μN ∗ μX ) (dν) + λ |ϕ (ν)| μX (dν) . 



aBM+2 E



Rm



Rm

Rm



β = Var



mX Rm

6BMHHv-

Rm



 ϕ (t) Y (t) dt

Var Rm

Rm

& N = 0N (dt) ZX (dν)& F∞   ϕ (t) N (dt) = |mX |2 |ϕ (ν)|2 μN (dν) (#2+mb2 ϕ ∈ BN ) .

ϕ (t) e



2iπν,t

 = Rm

  |ϕ (ν)|2 μN ∗ μX + λ2 μX + |mX |2 μN (dν) ,

b?QrBM; i?i {Y (t)}t∈Rm /KBib M 2ti2M/2/ "Q+?M2` bT2+i`H K2bm`2 ;Bp2M #v URyXR3VX 

jNk

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

G2KK RyX8Xj G2i N #2  bBKTH2 HQ+HHv #QmM/2/ rbb TQBMi T`Q+2bb /2}M2/ QM Rm M/ /KBiiBM;  "`iH2ii bT2+i`H K2bm`2 μN X G2i M2 #2 Bib b2+QM/ KQK2Mi K2bm`2X G2i {X(t)}t∈Rm #2  rbb `M/QK }2H/ rBi? *`Kû`ĜE?BM+?BM /2+QKTQ@ bBiBQM ZX M/ TQr2` bT2+i`H K2bm`2 μX X h?2M- 7Q` HH ϕ ∈ L2 (M2 )    ϕ (t) X(t)N (dt) = ϕ (t) e2iπ N (dt) ZX (dν) . URyXkyV Rm

Rm

Rm

S`QQ7X h?2 T`QQ7 Bb /QM2 BM i?2 mMBp`Bi2 +b2X h?2 KmHiBp`Bi2 +b2 7QHHQrb 2t+iHv i?2 bK2 HBM2bX h?2 H27i@?M/ bB/2 Q7 URyXkyV Bb   A= ϕ (Tn ) X(Tn ) = lim ϕ (Tn ) X(Tn )1[−c,+c] (Tn ) = lim A (c) c↑∞

n∈Z

c↑∞

n∈Z

r?2`2 i?2 HBKBi Bb BM L1C (P )X AM/22/ 

E [|A − A (c)|] ≤ E |ϕ (t) X(t)| N (dt) [−c,+c]   |ϕ (t)| E [|X(t)|] λdt ≤ λK = [−c,+c]

|ϕ (t)| dt

[−c,+c]

r?2`2 K = supt E [|X(t)|] < ∞ U#v a+?r`xǶb BM2[mHBiv- r2 ?p2 E [|X(t)|] ≤ 1 1 E |X(t)|2 2 = E |X (0)|2 2 VX h?2`27Q`2- bBM+2 ϕ ∈ L1C (R)- limc↑∞ E [|A − A (c)|] = 0X h?2 `B;?i@?M/ bB/2 Bb    B = lim ϕ (t) e2iπ N (dt) ZX (dν) = lim B (c) , c↑∞

R

c↑∞

[−c,+c]

r?2`2 i?2 HBKBi Bb BM L2C (P )X AM/22/ &  & 2 E |B − B (c)| = E && R

 & & & =E & R

ϕ (t) e

[−c,+c]

  &  & = E E && R

&2  & N (dt) ZX (dν)&& 

2iπνt

 &2 && && N & ϕ (t) e N (dt) ZX (dν)& & F∞ & [−c,+c]  &2 & ϕ (t) e2iπνt N (dt)&& μX (dν) . 

2iπνt

[−c,+c]

G2i ϕc (t) := ϕ (t) 1[−c,c] (t)X h?2M  &   & &2 &2  & & & & 2iπνt 2iπνt & & & E N (dt)& μX (dν) = E & ϕc (t) e N (dt)&& μX (dν) . & ϕc (t) e R

"mi

R

R

R

&  &2  2  & & 2iπνt E && ϕc (t) e N (dt)&& ≤ E |ϕc (t)| N (dt) R R  = |ϕc (t)| |ϕc (s)| M2 (dt × ds) , R×R

RyX8X _L.PJ aJSGAL:

jNj

 [mMiBiv i?i i2M/b iQ 0 b c ↑ ∞- #v /QKBMi2/ +QMp2`;2M+2X .QKBMi2/ +QM@ p2`;2M+2 TTHB2/ iQ i?2 }MBi2 K2bm`2 μX i?2M vB2H/b i?2 /2bB`2/ +QMp2`;2M+2 BM L2C (P )X "mi  ϕ (Tn ) X(Tn )1[−c,+c] A (c) = n∈Z

=



 ϕ (Tn )

  R

e R

n∈Z

=

 2iπνTn

ϕ (Tn ) e

ZX (dν) 1[−c,+c] (Tn ) 

2iπνTn

1[−c,+c] (Tn ) ZX (dν) = B (c) ,

n∈Z

r?2`2 i?2 7+i i?i i?2 bmKb BMpQHp2/ `2 }MBi2 ?b #22M iF2M BMiQ ++QmMiX h?2`27Q`2  A BM L1C (P ) lim A (c) = c↑∞ B BM L2C (P ) , 7`QK r?B+? Bi 7QHHQrb i?i A = B- XbX U`2+HH i?i B7  b2[m2M+2 Q7 `M/QK p`B@ #H2b +QMp2`;2b BM L1C (P ) Q` L2C (P ) iQ bQK2 `M/QK p`B#H2- QM2 +M 2ti`+i  bm#b2[m2M+2 i?i +QMp2`;2b HKQbi bm`2Hv iQ i?2 bK2 `M/QK p`B#H2VX  1tKTH2 RyX8X9, *Qt aKTHBM;X G2i N #2  *Qt T`Q+2bb rBi?  rbb BMi2MbBiv T`Q+2bb {λ (t)}t∈Rm rBi? bT2+i`H K2bm`2 μλ X h?2M- `2+HHBM; 7`QK 1tKTH2 NXkXRj i?2 2tT`2bbBQM μN (dν) = μλ (dν) + λdν- r2 Q#iBM  2  μY = μλ ∗ μX + λ2 μX + |mX |2 μλ + λ σX + |mX |2 m r?2`2 m Bb i?2 G2#2b;m2 K2bm`2X UAM Q`/2` iQ Q#iBM i?Bb 2tT`2bbBQM- Bi rb MQi2/ i?i μX ∗ m = μX X AM 7+i- 7Q` HH C ∈ B(Rm )

m (C − ν) μX (dν) (μX ∗ m )(C) = m R 2 =

m (C) μX (dν) = m (C)μX (Rm ) = m (C)σX .) Rm

HbQ 7`QK 1tKTH2 NXkXRj M/ h?2Q`2K RyX8XR- BN = L1C (Rm ) ∩ L2C (Rm ) = BY X AM i?2 T`iB+mH` +b2 r?2`2 i?2 bKTH2` N Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb M/ i?2 bKTH2/ T`Q+2bb Bb +2Mi2`2/2 m μY = λ2 μX + λσX

.

URyXkRV

1tKTH2 RyX8X8, _2;mH` aKTHBM;- hF2 RX _2;mH` bKTHBM; `272`b iQ i?2 +b2 r?2`2 i?2 bKTH2` Bb  ;`B/ UBM i?Bb 2tKTH2, QM i?2 `2H HBM2V rBi? TQBMib b2T`i2/ #v i?2 /BbiM+2 T X amTTQb2 7Q` bBKTHB+Biv

i?i i?2 bKTH2/ T`Q+2bb Bb +2Mi2`2/X TTHvBM; 6Q`KmH URyXR3V rBi? μN = T1 n∈Z\{0} ε Tn - M/ Q#b2`pBM; i?i

jN9

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

μX ∗ ε Tn (C) = μX (C − Tn )- QM2 }M/b BM i?2 +b2 r?2`2 i?2`2 2tBbib  TQr2` bT2+i`H /2MbBiv fX 7Q` i?2 bKTH2/ T`Q+2bb i?2 7QHHQrBM; 2tT`2bbBQM 7Q` i?2 bT2+i`H /2MbBiv Q7 i?2 bKTH2 +QK#  2   1 n . fX ν − fY (ν) = T T n∈Z h?2`27Q`2- i?2 bT2+i`H /2MbBiv Q7 i?2 bKTH2/ T`Q+2bb +M #2 2MiB`2Hv `2+Qp2`2/ 7`QK i?i Q7 i?2 bKTH2 +QK# T`QpB/2/ i?2 7Q`K2` Bb #M/@HBKBi2/- rBi? #M/ rB/i? 2B < T1 X

1tKTH2 RyX8Xe, SQBbbQM bKTHBM;- AX SQBbbQM bKTHBM; `272`b iQ i?2 +b2 r?2`2 i?2 bKTH2` Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? BMi2MbBiv λ ∈ (0, ∞)X lM/2` i?2 bK2 +QM/BiBQMb 7Q` i?2 bKTH2/ T`Q+2bb b BM i?2 T`2pBQmb 2tKTH2i?2 bT2+i`H K2bm`2 Q7 i?2 bKTH2 +QK# /KBib i?2 /2MbBiv 2 fY (ν) = λ2 fX (ν) + λσX .

h?2`27Q`2- r?i2p2` i?2 bKTHBM; 7`2[m2M+v νs = λ- i?2`2 Bb MQ HBbBM;- M/ i?2 bT2+i`mK Q7 i?2 bKTH2/ T`Q+2bb +M #2 `2+Qp2`2/ 7`QK i?i Q7 i?2 bKTH2 +QK#X

_2+QMbi`m+iBQM 7`QK aKTH2b AM i?2 Hbi irQ 2tKTH2b- r2 r2`2 BMi2`2bi2/ BM i?2 `2+Qp2`v Q7 i?2 bT2+i`H K2@ bm`2 Q7 i?2 bKTH2/ T`Q+2bb 7`QK i?i Q7 i?2 bKTH2 +QK#X q2 MQr +QMbB/2` i?2 T`Q#H2K Q7 `2+QMbi`m+iBM; i?2 T`Q+2bb Bib2H7 7`QK Bib bKTH2bX JQ`2 KQ/2biHv- r2 TT`QtBKi2 i?2 bKTH2/ T`Q+2bb #v  }Hi2`2/ p2`bBQM Q7 i?2 bKTH2 +QK#,  ϕ (t − s) Y (s) ds , Rm

r?2`2 ϕ ∈ L1C Rm ) ∩ L2C Rm )- M/ r2 HbQ bbmK2 i?i BN = L1C Rm ) ∩ L2C Rm )X h?2 /Bz2`2M+2 #2ir22M X (t) M/ Bib TT`QtBKiBQM- i?i Bb- i?2 `2+QMbi`m+iBQM 2``Q`Bb K2bm`2/ #v & &2  & & ε := E && ϕ (t − u) Y (u) du − X (t)&& . m R

h?2 7QHHQrBM; `2bmHi Bb bii2/ BM i?2 mMB/BK2MbBQMH +b2 7Q` 2b2 Q7 MQiiBQMX h?2Q`2K RyX8Xd h?2 `2+QMbi`m+iBQM 2``Q` Bb- r?2M i?2 bKTH2/ T`Q+2bb Bb +2M@ i2`2/,   2 |λϕ (ν) − 1| μX (dν) + λ |ϕ (ν)|2 (μX ∗ μλ ) (dν) . URyXkkV ε= Rm

Rm

RyX8X _L.PJ aJSGAL:

jN8 

S`QQ7X a22 1t2`+Bb2 RyXeXeX

q2 MQr ;Bp2 i?2 `2+QMbi`m+iBQM 2``Q`b 7Q` /Bz2`2Mi bKTHBM; b+?2K2bX q2 /2@ p2HQT i?2 +QKTmiiBQMb 7Q`  ;2M2`HBx2/ dz#M/@HBKBi2/Ǵ bKTH2/ T`Q+2bb- }Hi2`2/ rBi?  dz#b2@#M/Ǵ }Hi2` Q7 i`MbKBiiM+2 ϕX JQ`2 T`2+Bb2Hv- /2MQiBM; #v S i?2 bmTTQ`i UbbmK2/ Q7 G2#2b;m2 K2bm`2 2B < ∞V Q7 i?2 bT2+i`H K2bm`2 μX ϕ (ν) =

1 1S (ν) , λ

r?2`2 λ Bb i?2 BMi2MbBiv Q7 i?2 bKTH2 +QK#X 1tKTH2 RyX8X3, SQBbbQM aKTHBM;- hF2 kX h?2 bBimiBQM Bb i?i Q7 1tKTH2 RyX8XeX h?2 bT2+i`H K2bm`2 Q7 i?2 bKTH2 +QK# Bb i?2M ;Bp2M #v URyXkRVX h?2 `2+QMbi`m+iBQM 2``Q` Bb   2 ε= |λϕ (ν) − 1|2 μX (dν) + λσX |ϕ (ν)|2 (dν) . URyXkjV Rm

Rm

AM i?2 dz+HbbB+HǴ #M/@HBKBi2/ +b2 /2b+`B#2/ #Qp2- r2 ?p2   1 2 2 |ϕ (ν)|2 (dν) = λσX 1 (ν) dν, ε = λσX 2 S R R λ i?i Bb

2B · λ h?2`27Q`2- bKTHBM; i i?2 dzLv[mBbi `i2Ǵ λ = 2B ;Bp2b  p2`v TQQ` T2`7Q`KM+2MQi #2ii2` i?M i?2 2biBKi2 #b2/ QM MQ Q#b2`piBQM i HHX 2 ε = σX

h?Bb /Q2b MQi K2M- ?Qr2p2`- i?i #2HQr i?2 `i2 λ = 2B- i?2`2 Bb MQ BM7Q`KiBQM UQ` BM  b2Mb2 b i?2 `2bmHi bm;;2bib dzM2;iBp2 BM7Q`KiBQMǴV +QM+2`MBM; i?2 T`Q+2bb Bib2H7 +QMiBM2/ BM Bib bKTH2bX  #2ii2` +?QB+2 Q7  }Hi2` rQmH/ BM/22/ ;Bp2  HBM2` 2 2biBKi2 rBi? 2``Q` bKHH2` i?M σX X 6Q` BMbiM+2- B7 r2 H2i ϕ #2 `2H- r2 }M/ 7Q` i?2 `2+QMbi`m+iBQM 2``Q`  2 ε= (λϕ (ν) − 1)2 fX (ν) + λσX ϕ (ν)2 dν, R

r?2`2 Bi Bb bbmK2/ i?i i?2 bKTH2/ T`Q+2bb ?b i?2 TQr2` bT2+i`H /2MbBiv fX X  KBMBKmK Q++m`b 7Q` λfX (ν) ϕ (ν) = 2 2 λ fX (ν) + λσX M/ i?2M

 ε=

r?2`2

2 σX

 1−

R

λfX (ν)  fX (ν) dν 1 + λfX (ν)

fX (ν) fX (ν) fX (ν) :=  = 2  ) dν  σX f (ν X R

 ,

jNe

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

Bb i?2 MQ`KHBx2/ TQr2` bT2+i`H /2MbBivX h?Bb QTiBKH }Hi2` `2[mB`2b i?2 FMQrH2/;2 Q7 i?2 bT2+i`H /2MbBiv Q7 i?2 bKTH2/ T`Q+2bb- r?B+? Bb BM T`BM+BTH2 pBH#H2 7`QK i?2 bKTH2b- b b?QrM BM 1tKTH2 RyX8XeX 1tKTH2 RyX8XN, _2;mH` aKTHBM;- hF2 kX q?2M i?2 bKTH2/ +QK# Bb Q#iBM2/ #v `2;mH` bKTHBM; b BM 1tKTH2 RyX8X8- i?2 `2+QMbi`m+iBQM 2``Q` Bb &2  &  & &1 & ϕ (ν) − 1& μX (dν) + 1 ε= |ϕ (ν) − 1|2 dν URyXk9V & & T R R T Ur?2`2 r2 iQQF BMiQ ++QmMi i?2 7+i i?i ∗ μX = μX (R) VX AM i?2 #M/@HBKBi2/ +b2- B7 r2 +QMbB/2` T = 1/2B- i?i Bb- λ = 2B- 2[miBQM URyXk9V ;Bp2b M 2``Q` 2[mH iQ x2`QX h?2`27Q`2- i?2 T`Q+2bb Bb T2`72+iHv `2+QMbi`m+i2/ #v   X (t) = ϕ (t − s) X (s) N (ds) = X (Tn ) sinc (2B(t − Tn )) , R

n∈Z

r?B+? Bb i?2 mbmH `2+QMbi`m+iBQM 7Q`KmH FMQrM b i?2 a?MMQMĜLv[mBbi 7Q`KmHX 1tKTH2 RyX8XRy, CBii2` BM _2;mH` Lv[mBbi aKTHBM;X h?2 `2+QMbi`m+@ iBQM 2``Q` 7`QK mMB7Q`K bKTH2b BM i?2 T`2b2M+2 Q7 DBii2` Bb Q#iBM2/ #v THm;;BM; μY ;Bp2M #v UNX99V BMiQ i?2 2``Q` 7Q`KmH URyXkkVX h?2 T`2pBQmb 2tKTH2 b?Qr2/ i?i rBi?BM i?2 dz+HbbB+HǴ bKTHBM; 7`K2rQ`F i?2 T`Q+2bb Kv #2 T2`72+iHv `2+QM@ bi`m+i2/X LQr- BM i?2 T`2b2M+2 Q7 DBii2`Rj i?Bb Bb MQi TQbbB#H2 M/ i?2 `2+QMbi`m+iBQM 2``Q` Bb ;Bp2M #v  B      1 2 ε= σX 1 − |ψZ |2 ∗ fX (ν) dν , URyXk8V 2B −B r?2`2 fX Bb i?2 MQ`KHBx2/ TQr2` bT2+i`H /2MbBiv Q7 i?2 T`Q+2bb X (t)X

RyXe 1t2`+Bb2b 1t2`+Bb2 RyXeXRX .2iBHb 7Q` G2KK RyXRXj :Bp2 i?2 /2iBHb Q7 i?2 T`QQ7 Q7 2tBbi2M+2 M/ mMB[m2M2bb Q7 {f (t)}t≥0 biBb7vBM; UV Q7 G2KK RyXRXjX 1t2`+Bb2 RyXeXkX  *?`+i2`BxiBQM Q7 Z(t) h?2 MQiiBQM M/ bbmKTiBQMb `2 i?Qb2 Q7 G2KK RyXRXjX a?Qr i?i- 7Q` Z(t) iQ #2 i?2 +QM/BiBQMH 2tT2+iiBQM Q7 Z(t) ;Bp2M Ot - Bi Bb M2+2bb`v M/ bm{+B2Mi i?i   E [Z(t)U (t)] = E Z(t)U (t) () Rj

1`Hv rQ`F , ("HF`Bb?MM- RNek) M/ ("`QrM- RNej)X

RyXeX 1s1_*Aa1a

jNd

7Q` HH #QmM/2/ Ot @K`iBM;H2b {U (t)}t≥0 X 1t2`+Bb2 RyXeXjX "B`i?@M/@/2i? S`Q+2bb 1biBKi2/ 7`QK i?2 .2i? S`Q+2bb G2i {X(t)}t≥0 #2  #B`i?@M/@/2i? T`Q+2bb- i?i Bb- M ?K+ rBi? bii2 bT+2 E = N- M/ r?Qb2 MQM@MmHH 2Mi`B2b Q7 i?2 BM}MBi2bBKH ;2M2`iQ` `2 qn,n+1 = λn M/ qn,n−1 = μn 1{n≥1} . h?2 Q#b2`piBQM Bb i?2 TQBMi T`Q+2bb N i?i +QmMib i?2 /QrMr`/ i`MbBiBQMb Ui?2 iBK2b t bm+? i?i X(t) = X(t−) − 1VX .2`Bp2 BM /2iBH i?2 }Hi2`BM; 2[miBQMb 

t

Zn (t) = P (X(0) = n) +

(Zn−1 (s)1{n≥0} λn−1 0

− Zn (s)(λn + μn 1{n≥1} ) + Zn+1 (s)μn+1 ) ds       N (ds) − ds . μj Zj (s−)) × μn+1 Zn+1 (s−) − Zn (s−)( + (0,t] j≥1 μj Zj (s−) j≥1

1t2`+Bb2 RyXeX9X SQBMi S`Q+2bb JQ/mHi2/ #v  h2H2;`T? aB;MH G2i {θ(t)}t≥0 #2  +QMiBMmQmb@iBK2 ?K+ rBi? bii2 bT+2 E = {0, 1} M/ BM}MBi2b@ BKH +?`+i2`BbiB+b n(1 − p) q0,1 = n , , q1,0 = p r?2`2 p ∈ (0, 1)X h?Bb ?K+ Bb bbmK2/ biiBQM`vX G2i N #2  bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb QM R+ rBi? i?2 Ftθ ∨ FtN @BMi2MbBiv λ(t) := cθ(t)X a?Qr i?i i?2 2biBKi2 θ(t) := E θ(t) | FtN 2pQHp2b ++Q`/BM; iQ θ(t) = p +

n p



t

(p − θ(s)) ds 0



+ (0,t]

cθ(s−)(1 − θ(s−)) 1 + cθ(s−)

(N (ds) − (λ + cθ(s) ds) .

1t2`+Bb2 RyXeX8X *Qt S`Q+2bb S`Qp2 i?2 bii2K2Mi Q7 1tKTH2 RyXRXRRX 1t2`+Bb2 RyXeXeX _2+QMbi`m+iBQM 1``Q` S`Qp2 h?2Q`2K RyX8XdX 1t2`+Bb2 RyXeXdX >B;? _i2 SQBbbQM aKTHBM; AM 1tKTH2 RyX8X3 b?Qr i?i i?2 2``Q` ε i2M/b iQ 0 b i?2 bKTHBM; `i2 λ i2M/b iQ BM}MBivX

jN3

*>Sh1_ RyX AL6P_JhAPL *PLh1Lh P6 SPALh S_P*1aa1a

1t2`+Bb2 RyXeX3X S`[email protected]/2Mi aKTHBM; *QMbB/2` i?2 ;2M2`H bBimiBQM Q7 `M/QK bKTHBM;- b BM am#b2+iBQM RyX8X q2 bmTTQb2 MQr i?i i?2 bKTHBM; `i2 /2T2M/b QM i?2 T`Q+2bbX JQ`2 T`2+Bb2Hv- i?2 KQ/2H 7Q` i?2 bKTH2` Bb MQr  *Qt T`Q+2bb QM Rd rBi? i?2 +QM/BiBQMH BMi2MbBiv Q7 i?2 7Q`K λ (t) = λ (t, X) , r?2`2 X `2T`2b2Mib  biiBQM`v biQ+?biB+ T`Q+2bb {X(t)}t∈Rd X U6Q` BMbiM+2- BM & &2 & & i?2 mMBp`Bi2 +b2- λ (t) = |X(t)|2 - λ (t) = &X˙ (t)& - r?2`2 X˙ Bb i?2 /2`BpiBp2 i t Q7 t → X(t)X JQ`2 +QKTH2t 7mM+iBQMHb +M #2 +QMbB/2`2/XV bbmK2 i?i E X(t)2 λ (t, X)2 < ∞ 7Q` HH t ∈ Rm - M/ i?i {λ (t)}t∈Rm Bb  HQ+HHv BMi2;`#H2 T`Q+2bbX G2i μZ #2 i?2 TQr2` bT2+i`mK Q7 i?2 biiBQM`v T`Q+2bb Z (t) = X(t)λ (t) bbmK2/ biiBQM`v Ub Bb i?2 +b2 BM i?2 2tKTH2b #Qp2VX a?Qr i?i μY (dν) = μZ (dν) + X 2 λdν , r?2`2 X 2 λ := E [X(t)2 λ (t)] UBM/2T2M/2Mi Q7 tVX

*?Ti2` RR SQBMi S`Q+2bb2b BM Zm2m2BM; Zm2m2BM; i?2Q`v Bb Q7 BMi2`2bi 7Q` i?2 T2`7Q`KM+2 2pHmiBQM Q7 b2`pB+2 bvbi2Kb 72im`BM; i?2 irQ `2Hi2/ T?2MQK2M Q7 +QM;2biBQM M/ /2Hv- r?B+? +QmH/ #2  MmBbM+2X h?Bb Bb i?2 +b2 7Q` +QKKmMB+iBQMb M/ +QKTmi2` bvbi2Kb- BM //BiBQM iQ i?2 +HbbB+H bvbi2Kb Q7 QT2`iBQMb `2b2`+? 72im`BM; iB+F2i #QQi?b- 7`22rv iQHHb M/ i?2 HBF2X 7i2` `2+HHBM; i?2 #bB+ i?2Q`v Q7 [m2m2b- KBMHv J`FQpBM- i?Bb +?Ti2` rBHH +QM+2Mi`i2 QM i?2 +QMi`B#miBQM Q7 TQBMi T`Q+2bb i?2Q`v- KBMHv i?2 biQ+?biB+ BMi2MbBiv i?2Q`v M/ i?2 SHK i?2Q`v QM i?2 HBM2- iQ i?2 7mM/K2MiH `2bmHib bm+? b GBiiH2Ƕb 7Q`KmH M/ i?2 Tbi T`QT2`iv M/ i?2B` TTHB+iBQMbX

RRXR

 _2pB2r Q7 J`FQpBM Zm2m2BM; h?2Q`v

h?Bb b2+iBQM Bb  bmKK`v Q7 i?2 dz2H2K2Mi`v i?2Q`vǴ Q7 [m2m2b- i?i Bb- i?2 iQTB+b i?i +M #2 i`2i2/ mbBM; i?2 2H2K2Mi`v i?2Q`v Q7 J`FQp +?BMbX AbQHi2/ J`FQpBM Zm2m2b >2`2 Bb  bKTH2 Q7 i?2 bBKTH2bi [m2m2BM; KQ/2HbX 1tKTH2 RRXRXR, JfJfRf∞f7B7QX *QMbB/2`  iB+F2i #QQi? rBi?  bBM;H2 ii2M@ /Mi- Q` b2`p2`X *mbiQK2`b rBi BM HBM2 BM 7`QMi Q7 i?2 #QQi?- M/ i?2 7+BHBiv Bb bQ H`;2 i?i MQ #QmM/ Bb BKTQb2/ QM i?2 MmK#2` Q7 +mbiQK2`b rBiBM; 7Q` b2`pB+2X AM Qi?2` rQ`/b- i?2 rBiBM; `QQK ?b BM}MBi2 +T+BivX am+?  bvbi2K rBHH #2 +HH2/  1/∞ b2`pB+2 bvbi2K- r?2`2 1 Bb 7Q` i?2 MmK#2` Q7 b2`p2`b- M/ ∞ Bb 7Q` i?2 +T+Biv Q7 i?2 rBiBM; `QQKX *mbiQK2` ``BpHb `2 KQ/2HH2/ #v  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb {Tn }n≥1 QM i?2 TQbBiBp2 ?H7@HBM2- Q7 TQbBiBp2 BMi2MbBiv λX *mbiQK2` n ``BpBM; i iBK2 Tn #`BM;b  b2`pB+2 `2[m2bi σn - r?B+? K2Mb i?i i?2 b2`p2` rBHH iF2 σn mMBib Q7 iBK2 7Q` T`Q@ +2bbBM; i?Bb `2[m2biX h?2 b2`pB+2 b2[m2M+2 {σn }n≥1 Bb bbmK2/ BB/- rBi? 2tTQM2MiBH /Bbi`B#miBQM Q7 K2M μ−1 X HbQ- i?2 ``BpH b2[m2M+2 M/ i?2 b2`pB+2 b2[m2M+2 `2 bbmK2/ iQ #2 BM/2T2M/2MiX am+?  Tii2`M Q7 ``BpHb Bb +HH2/ M JfJ BMTmi T`Q+2bbX AM i?Bb MQiiBQM- BMi`Q/m+2/ #v .pB/ E2M/HH-R J biM/b 7Q` dzJ`FQ@ R

(E2M/HH- RN8j)X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_11

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*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

pBMǴX UAM/22/- i?2 SQBbbQM T`Q+2bb Bb J`FQpBM- M/ 2tTQM2MiBH /Bbi`B#miBQMb `2 BMiBKi2Hv +QMM2+i2/ rBi? i?2 J`FQp T`QT2`ivXV h?2 b2`p2` ii2M/b QM2 +mbiQK2` i  iBK2 M/ /Q2b MQi `2KBM B/H2 b HQM; b i?2`2 Bb i H2bi QM2 +mbiQK2` BM i?2 bvbi2K UiB+F2i #QQi? THmb rBiBM; `QQKVX PM+2 i?2 b2`pB+2 Q7  +mbiQK2` Bb bi`i2/ Bi +MMQi #2 BMi2``mTi2/ #27Q`2 +QKTH2iBQMX h?2 #Qp2 bvbi2K Bb +HH2/ M JfJfRf∞ [m2m2X Aib /2b+`BTiBQM +QmH/ #2 +QK@ TH2K2Mi2/ #v M BM/B+iBQM Q7 i?2 b2`pB+2 /Bb+BTHBM2, 7Q` BMbiM+2- 7B7Q U}`bi@BM@ }`bi@QmiV- r?2`2 i?2 b2`p2`- 7i2` +QKTH2iBQM Q7  b2`pB+2- +?QQb2b ?Bb M2ti +mbiQK2` i i?2 ?2/ Q7 i?2 HBM2X h?2 biQ+?biB+ T`Q+2bb {X(t)}t≥0 - r?2`2 X(t) Bb i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 t- Bb +HH2/ i?2 +QM;2biBQM T`Q+2bbX Ai Bb  +QMiBMmQmb@ iBK2 ?K+ rBi? bii2 bT+2 E = N M/ BM}MBi2bBKH ;2M2`iQ` Ub22 i?2 `2K`F #2HQrV qi,i+1 = λ , qi,i−1 = μ1{i≥1} . Ub mbmH- QMHv i?2 Qz@/B;QMH 2Mi`B2b Q7 i?2 BM}MBi2bBKH ;2M2`iQ` Ki`Bt `2 ;Bp2M M/ i?2 MmHH QM2b `2 QKBii2/XV

_2K`F RRXRXk h?2 7Q`KH T`QQ7b Q7 J`FQpBMBiv M/ Q7 i?2 7Q`KmHb ;BpBM; i?2 BM}MBi2bBKH ;2M2`iQ`b Q7 i?2 [m2m2BM; T`Q+2bb2b M/ Q7 i?2B` M2irQ`Fb rBHH #2 ;Bp2M BM a2+iBQM RRXkX

1tKTH2 RRXRXj, JfJfEf∞f7B7QX h?Bb Bb i?2 JfJfRf∞f7B7Q [m2m2- 2t+2Ti i?i i?2`2 `2 MQr K b2`p2`bX h?2 +QM;2biBQM T`Q+2bb Bb  +QMiBMmQmb@iBK2 ?K+ rBi? bii2 bT+2 E = N M/ BM}MBi2bBKH ;2M2`iQ` qi,i+1 = λ , qi,i−1 = inf(i, K)μ .

1tKTH2 RRXRX9, JfJfEfy- Q` 1`HM; [m2m2Xk h?Bb [m2m2 Bb +HH2/ i?2 1`HM; [m2m2 BM ?QMQ` Q7 i?2 .MBb? i2H2T?QM2 2M;BM22`- r?Q mb2/ Bi b  KQ/2H 7Q`  i2H2T?QM2 brBi+? rBi? K HBM2b- Q` +?MM2Hb,  +mbiQK2` }M/BM;  7`22 HBM2 Bb +QMM2+i2/ M/ ?QH/b i?2 HBM2 7Q` i?2 iBK2 Q7  +QMp2`biBQMX  +mbiQK2` 2Mi2`BM; i?2 bvbi2K Ui?i Bb- ``BpBM; M/ b22BM; QM2 Q` KQ`2 B/H2 +?MM2HbV rBHH #2 bbB;M2/  7`22 +?MM2H i `M/QKX  +mbiQK2` }M/BM; HH +?MM2Hb #mbv Bb `2D2+i2/ M/ HQbi UQ`- T2`?Tb- `Qmi2/ iQ MQi?2` brBi+? +2Mi2`VX h?2`27Q`2 i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i Mv ;Bp2M iBK2 Bb H2bb i?M Q` 2[mH iQ KX Ai Bb  [m2m2 rBi? #HQ+FBM;X 1`HM; rb #H2 iQ Q#iBM ?Bb 7KQmb #HQ+FBM; 7Q`KmH ;BpBM; i?2 T`Q##BHBiv i?i- BM biiBQM`v `2;BK2-  ;Bp2M +mbiQK2` }M/b HH HBM2b #mbvX h?Bb Bb i?2 }`bi 7Q`KmH Q7 [m2m2BM; i?2Q`v Ub22 1tKTH2 RRXRXN #2HQrVX k

(1`HM;- RNRd)X

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

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>2`2- X(t) Bb i?2 MmK#2` Q7 #mbv HBM2b UM/ i?2`27Q`2 i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2KV i iBK2 tX h?2 +QM;2biBQM T`Q+2bb Bb  +QMiBMmQmb@iBK2 ?K+ rBi? bii2 bT+2 E = {0, 1, . . . , K} M/ BM}MBi2bBKH ;2M2`iQ` A ;Bp2M #v qi,i+1 = λ1{0≤i≤K−1} , qi,i−1 = iμ1{1≤i≤K} .

1tKTH2 RRXRX8, JfJfRf∞fHB7Q T`22KTiBp2 `2bmK2X h?Bb [m2m2BM; bvb@ i2K `2+2Bp2b  /2b+`BTiBQM bBKBH` iQ i?i Q7  JfJfRf∞f7B7Q- 2t+2Ti 7Q` i?2 b2`pB+2 /Bb+BTHBM2 r?B+? MQr #2+QK2b HB7Q T`22KTiBp2 `2bmK2 Ui?2 ##`2pBiBQM HB7Q biM/b 7Q` dzGbi AM 6B`bi PmiǴVX  +mbiQK2` mTQM ``BpH ;Q2b `B;?i rv iQ i?2 iB+F2i #QQi?- i?2 +mbiQK2` T`2b2MiHv `2+2BpBM; b2`pB+2 #2BM; b2Mi #+F iQ i?2 rBiBM; `QQK- r?2`2 ?2 rBHH biM/ BM 7`QMi Q7 i?2 HBM2 Ui H2bi mMiBH i?2 iBK2 r?2M MQi?2` `m/2 +mbiQK2` b?Qrb mT- b2M/BM; i?2 }`bi `m/2 +mbiQK2` iQ i?2 7`QMi Q7 i?2 [m2m2- M/ bQ QMVX h?Bb ivT2 Q7 /Bb+BTHBM2 Bb +HH2/ T`22KTiBp2X h?2 T?`b2 dzT`22KTiBp2 `2bmK2Ǵ K2Mb i?i  T`22KTi2/ +mbiQK2` /Q2b MQi ?p2 iQ bi`i 7`QK b+`i+?, r?2M i?2 b2`p2` b22b ?BK M2ti iBK2- ?2 rBHH `2bmK2 rQ`F r?2`2 Bi rb H27iXj h?2 +QM;2biBQM T`Q+2bb Q7 i?Bb [m2m2 Bb 2t+iHv Q7 i?2 bK2 Mim`2 b i?i Q7 i?2 JfJfRf∞f7B7Q [m2m2X AM T`iB+mH` Bi ?b i?2 bK2 BM}MBi2bBKH ;2M2`iQ`X 1tKTH2 RRXRXe, JfJf∞X h?Bb Bb M JfJf∞f∞ [m2m2- #mi i?2 b2+QM/ ∞ Bb QKBii2/ M/ r2 i?2M bT2F Q7 M JfJf∞ [m2m2X h?Bb Bb MQi `2HHv  [m2m2BM; bvbi2K- #mi  Tm`2 /2Hv bvbi2KX h?2 ``BpH T`Q+2bb- i?2 b2`pB+2 iBK2b b2[m2M+2M/ i?2 rBiBM; `QQK `2 b BM i?2 JfJfRf∞ KQ/2H- #mi MQr i?2`2 Bb M BM}MBiv Q7 b2`p2`b- M/ i?2`27Q`2 MQ [m2m2BM;- bBM+2 MvQM2 2Mi2`BM; i?2 bvbi2K }M/b M B/H2 b2`p2` Dmbi 7Q` ?BKb2H7X h?2 +QM;2biBQM T`Q+2bb Bb  +QMiBMmQmb@iBK2 ?K+ rBi? bii2 bT+2 E = N M/ BM}MBi2bBKH ;2M2`iQ` qi,i+1 = λ , qi,i−1 = iμ .

1tKTH2 RRXRXd, JfJfRf∞f7B7Q [m2m2 rBi? BMbiMiM2Qmb 722/#+FX h?Bb Bb M JfJfRf∞ [m2m2 r?2`2  +mbiQK2` }MBb?BM; b2`pB+2 2Bi?2` H2p2b i?2 bvbi2K rBi? T`Q##BHBiv 1 − p Q` Bb BKK2/Bi2Hv `2+v+H2/ rBi? T`Q##BHBiv p i i?2 2M/ Q7 i?2 rBiBM; HBM2 Q` i i?2 b2`pB+2 #QQi? B7 i?2`2 Bb M 2KTiv rBiBM; HBM2rBi?  M2r BM/2T2M/2Mi 2tTQM2MiBH b2`pB+2 `2[m2biX AM i?Bb 2tKTH2- i?2 iBK2b Q7 j h?Bb ivT2 Q7 /Bb+BTHBM2 Bb MQi b mM7B` b Bi Kv TT2`X 6B`bi Q7 HH- HH +mbiQK2`b #2BM; 2[mHHv `m/2- i?2v 2M/m`2 b Km+? b i?2v ?m`iX  +mbiQK2` rBi?  H`;2 b2`pB+2 `2[m2bi bT2M/b  H`;2` iBK2 i i?2 iB+F2i #QQi? M/ Bb i?2`27Q`2 KQ`2 2tTQb2/ i?M  +mbiQK2` rBi?  KQ/2bi `2[m2bi- M/ Bi Bb T`2+Bb2Hv BM i?Bb b2Mb2 i?i i?2 /Bb+BTHBM2 Bb 7B`X Ai KF2b HQM;2` `2[m2bib r?Q `2 `2bTQMbB#H2 7Q` +QM;2biBQM rBi HQM;2` BM i?2 bvbi2KX h?2 T`2+Bb2 `2bmHi Bb i?i i?2 2tT2+i2/ bQDQm`M iBK2 Q7  +mbiQK2` BM i?2 bvbi2K- ;Bp2M i?i Bib b2`pB+2 `2[m2bi Bb x- Bb 2[mH iQ x/(1 − ρ)r?2`2 ρ = λ/μX a22 (qQHz- RN3N)X

9yk

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

b2`pB+2 +QKTH2iBQM Q7 `2+v+H2/ +mbiQK2`b /Q MQi +Q``2bTQM/ iQ  ;2MmBM2 i`MbBiBQM Q7 i?2 +QM;2biBQM T`Q+2bbX h?2 +QM;2biBQM T`Q+2bb Q7 i?2 JfJfRf∞f7B7Q [m2m2 rBi? BMbiMiM2Qmb 722/#+F Bb  +QMiBMmQmb@iBK2 ?K+ rBi? bii2 bT+2 E = N M/ BM}MBi2bBKH ;2M2`iQ` ;Bp2M #v qi,i+1 = λ , qi,i−1 = (1 − p)μ1{i≥1} .

AM i?2 [m2m2BM; HBi2`im`2- 2bT2+BHHv r?2M TTHB+iBQMb iQ +QKKmMB+iBQMb M/ +QKTmi2` M2irQ`Fb `2 +QMbB/2`2/-  [m2m2BM; bvbi2K Bb `2T`2b2Mi2/ #v i?2 TB+iQ;`K Q7 i?2 M2ti };m`2- r?2`2 i?2 BMTmi ``Qr `2T`2b2Mib i?2 ``BpH bi`2K Q7 DQ#b U+mbiQK2`bV- i?2 QmiTmi ``Qr `2T`2b2Mib i?2 bi`2K Q7 +QKTH2i2/ DQ#b Ub2`p2/ +mbiQK2`bV- i?2 +B`+H2 Bb  T`Q+2bbQ` Ui?2 b2`pB+2 bvbi2KV- M/ i?2 bi+F Bb  #mz2` U rBiBM; `QQK- r?2`2 +mbiQK2`b rBi 7Q`  b2`p2` iQ #2 7`22VX 1 μ

c SQBbbQM λ

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K T`Q+2bbQ` #mz2` UrBiBM; `QQKV UiB+F2i #QQi?V h?2 ?QHv TB+iQ;`K Q7 [m2m2BM; i?2Q`v

h?Bb TB+iQ;`K +M #2 `B+?Hv /Q`M2/X 6Q` BMbiM+2- BM i?2 };m`2- r2 ?p2  bvbi2K rBi? K b2`p2`b-  rBiBM; `QQK Q7 +T+Biv c- λ Bb i?2 ``BpH `i2 Q7 +mbiQK2`bM/ i?2 /2`BpiBQM b?Qrb i?i i?2 +mbiQK2`b }M/BM; HH K b2`p2`b #mbv M/  7mHH rBiBM; `QQK `2 `2D2+i2/X HbQ- i?2`2 Bb M BM/B+iBQM Q7 i?2 p2`;2 b2`pB+2 iBK2- μ−1 - M/ Q7 i?2 7+i i?i i?2 BM+QKBM; bi`2K Bb  SQBbbQM T`Q+2bbX AM i?Bb TB+iQ;`K- QM2 rQmH/ iF2 Bi b BKTHB+Bi i?i i?2 b2`pB+2 iBK2b b2[m2M+2 Bb BB/ M/ BM/2T2M/2Mi Q7 i?2 ``BpH T`Q+2bb- mMH2bb Qi?2`rBb2 2tTHB+BiHv K2MiBQM2/X PM2 bQK2iBK2b HbQ BM/B+i2b i?2 b2`pB+2 /Bb+BTHBM2 UHB7Q- 7B7Q- 2i+XVX >2m`BbiB+b h?2 `B;Q`Qmb T`QQ7 i?i i?2 +QM;2biBQM T`Q+2bb2b Q7 i?2 [m2m2BM; bvbi2Kb +QM@ bB/2`2/ #Qp2 `2 +QMiBMmQmb@iBK2 ?K+ rBi? i?2 BM/B+i2/ BM}MBi2bBKH ;2M2`@ iQ`b +M #2 7QmM/ BM am#b2+iBQM RRXkX i i?Bb TQBMi- r2 b?HH #2 +QMi2Mi rBi?  ?2m`BbiB+ /2`BpiBQM Q7 i?2 BM}MBi2bBKH ;2M2`iQ`X *QMbB/2` 7Q` BMbiM+2 i?2 [m2m2 JfJfEf+- rBi? 7B7Q /Bb+BTHBM2X >2`2 c Bb i?2 +T+Biv Q7 i?2 rBiBM; `QQK- M/ i?2`27Q`2 M ``BpBM; +mbiQK2` b22BM; K + c +mbiQK2`b BM b2`pB+2 Bb `2D2+i2/- Q` `Qmi2/ 2Hb2r?2`2X i  ;Bp2M iBK2 t- i?2`2 `2  +2`iBM MmK#2` Q7 2tTQM2MiBH `M/QK p`B#H2b dzBM +iBpBivǴ- QM2 7Q` 2+? Q7 i?2 +mbiQK2`b T`2b2MiHv `2+2BpBM; b2`pB+2- M/ QM2 +Q``2bTQM/BM; iQ i?2 iBK2 iQ i?2 M2ti ``BpH Ur?B+? Kv #2 `2@ 7mb2/ B7 i?2 bvbi2K Bb 7mHHVX A7 X(t) = i < K- 2+? Q7 i?2 i +mbiQK2`b Bb ii2M/2/ #v  b2`p2`- M/ r2 ?p2 `M/QK p`B#H2b X1 , . . . , Xi `2T`2b2MiBM; i?2 `2KBMBM;

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

9yj

iBK2b mMiBH +QKTH2iBQM Q7 b2`pB+2X "v i?2 H+F Q7 K2KQ`v T`QT2`iv Q7 i?2 2tTQM2M@ iBH `M/QK p`B#H2b- i?2b2 `2 BB/ 2tTQM2MiBH `M/QK p`B#H2b rBi? K2M μ−1 X 6Q` bBKBH` `2bQMb- i?2 iBK2 iQ i?2 M2ti ``BpH Bb M 2tTQM2MiBH `M/QK p`B#H2 X0 rBi? K2M λ−1 - BM/2T2M/2Mi Q7 X1 , . . . , Xi X h?2`27Q`2 P (X(t + h) = i + 1 | X(t) = i) = P (X0 ≤ h, X1 > h, . . . , Xi > h) + o(h), iFBM; BMiQ ++QmMi i?i 7Q` BM}MBi2bBKH h i?2`2 Bb i KQbi QM2 Q7 i?2 2tTQM2MiBHb X0 , X1 , . . . , Xi BM (0, h)X LQr- iQ i?2 }`bi Q`/2` BM hP (X0 ≤ h, X1 > h, . . . , Xi > h) = (1 − eλh ) × eμh × . . . × eμh  λh. _2+HH i?2 K2MBM; Q7 qi,i+1 P (X(t + h) = i + 1 | X(t) = i) = qi,i+1 h + o(h). h?2`27Q`2- qi,i+1 = λX aBKBH`Hv- B7 1 ≤ i ≤ KP (X(t + h) = i − 1 | X(t) = i) = P (X0 > h, inf {X1 , . . . , Xi } ≤ h) + o(h). "mi inf {X1 , . . . , Xi } Bb M 2tTQM2MiBH `M/QK p`B#H2 Q7 K2M (iμ)−1 - M/ i?2`2@ 7Q`2 P (X0 > h, inf {X1 , . . . , Xi } ≤ h)  iμh. *QKT`BM; rBi? P (X(t + h) = i − 1 | X(t) = i) = qi,i−1 h + o(h) r2 ?p2 i?i qi,i−1 = iμX A7 i ≥ K i?2`2 `2 2t+iHv K +mbiQK2`b BM b2`pB+2- M/ i?2 bK2 ?2m`BbiB+b b #Qp2 ;Bp2 qi,i−1 = KμX AM i?2 +b2 i = 0- i?2`2 Bb Q7 +Qm`b2 QMHv QM2 +iBp2 2tTQM2MiBH- MK2Hv X0 - i?2 `2KBMBM; iBK2 iQ i?2 M2ti ``BpHX q2 ?p2 qi,i−1 = 0- M/ qi,i+1 = λX AM i?2 +b2 i = c + K- MQ ``BpHb BM i?2 bvbi2K `2 TQbbB#H2- bBM+2  TQi2MiBH +mbiQK2` rBHH MQi #2 ++2Ti2/X h?2`27Q`2 BM i?Bb +b2qi,i+1 = 0X AM bmKK`vqi,i+1 = λ 1{iSh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

h?2 i`2iK2Mi /Bz2`b BM i?2 +QKTmiiBQM Q7 qi,i−1 7Q` i > 0X q2 ?p2 iQ BMi`Q/m+2  {0, 1}@pHm2/ `M/QK p`B#H2 Z- BM/2T2M/2Mi Q7 X0 , X1 - rBi? P (Z = 1) = pBM/B+iBM; r?2i?2` i?2 M2ti +mbiQK2` iQ +QKTH2i2 b2`pB+2 rBHH #2 `2+v+H2/ UZ = 1V Q` MQi UZ = 0VX >2`2 qi,i−1 h  P (X(t + h) = i − 1 | X(t) = i)  P (X0 > h, X1 ≤ h, Z = 0) = P (X0 > h, X1 ≤ h)P (Z = 0)  μh × (1 − p), r?2`2 i?2 bvK#QH  BM/B+i2b 2[mHBiv mT iQ i?2 }`bi Q`/2` BM hX h?2`27Q`2 qi,i−1 = μ(1 − p) 1{i ≥ 1}X h?2 *QM;2biBQM S`Q+2bb b  "B`i?@M/@/2i? S`Q+2bb h?2 BM}MBi2bBKH ;2M2`iQ` Q7 i?2 +QM;2biBQM T`Q+2bb Q7 M JfJfRf∞ [m2m2 Bb BM/2T2M/2Mi Q7 i?2 b2`pB+2 /Bb+BTHBM2 B7 r2 +QMbB/2` QMHv b2`pB+2 /Bb+BTHBM2b bm+? i?i i?2 b2`p2` rQ`Fb i 7mHH bT22/- 2[mH iQ R- r?2M2p2` i?2`2 Bb i H2bi QM2 +mbiQK2` BM i?2 bvbi2KX h?Bb ;2M2`H 7+i Bb /m2 iQ i?2 H+F Q7 K2KQ`v Q7 2tTQM2MiBH p`B#H2bM/ Bi +M #2 +?2+F2/ mbBM; i?2 bK2 ?2m`BbiB+b b BM i?2 7B7Q +b2 7Q` /Bb+BTHBM2b rBi?Qmi BMi2``mTiBQM Q7 b2`pB+2- bm+? b HB7Q MQM@T`22KTiBp2- `M/QK Ui?2 M2ti +mbiQK2` Bb +?Qb2M `M/QKHv BM i?2 rBiBM; `QQKVX AM i?2 +b2 Q7 BMi2``mTiBQM Q7 b2`pB+2- rBi? `2bmK2/ b2`pB+2 Ui?i Bb- i?2 b2`pB+2 H`2/v T`QpB/2/ Bb MQi #QHBb?2/ #v i?2 T`22KTiBQMV i?2 ?2m`BbiB+b rBHH ;Bp2 ;BM i?2 bK2 `2bmHib- #2+mb2 i?2 `2KBMBM; iBK2 Q7 M BMi2``mTi2/ 2tTQM2MiBH `M/QK p`B#H2 Bb M 2tTQM2MiBH `M/QK p`B#H2 rBi? i?2 bK2 K2MX A7 7i2` T`22KTiBQM i?2 b2`pB+2 Bb bi`i2/ 7`QK i?2 #2;BMMBM;- ;BM MQi?BM; +?M;2b b 7` b i?2 BM}MBi2bBKH ;2M2`iQ` Bb +QM+2`M2/X h?2 +QM;2biBQM T`Q+2bb2b U+QmMiBM; i?2 +mbiQK2`b BM i?2 bvbi2KV Q7 i?2 [m2m2@ BM; bvbi2Kb i?i r2 Dmbi +QMbB/2`2/ `2 bT2+BH +b2b Q7 #B`i?@M/@/2i? T`Q+2bb2b rBi? M BM}MBi2bBKH ;2M2`iQ` Q7 i?2 7Q`K ⎛ ⎞ −λ0 λ0 0 0 ··· ⎜ μ1 −(λ1 + μ1 ) λ1 0 · · ·⎟ ⎜ ⎟ A=⎜ 0 μ2 −(λ2 + μ2 ) λ2 · · ·⎟ ⎝ ⎠ XX XX XX XX X X X X r?2`2 i?2 bii2 bT+2 Bb E = N- Q` E = {0, 1, . . . , N } 7Q` }MBi2 N X 6Q` HH i?2 [m2m2BM; T`Q+2bb2b r2 +QMbB/2` BM i?2 T`2b2Mi b2+iBQM- λi > 0 7Q` HH i ∈ E- 2t+2Ti i = N r?2M E = {0, 1, . . . , N }- M/ μi > 0 7Q` HH i ∈ E- 2t+2Ti i = 0X h?2b2 +QM/BiBQMb Q#pBQmbHv ;m`Mi22 B``2/m+B#BHBivX q2 `2+HH i?2 ;2M2`H `2bmHi QM #B`i?@ M/@/2i? T`Q+2bb2b- MQiBM; }`bi i?i HH i?2 +?BMb `BbBM; 7`QK i?2 2tKTH2b #Qp2 `2 `2;mH` DmKT T`Q+2bb2b U1t2`+Bb2 RRXeXRVX h?2 2`;Q/B+Biv +QM/BiBQM Bb, i  λn−1 i∈E i≥1

BM r?B+? +b2- 7Q` i ≥ 1-

n=1

μn

< ∞,

URRXRV

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u π(i) = π(0)

i  λn−1 n=1

μn

,

9y8 URRXkV

r?2`2 π(0) Bb Q#iBM2/ #v MQ`KHBxiBQMX h?2 2`;Q/B+Biv +QM/BiBQM Bb- Q7 +Qm`b2miQKiB+HHv biBb}2/ r?2M i?2 bii2 bT+2 Bb }MBi2X 1tKTH2 RRXRX3, JfJfRf∞X 6Q` i?2 JfJfRf∞ [m2m2- E = N- λi ≡ λ > 0  i M/ μi = μ > 0 7Q` HH i ≥ 1X h?2 2`;Q/B+Biv +QM/BiBQM `2/b i≥1 μλ < ∞- i?i Bbλ ρ := < 1 . μ h?Bb +QM/BiBQM Bb Mim`H- bBM+2 ρ = λE[σ1 ]- i?2 i`{+ BMi2MbBiv- Bb i?2 p2`;2 `i2 Q7 rQ`F 2Mi2`BM; i?2 bvbi2K T2` mMBi iBK2- M/ b?QmH/ MQi 2t+22/ i?2 KtBKH bT22/ Q7 b2`pB+2- 2[mH iQ 1X h?2 bQHmiBQM Q7 i?2 #HM+2 2[miBQM Bb π(i) = (1 − ρ)ρi .

1tKTH2 RRXRXN, JfJfEfy- Q` 1`HM; [m2m2X 6Q` i?2 JfJfEfy [m2m2 U1`HM; [m2m2V E = {0, . . . , K} M/ i?2 bQHmiBQM Q7 i?2 #HM+2 2[miBQMb Bb ρi /i! π(i) = K n n=0 ρ /n! 7Q` 0 ≤ i ≤ KX AM T`iB+mH`ρK /K! π(K) = K n n=0 ρ /n! Bb i?2 #HQ+FBM; T`Q##BHBiv- i?2 T`Q##BHBiv Q7 }M/BM; HH +?MM2Hb #mbvX h?2 +Q``2@ bTQM/BM; 7Q`KmH Bb +HH2/ i?2 1`HM; #HQ+FBM; 7Q`KmHX h?2 /Bbi`B#miBQM π Bb +HH2/  SQBbbQM /Bbi`B#miBQM i`mM+i2/ i K- bBM+2 π(i) = P (Z = i | Z ≤ K), r?2`2 Z Bb  SQBbbQM `M/QK p`B#H2 rBi? K2M ρX 1tKTH2 RRXRXRy, JfJf∞X 6Q` i?2 JfJf∞ [m2m2- i?2 2`;Q/B+Biv +QM/BiBQM

ρi URRXRV Bb ∞ i=1 i! < ∞ M/ Bb Hrvb biBb}2/ b HQM; b ρ < ∞X h?2 biiBQM`v /Bbi`B#miBQM Bb i?2 SQBbbQM /Bbi`B#miBQM rBi? K2M 2[mH iQ i?2 i`{+ BMi2MbBiv ρπ(i) = e−ρ

ρi . i!

9ye

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

1tKTH2 RRXRXRR, JfJfEf∞X 6Q` i?2 JfJfEf∞ [m2m2- i?2 2`;Q/B+Biv +QM@ /BiBQM URRXRV Bb biBb}2/ QMHv B7 ρ :=

λ < K. μ

Ai bvb i?i i?2 p2`;2 BM+QKBM; rQ`F T2` mMBi iBK2 +MMQi 2t+22/ i?2 KtBKH b2`pB+2 bT22/- r?B+? Bb K r?2M HH b2`p2`b `2 #mbvX h?2 biiBQM`v /Bbi`B#miBQM Bb i?2M- 7Q` 0 ≤ i ≤ Kρi π(i) = π(0) , i! M/ r?2M i ≥ KρK  ρ i−K , π(i) = π(0) s! K r?2`2 s−1 i  ρ ρK 1 + . π(0)−1 = i! K! 1 − ρ/s i=0 AM i?Bb bvbi2K- i?2 T`Q##BHBiv Q7 rBiBM; Bb i?2 T`Q##BHBiv Q7 2Mi2`BM; i?2 bvbi2K r?2M i?2 K b2`p2`b `2 #mbv- i?i Bb π(≥ K) := i≥K π(i)X PM2 Q#iBMb i?2 1`HM; rBiBM; 7Q`KmH π(≥ K) =

ρs 1 K! 1−ρ/K

K−1 ρi ρK 1 i=0 i! + K! 1−ρ/K

.

h?2 JfJfRf∞ hM/2K q2 MQr im`M iQ i?2 bim/v Q7 [m2m2BM; M2irQ`Fb- bi`iBM; rBi? i?2 bBKTH2bi QM2- i?2 iM/2K Q7 JfJfRf∞ [m2m2bX h?2 b2KBMH `2bmHi ?2`2 Bb "m`F2Ƕb QmiTmi i?2Q`2KX9 Ai bvb BM T`iB+mH` i?i i?2 TQBMi T`Q+2bb Q7 i?2 /2T`im`2 iBK2b Q7 M JfJfRf∞ [m2m2 BM 2[mBHB#`BmK Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb rBi? i?2 bK2 BMi2MbBiv b i?2 ``BpH T`Q+2bbX AM 7+i- i?2 `2bmHi 2ti2M/b iQ KQ`2 ;2M2`H #B`i?@M/@/2i? T`Q+2bb2bX8 *QMbB/2`  #B`i?@M/@/2i? T`Q+2bb QM N rBi? #B`i? T`K2i2`b Q7 i?2 7Q`K λi ≡ λ, M/ bmTTQb2 i?i μi > 0 7Q` HH i ≥ 1X h?2 +Q``2bTQM/BM; +?BM Bb B``2/m+B#H2- M/ r2 b?HH bbmK2 i?i Bi Bb 2`;Q/B+X i 2[mBHB#`BmK- i?Bb +?BM Bb `2p2`bB#H2 bBM+2 i?2 /2iBH2/ #HM+2 2[miBQMb π(i + 1)μi+1 = π(i)λ Ui ∈ NV `2 biBb}2/- b +M #2 `2/BHv +?2+F2/ mbBM; i?2 2tTHB+Bi 7Q`K Q7 i?2 biiBQM`v /Bbi`B#miBQMX h?2 mTr`/ i`MbBiBQMb `2 /m2 iQ ``BpBM; +mbiQK2`b M/ i?2 /QrMr`/ i`M@ bBiBQMb `2 /m2 iQ /2T`iBM; +mbiQK2`bX amTTQb2 i?i i?2 [m2m2 Bb BM 2[mBHB#`BmKX 9 8

("m`F2- RN8e)X h?2 T`QQ7 #2HQr- mbBM; i?2 `2p2`bB#BHBiv `;mK2Mi- Bb i?i Q7 (_2B+?- RN8d)X

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

9yd

AM T`iB+mH`- i?2 `2p2`b2/ T`Q+2bb ?b i?2 bK2 /Bbi`B#miBQM b i?2 /B`2+i T`Q+2bbXe q?2M iBK2 Bb `2p2`b2/- /2T`im`2b #2+QK2 ``BpHb M/ i?2`27Q`2- BM pB2r Q7 i?2 `2p2`bB#BHBiv T`QT2`iv- i?2 `2p2`b2/ T`Q+2bb Q7 /2T`im`2b Bb M ?TTX LQr- i?2 T`Q#@ #BHBbiB+ Mim`2 Q7 M ?TT /Q2b MQi +?M;2 r?2M iBK2 Bb `2p2`b2/X h?2`27Q`2- i?2 /2T`im`2 T`Q+2bb Bb  SQBbbQM T`Q+2bbX HbQ- bBM+2 BM /B`2+i iBK2- 7Q` Mv iBK2 t ≥ 0- i?2 bii2 X(t) Bb BM/2T2M/2Mi Q7 i?2 7mim`2 i iBK2 t Q7 i?2 ``BpH T`Q+2bb UJ`FQp T`QT2`iv Q7 SQBbbQM T`Q+2bb2bVBi 7QHHQrb 7`QK `2p2`bB#BHBiv i?i X(t) Bb BM/2T2M/2Mi Q7 i?2 Tbi i iBK2 t Q7 i?2 /2T`im`2 T`Q+2bbX X(t)

t ``BpHb Q7 i?2 iBK2@`2p2`b2/ T`Q+2bb S`QQ7 Q7 "m`F2Ƕb i?2Q`2K #v `2p2`bB#BHBiv

1tKTH2 RRXRXRk, hM/2K Q7 JfJfRf∞ [m2m2bX *QMbB/2` irQ JfJfRf∞ BM bm++2bbBQM- BM i?2 b2Mb2 i?i i?2 /2T`im`2 T`Q+2bb 7`QK i?2 }`bi QM2 Bb i?2 ``BpH T`Q+2bb Q7 i?2 b2+QM/ QM2X h?2 TQBMi T`Q+2bb Q7 ``BpHb BMiQ i?2 }`bi [m2m2 Bb M ?TT Q7 BMi2MbBiv λX h?2 b2`pB+2 b2[m2M+2b `2 BM/2T2M/2Mi BB/ 2tTQM2MiBH b2[m2M+2b rBi? K2M 1/μ1 M/ 1/μ2 `2bT2+iBp2HvX "Qi? b2`pB+2 b2[m2M+2b `2 BM/2T2M/2Mi Q7 i?2 ``BpH T`Q+2bbX G2i X1 (t) M/ X2 (t) #2 i?2 MmK#2` Q7 +mbiQK2`b BM i?2 }`bi M/ i?2 b2+QM/ [m2m2BM; bvbi2K `2bT2+iBp2HvX



X1 (t)

∞ 1

X2 (t) 1

e ai`B+iHv bT2FBM;- i?2 `2p2`b2/ +QM;2biBQM T`Q+2bb Bb H27i@+QMiBMmQmb r?2`2b i?2 Q`B;BMH T`Q+2bb Bb `B;?i@+QMiBMmQmbX >Qr2p2`- +?M;BM; i?2 pHm2b i- bv- i?2 k@i? /Bb+QMiBMmBiv TQBMi /Q2b MQi +?M;2 i?2 T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 `2p2`b2/ T`Q+2bb- bBM+2 i?2 HQ+iBQM Q7 bm+? /Bb+QMiBMmBiv /KBib  T`Q##BHBiv /2MbBivX

9y3

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

i iBK2 t- X2 (t) /2T2M/b QM i?2 /2T`im`2 T`Q+2bb Q7 i?2 }`bi [m2m2 #27Q`2 iBK2 t M/ QM i?2 b2+QM/ b2`pB+2 b2[m2M+2X aBM+2 #v "m`F2Ƕb i?2Q`2K M/ i?2 BM/2T2M@ /2M+2 T`QT2`iv Q7 i?2 b2`pB+2 b2[m2M+2 i?2 Hii2` `2 BM/2T2M/2Mi Q7 X1 (t)- Bi 7QH@ HQrb i?i X1 (t) M/ X2 (t) `2 BM/2T2M/2MiX "Qi? [m2m2b `2 BM BbQHiBQM JfJfRf∞ rBi? i`{+ BMi2MbBiB2b ρ1 = λ/μ1 M/ ρ2 = λ/μ2 - M/ i?2`27Q`2  M2+2bb`v M/ bm{+B2Mi +QM/BiBQM Q7 2`;Q/B+Biv Q7 i?2 +QMiBMmQmb@iBK2 ?K+ {(X1 (t), X2 (t))}t≥0 Bb λ < inf(μ1 , μ2 ) M/ Bib biiBQM`v /Bbi`B#miBQM i?2M iF2b i?2 T`Q/m+i 7Q`K π(n1 , n2 ) = π1 (n1 )π2 (n2 ) = (1 − ρ1 )ρn1 1 (1 − ρ2 )ρn1 2

(n1 , n2 ≥ 0) .

h?2 C+FbQM L2irQ`F h?Bb Bb M QT2M M2irQ`F Q7 BMi2`+QMM2+i2/ [m2m2bXd h?2`2 `2 K biiBQMb- M/ 2+? biiBQM ?b  Rf∞ b2`pB+2 bvbi2K- i?i Bb-  mMB[m2 b2`p2` rQ`FBM; i mMBi bT22/ M/ M BM}MBi2 rBiBM; `QQKX h?2`2 `2 irQ ivT2b Q7 +mbiQK2`b [m2m2BM; i  ;Bp2M biiBQM, URV i?Qb2 r?B+? `2 72/@#+F- i?i Bb- r?Q ?p2 `2+2Bp2/ b2`pB+2 BM MQi?2` Q` i?2 bK2 biiBQM M/ `2 `2`Qmi2/ iQ i?2 ;Bp2M biiBQM 7Q` KQ`2 b2`pB+2- M/ UkV i?Qb2 r?Q `2 2Mi2`BM; i?2 M2irQ`F 7Q` i?2 }`bi iBK2 Ub22 i?2 };m`2 #2HQrVX ¯i λ

ri

rij rji ¯j λ

rj

C+FbQM M2irQ`F ¯i - rBi? h?2 2tQ;2MQmb ``BpHb BMiQ biiBQM i U1 ≤ i ≤ KV 7Q`K M ?TT- /2MQi2/ #v N ¯ BMi2MbBiv λi ∈ [0, ∞)X h?2 b2[m2M+2 Q7 b2`pB+2 iBK2b i biiBQM i `2 2tTQM2MiBH `M/QK p`B#H2b Q7 K2M 1/μi ∈ (0, ∞)X h?2 b2`pB+2 iBK2b BM i?2 bK2 M/ BM /Bz2`2Mi biiBQMb `2 BM/2T2M/2Mi M/ ¯i - M/ i?2 Hii2` ?TTb `2 BM/2T2M/2Mi BM/2T2M/2Mi Q7 i?2 2tQ;2MQmb BMTmi ?TTb N Q7 QM2 MQi?2`X h?2 `QmiBM; Bb Q7 i?2 "2`MQmHHB ivT2X 1+? +mbiQK2` Dmbi +QKTH2iBM; b2`pB+2 BM biiBQM i iQbb2b  (K +1)@7+2/ /B2 rBi? T`Q##BHBiB2b ri,1 , . . . , ri,K , ri rBi? i?2 2z2+i i?i i?2 +mbiQK2` Bb b2Mi iQ biiBQM j rBi? T`Q##BHBiv rij Q` H2p2b i?2 bvbi2K rBi?

T`Q##BHBiv ri = 1 − K j=1 rij X h?2 Ki`Bt d

(C+FbQM- RNej)X

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

9yN

R = {rij }1≤i,j≤K Bb i?2 `QmiBM; Ki`BtX h?2 bm++2bbBp2 iQbb2b Q7 i?2 `QmiBM; /B+2 Q7 HH biiBQMb `2 BM/2T2M/2Mi- M/ BM/2T2M/2Mi Q7 i?2 2tQ;2M2Qmb ``BpH T`Q+2bb2b M/ Q7 HH i?2 b2`pB+2 iBK2bX h?Bb Bb i?2 Q`B;BMH C+FbQM KQ/2H- r?B+? +M #2 2M`B+?2/ #v i?2 BMi`Q/m+iBQM Q7 b2`pB+2 bT22/bX A7 i?2`2 `2 ni +mbiQK2`b BM biiBQM i U1 ≤ i ≤ KV- i?2 b2`p2` rQ`Fb i bT22/ φi (ni )- r?2`2 φi (0) = 0 M/ φi (ni ) > 0 7Q` HH ni ≥ 1X G2i Xi (t) #2 i?2 MmK#2` Q7 +mbiQK2`b BM biiBQM i i iBK2 t- M/ H2i X(t) := (X1 (t), . . . , Xk (t)). Ai +M #2 T`Qp2/ UmbBM; i?2 K2i?Q/b Q7 a2+iBQM RRXkV i?i i?2 T`Q+2bb {X(t)}t≥0 Bb  `2;mH` DmKT ?K+ rBi? bii2 bT+2 E = NK M/ rBi? BM}MBi2bBKH ;2M2`iQ` A = {qn,n }- r?2`2 HH i?2 MQM@MmHH Qz@/B;QMH i2`Kb `2 ¯i, qn,n+ei = λ qn,n−ei = μi φi (ni )ri 1{ni >0} , qn,n−ei +ej = μi φi (ni )rij 1{ni >0} , r?2`2 n = (n1 , . . . , nK ) ∈ E = NK M/ r?2`2 ei Bb i?2 ii? p2+iQ` Q7 i?2 +MQMB+H #bBb Q7 RK X h?Bb ;2M2`iQ` Bb BM/2T2M/2Mi Q7 i?2 b2`pB+2 bi`i2;věHB7Q- 7B7Q- Q` T`Q+2bbQ`@b?`BM;X h?2 7Q`K Q7 i?2 BM}MBi2bBKH ;2M2`iQ` +M #2 Q#iBM2/ 7`QK  7mHH /2b+`BTiBQM Q7 i?2 M2irQ`F b  SQBbbQM bvbi2K Ub22 a2+iBQM RRXkVX >Qr2p2`- r2 b?HH #2 +QMi2Mi rBi? ?2m`BbiB+ `;mK2MibX 6Q` BMbiM+2- i?2 2tT`2bbBQM 7Q` qn,n−ei +ej 7QHHQrb 7`QK i?2 BMimBiBp2 +QMbB/2`iBQMb #2HQrX A7 i?2 bii2 i iBK2 t Bb n-  i`Mb72` 7`QK biiBQM i iQ biiBQM j `2[mB`2b i?i i?2 2tTQM2MiBH `M/QK p`B#H2 rBi? K2M 1/μi `2T`2b2MiBM; i?2 `2[mB`2/ b2`pB+2 Q7 i?2 +mbiQK2` #2BM; b2`p2/ i biiBQM i i iBK2 t #2 i2`KBMi2/ #2ir22M iBK2b t M/ t + h UT`Q##BHBiv μi φi (ni )h mT iQ i?2 }`bi Q`/2` BM hV M/ i?i i?2 +Q``2bTQM/BM; +mbiQK2` #2 `Qmi2/ iQ biiBQM j UT`Q##BHBiv rij )X bbmK2 i?2 +?BM {X(t)}t≥0 B``2/m+B#H2X h?Bb Bb i?2 +b2 r?2M (α) 7Q` HH j ∈ {1, 2, . . . , K}- i?2`2 2tBbi i, i1 , . . . , im ∈ {1, 2, . . . , K} bm+? i?i ¯ i rii ri i · · · rim j > 0 , λ 1 1 2 M/ (β) 7Q` HH j ∈ {1, 2, . . . , K}- i?2`2 2tBbi j1 , j2 , . . . , j , k ∈ {1, 2, . . . , K} bm+? i?i rjj1 rj1 j2 · · · rj k rk > 0 . _2+HHBM; i?i μi > 0 7Q` HH biiBQMb- +QM/BiBQM (α) bvb i?i Mv biiBQM Bb 2tQ;2MQmbHv bmTTHB2/ r?2`2b +QM/BiBQM (β) bvb i?i Mv biiBQM ?b M QmiH2iX *QMbB/2` MQr i?2 (K + 1) × (K + 1) Ki`Bt

9Ry

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: ⎛

··· ···

r11 r12 ⎜ r21 r22 ⎜ XX X ˜ =⎜ R ⎜ XX X ⎜ ⎝rK1 rK2 0 0

r1K r2K XX X

··· ···

rKK 0

⎞ r1 r2 ⎟ ⎟ XX ⎟ . X ⎟ ⎟ rK ⎠ 1

Ai +M #2 BMi2`T`2i2/ b i?2 i`MbBiBQM Ki`Bt Q7 M ?K+ QM i?2 }MBi2 bii2 bT+2 {1, . . . , K, K+1}X *QM/BiBQM (β) BKTHB2b i?i bii2 K+1 Bb #bQ`#BM; M/ i?2 bii2b 1, . . . , K `2 i`MbB2MiX AM T`iB+mH`- 7Q` HH i, j BM i?2 i`MbB2Mi b2i- limn↑∞ pij (n) = 0- i?i Bb- limn↑∞ Rn = 0X aBM+2 (I + R + · · · + Rn )(I − R) = I − Rn+1 , i?2 BMp2`b2 (1 − R)−1 2tBbib M/ Bb 2[mH iQ ( i?2 bvbi2K Q7 2[miBQMb ¯i + λi = λ

K 

λj rji

∞ n=0

Rn )X h?2`27Q`2 i?2 bQHmiBQM Q7

(1 ≤ i ≤ K) ,

URRXjV

j=1

¯ T + λT R- ?b  mMB[m2 bQHmiBQM i?i Bb- rBi? Q#pBQmb MQiiBQM- λT = λ ¯ T (1 − R)−1 , λT = λ r?2`2 i?2 Hii2` 2tT`2bbBQM b?Qrb i?i i?Bb bQHmiBQM Bb MQM@M2;iBp2X 1[miBQMb URRXjV `2 +HH2/ i?2 i`{+ 2[miBQMb- #2+mb2 i?2v ;Bp2  M2+2bb`v `2HiBQM #2@ ir22M i?2 p2`;2 MmK#2`b Q7 +mbiQK2`b λi 2Mi2`BM; biiBQM i BM bi2/v bii2 B7 i?2 ¯ i THmb i?2 M2irQ`F Bb 2`;Q/B+X AM/22/- λi Bb 2[mH iQ i?2 2tQ;2M2Qmb `i2 Q7 ``BpHb λ bmK Q7 HH p2`;2 `i2b Q7 i`Mb72` 7`QK Qi?2` biiBQMbX 6`QK biiBQM j- i?2 +Q``2@ bTQM/BM; `i2 Bb αj rji - r?2`2 αj Bb i?2 p2`;2 `i2 Q7 +mbiQK2`b }MBb?BM; b2`pB+2 BM biiBQM jX "mi i 2[mBHB#`BmK αj = λj - bBM+2 i?2 p2`;2 MmK#2` Q7 +mbiQK2`b BM biiBQM j `2KBMb +QMbiMi- r?2M+2 i?2 i`{+ 2[miBQMb URRXjVX h?Bb ?2m`BbiB+ `;mK2Mi Bb MQi M22/2/- M/ BM T`iB+mH`- QM2 M22/ MQi ii2KTi iQ B/2MiB7v λi BM URRXjV b i?2 p2`;2 BM+QKBM; ``BpH `i2 BM biiBQM iX h?Bb B/2MiB}+iBQM Bb TQbbB#H2 B7 2[mBHB#`BmK Bb ;m`Mi22/ U1t2`+Bb2 RRXeXRRVX >Qr2p2`2p2M BM i?2 MQM@2`;Q/B+ +b2b- i?2 i`{+ 2[miBQMb ?p2  mMB[m2 bQHmiBQMX q2 MQr +QMbB/2` i?2 +b2 r?2`2 i?2 b2`pB+2 bT22/b Q7 HH i?2 b2`p2`b `2 2[mH iQ R M/ `272` i?2 `2/2` iQ 1t2`+Bb2 RRXeX9 7Q` i?2 ;2M2`H +b2X h?2 ;HQ#H #HM+2 2[miBQMb `2  K  K K    ¯ i 1{n >0} + π(n) (λi + μi ri 1{n >0} ) = π(n − ei )λ π(n + ei )μi ri i

i=1

i

i=1

+

K  K 

i=1

π(n + ei − ej )μi rij 1{nj >0} .

i=1 j=1

Ai im`Mb Qmi i?i B7 i?2 bQHmiBQM Q7 i?2 i`{+ 2[miBQM biBb}2b

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u ρi =

9RR

λi < 1 (1 ≤ i ≤ K) , μi

URRX9V

i?2M i?2 M2irQ`F Bb 2`;Q/B+ rBi? biiBQM`v /Bbi`B#miBQM π(n) =

K 

URRX8V

πi (ni ) ,

i=1

r?2`2 πi Bb i?2 biiBQM`v /Bbi`B#miBQM Q7 M JfJfRf∞ [m2m2 rBi? i`{+ BMi2MbBiv ρi πi (ni ) = ρni i (1 − ρi ) . URRXeV ˜ QM hQ T`Qp2 i?Bb- TTHv i?2 `2p2`bH i2bi U1t2`+Bb2 RRXeXkVX .2}M2 i?2 ;2M2`iQ` A K E = N #v π(n)˜ qn,n = π(n )qn ,n M/ +?2+F i?i



>2`2 qn =

q˜n,n = qn .

K 

λi + μi 1{ni >0} .

i=1

˜ Bb ;Bp2M #v h?2 ;2M2`iQ` A π(n)˜ qn,n+ei = π(n + ei )qn+ei ,n = π(n + ei )μi ri , ¯ i 1{n >0} , π(n)˜ qn,n−ei = π(n − ei )qn−ei ,n = π(n − ei )λ i π(n)˜ qn,n+ei −ej = π(n + ei − ej )qn+ei −ej ,n = π(n + ei − ej )μi rij 1{nj >0} , M/ i?2`27Q`2- iFBM; BMiQ ++QmMi i?2 bT2+B}+ 7Q`K Q7 π(n) ;Bp2M #v URRX8V M/ URRXeVq˜n,n+ei = ρi μi ri = λi ri , ¯i λ q˜n,n−ei = 1{n >0} , ρi i ρi 1 μi rij 1{nj >0} = λi rij 1{nj >0} . q˜n,n+ei −ej = ρj ρj Ai Kmbi #2 p2`B}2/ i?i K 

¯ i + μi 1{r >0} ) = (λ i

i=1

K  i=1

"v i?2 i`{+ 2[miBQMi?2 T`2pBQmb 2[mHBiv Bb

K i=1

K   i=1



¯i λ 1 λi ri + 1{ni >0} + ρi ρi

K 

 λj rji

 1{ni >0}

.

i=1

¯ i - M/ i?2`27Q`2 i?2 `B;?i@?M/ bB/2 Q7 λj rji = λi − λ

1 λi ri + λi 1{ni >0} ρi

M/ Bi `2KBMb iQ +?2+F i?i



 =

K  i=1

(λi ri + μi 1{ni >0} ),

9Rk

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: K 

¯i = λ

i=1

K 

λi ri .

i=1

6Q` i?Bb- Bi bm{+2b iQ bmK i?2 i`{+ 2[miBQMb URRXjVX 1tKTH2 RRXRXRj, h?2 bTHBiiBM; T`/QtX3 h?2`2 Bb M TT`2Mi T`/Qt `2HiBp2 iQ i?2 [m2biBQM Q7 ~Qrb BM  [m2m2BM; M2irQ`F- BM 7+i i?2 bBKTH2bi MQM@ +b+/2 C+FbQM M2irQ`F, i?2 M/M/1/∞ [m2m2 rBi? BMbiMiM2Qmb 722/#+F rBi? `2+v+HBM; T`Q##BHBiv 12 X AM i?Bb +b2- i?2 QmiTmi ~Qr 7`QK i?2 bvbi2K U+mbiQK2`b H2pBM; 7Q` ;QQ/V Bb SQBbbQM- #mi i?2 `2+v+H2/ ~Qr Bb MQi SQBbbQMX h?2 TT`2Mi T`/Qt Ui?2 SQBbbQM bTHBiiBM; T`/QtV Bb i?i r2 ?p2  TQBMi T`Q+2bb U+mbiQK2`b Dmbi H2pBM; i?2 b2`pB+2 #QQi?V r?B+? Bb bTHBi BM irQ mbBM; M BB/ b2[m2M+2 Q7 7B` +QBM iQbb2b- M/ ?H7 Q7 Bi `2KBMb SQBbbQM r?BH2 i?2 Qi?2` ?H7 Bb MQi SQBbbQMX h?2 T`QQ7 i?i i?2 722/#+F TQBMi T`Q+2bb Bb MQi M ?TT Bb b 7QHHQrbX h?2 }Hi2`BM; 2[miBQM 7Q` Z0 (t) := 1{X(t)=0} rBi? `2bT2+i iQ i?2 Tbi Q7 i?2 722/#+F TQBMi T`Q+2bb Bb U2t2`+Bb2- mb2 i?2 `2bmHib Q7 a2+iBQM RyXRV, 



t

Z0F (s)μ(1 − p) ds −

Z0F (t) = Z0F (0) + 0

Z0F (s−)(N (ds) − μp(1 − Z0F (s)) ds) . (0,t]

amTTQb2 i?2 722/#+F ~Qr r2`2 SQBbbQM- i?2M Bib FtF @T`2/B+i#H2 BMi2MbBiv μp(1 − Z0F (t−)) rQmH/ #2  +QMbiMi ξX h?2`27Q`2- #v mMB[m2M2bb Q7 i?2 FtF @ T`2/B+i#H2 BMi2MbBiv- P (dω)ξ dt@X2X M/ P (dω) N (dt)@X2X Uh?2Q`2K 8XRXj3V μp(1 − Z0F (t−, ω)) = ξ . .2MQiBM; #v τ1 i?2 }`bi TQbBiBp2 2p2Mi iBK2 Q7 F - Z0F (τ1 ) = 0 Ui?Bb Bb BMimBiBp2Hv Q#pBQmb- Q` 2Hb2- mb2 i?2 2tT`2bbBQM 7Q` i?2 ;BM iQ Q#iBM Z0F (τ1 ) = Z0F (τ1 −) + K0F (τ1 ) = Z0F (τ1 −) − Z0F (τ1 −) = 0 ). P#b2`pBM; i?i Z0F (t−) = Z0F (t) B7 t Bb MQi M 2p2Mi iBK2 Q7 N - r2 ?p2 i?i Z0F (t) =

μp − ξ μp

M/ i?2`27Q`2- #v `B;?i@+QMiBMmBivZ0F (τ1 ) =

μp − ξ . μp

X h?Bb BM im`M BKTHB2b i?i P @XbX- Z0F (t) = 0 7Q` HH t ≥ 0X h?2`27Q`2 0 = μp−ξ μp hFBM; 2tT2+iiBQMb- 7Q` HH t ≥ 0- P (X(t) = 0) = 0X h?Bb Bb BM +QMi`/B+iBQM rBi? λ P (X(t) = 0) = μ(1−p) X 3

("`ûKm/- RNd3)c b22 HbQ (qH`M/ M/ o`Bv- RNd3)X

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

9Rj

h?2 :Q`/QMĜL2r2HH L2irQ`F h?Bb Bb  +HQb2/ C+FbQM M2irQ`F- rBi? i?2 7QHHQrBM; bT2+B}+Biv,N ¯ i = 0, ri = 0 λ

(1 ≤ i ≤ K) .

URRXdV

AM Qi?2` rQ`/b- i?2`2 Bb MQ BMH2i M/ MQ QmiH2i- M/ i?2`27Q`2 i?2 MmK#2` N Q7 +mbiQK2`b BM i?2 M2irQ`F `2KBMb +QMbiMiX h?2 bii2 bT+2 Bb E = {(n1 , . . . , nK ) ∈ NK ;

K 

ni = N } .

i=1

h?2 i`{+ 2[miBQMb `2 MQr

λT = λT R ,

URRX3V

M/ bBM+2 R Bb  biQ+?biB+ Ki`Bt bbmK2/ B``2/m+B#H2- Bi ?b M BM}MBiv Q7 bQHm@ iBQMb- HH KmHiBTH2b Q7 i?2 bK2 p2+iQ` Ui?2 biiBQM`v /Bbi`B#miBQM Q7 RVX Ai Bb i`m2 i?i i?2 p2+iQ` Q7 p2`;2 i`{+b i?`Qm;? i?2 biiBQMb Bb  bQHmiBQM Q7 URRX3VX q2 /Q MQi FMQr r?B+? QM2- BM +QMi`bi rBi? i?2 QT2M M2irQ`F- r?2`2 i?2 bQHmiBQM Q7 i?2 i`{+ 2[miBQM Bb mMB[m2X >Qr2p2`- 7Q` Mv TQbBiBp2 bQHmiBQMr2 b22 #v BMbT2+iBQM i?i- H2iiBM; ρi := λi /μi i?2M

(1 ≤ i ≤ K) ,

 1 ρn i G(N, K) i=1 i K

π(n) :=

(n ∈ E)

URRXNV

Bb  biiBQM`v bQHmiBQMX >2`2 G(N, K) Bb i?2 MQ`KHBxBM; 7+iQ` G(N, K) =



K 

n∈NK n1 +···+nK =N

i=1

ρni i .

URRXRyV

LQi2 i?i mM/2` i?2 B``2/m+B#BHBiv bbmKTiBQM 7Q` i?2 `QmiBM; Ki`Bt R- i?2 +?BM {X(t)}t≥0 Bb Bib2H7 B``2/m+B#H2- M/ bBM+2 i?2 bii2 bT+2 Bb }MBi2- Bi Bb TQbBiBp2 `2+m``2Mi rBi?  mMB[m2 biiBQM`v /Bbi`B#miBQMX AM T`iB+mH`- 7Q` +HQb2/ M2irQ`Fbi?2`2 Bb MQ 2`;Q/B+Biv +QM/BiBQMX  T`+iB+H T`Q#H2K rBi? +HQb2/ C+FbQM M2irQ`Fb `2bB/2b BM i?2 +QKTmiiBQM Q7 i?2 MQ`KHBxBM; 7+iQ` G(N, K)X "`mi2 7Q`+2 bmKKiBQM pB URRXRyV Bb T`+iB+HHv BM72bB#H2 7Q` H`;2 TQTmHiBQMb M/fQ` H`;2 M2irQ`FbX AMbi2/- QM2 Kv T`Q+22/ b 7QHHQrbX .2}M2 G(j, ) iQ #2 i?2 +Q2{+B2Mi Q7 z j BM i?2 TQr2` b2`B2b /2p2HQTK2Mi Q7  ∞       1 ni ni . = ρi z g (z) = 1 − ρi z n =0 i=1 i=1 i

N

(:Q`/QM M/ L2r2HH- RNed)X

9R9

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

h?2 MQ`KHBxBM; 7+iQ` Bb BM/22/ 2[mH iQ G(N, K)X aBM+2 g (z) = g−1 (z) + ρ zg (z), r2 }M/ i?2 `2+m``2M+2 2[miBQM G(j, ) = G(j, − 1) + ρ G(j − 1, )

URRXRRV

rBi? i?2 BMBiBH +QM/BiBQMb G(j, 1) = ρj1 (j ≥ 0),

URRXRkV

G(0, ) = 1 ( ≥ 1). h?Bb T`QpB/2b M H;Q`Bi?K 7Q` +QKTmiBM; i?2 MQ`KHBxBM; 7+iQ`X Ai Bb HbQ Q7 BMi2`2bi iQ #2 #H2 iQ +QKTmi2 i?2 miBHBxiBQM Q7 b2`p2` i- /2}M2/ #v URRXRjV

Ui (N, K) := P (Xi (t) > 0) .

h?Bb +M #2 mb27mH- 7Q` BMbiM+2 BM +QKTmiBM; i?2 p2`;2 i?`Qm;?Tmi 7`QK biiBQM i iQ biiBQM j- i?i Bb- i?2 p2`;2 MmK#2` Q7 +mbiQK2`b i`Mb72``2/ 7`QK biiBQM i iQ biiBQM j BM QM2 mMBi Q7 iBK2, URRXR9V

dij (N, K) = μi Ui (N, K)rij . aBM+2 Ui (N, K) =



π(n) =

n1 +···+nK =N ni >0

1 G(N, K)



K 

n1 +···+nK =N ni >0

j=1

n

ρj j ,

r2 b22 i?i G(N, K)Ui (N, K) Bb i?2 +Q2{+B2Mi Q7 z N BM ⎛ ⎛ ⎞⎞   K ∞ ∞    ⎜ ⎝ nj nj ⎟ n n i i . ρj z ⎠⎠ ρi z g˜N,i (z) = ⎝ j=1 j=i

nj =0

ni =1

LQrg˜N,i (z) = gN (z)ρi z, M/ i?2`27Q`2 G(N, K)Ui (N, K) = ρi G(N − 1, K)- i?i BbUi (N, K) = ρi

G(N, K) . G(N − 1, K)

URRXR8V

J`FQpBM MHvbBb Q7 LQM@J`FQpBM Zm2m2b J`FQpBM KQ/2Hb `2 2bv iQ ?M/H2- #mi i?2v `2 MQi Hrvb BM ;`22K2Mi rBi? `2HBivX h?2 SQBbbQMBM bbmKTiBQM 7Q` i?2 BMTmi Bb p2`v Q7i2M DmbiB}2/X >Qr2p2`i?2 2tTQM2MiBH bbmKTiBQM 7Q` i?2 b2`pB+2 iBK2 /Bbi`B#miBQM Bb mbmHHv mM`2HBbiB+X >Qr2p2`- i?2 bim/v Q7 bm+? KQ/2Hb +M #2 `2+bi BM i?2 7`K2rQ`F Q7 /Bb+`2i2@iBK2 ?K+bXRy Ry

(E2M/HH- RN8j)X

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

9R8

1tKTH2 RRXRXR9, Jf:AfRf∞f7B7QX h?Bb [m2m2 Bb 2t+iHv HBF2 M JfJfRf∞f7B7Q [m2m2- QMHv rBi?  ;2M2`H /Bbi`B#miBQM 7Q` i?2 BB/ b2`pB+2 iBK2b b2[m2M+2, G(x) := P (σ1 ≤ x)X h?2 +Q``2bTQM/BM; +QM;2biBQM T`Q+2bb {X(t)}t≥0 Bb MQ HQM;2` J`FQpBMX AMbi2/- r2 +QMbB/2` i?2 T`Q+2bb {Xn }n≥0 - r?2`2 τ0 = 0 M/ 7Q` n ≥ 1 Xn := X(τn ) , r?2`2 τn Bb i?2 ni? /2T`im`2 iBK2 Ub22 i?2 };m`2 #2HQrVX aBM+2 i?2 +QM;2biBQM T`Q@ +2bb Bb iF2M iQ #2 `B;?i@+QMiBMmQmb- Xn Bb i?2 MmK#2` Q7 +mbiQK2`b i?i +mbiQK2` n H2p2b #2?BM/ ?BK mTQM +QKTH2iBM; b2`pB+2X

Xn+1

Xn+1

Xn Xn = 0 τn+1

τn 

τn

Tn

τn+1

# h?2 2K#2//2/ T`Q+2bb i /2T`im`2 iBK2b

 ;HM+2 i i?2 };m`2 #Qp2 `2p2Hb i?i Xn+1 = (Xn − 1)+ + Zn+1 ,

URRXReV

r?2`2 Zn+1 Bb i?2 MmK#2` Q7 +mbiQK2`b ``BpBM; BM i?2 BMi2`pH (αn , αn + σn+1 ]r?2`2 αn = τn B7 Xn > 0 M/ αn = Tn B7 Xn = 0X h?2 iBK2 αn Bb M FtA ∨ Fnσ @biQTTBM; iBK2 Ur?2`2 A Bb i?2 ``BpH SQBbbQM T`Q+2bbM/ Fnσ Bb i?2 σ@}2H/ ;2M2`i2/ #v i?2 }`bi n b2`pB+2 `2[m2bib σ1 , . . . , σn VX HbQ- i?2 (n + 1)bi b2`pB+2 `2[m2bi σn+1 Bb BM/2T2M/2Mi Q7 i?2 ``BpH T`Q+2bb #27Q`2 αn M/ Q7 i?2 T`2pBQmb `2[m2bib (σ1 , . . . , σn )X AM T`iB+mH`- #v i?2 bi`QM; J`FQp T`QT2`iv Q7 ?TTbZn+1 = A(αn , αn + σn+1 ] Bb BM/2T2M/2Mi Q7 X0 , . . . , Xn - M/ {Zn }n≥1 Bb M BB/ b2[m2M+2 /Bbi`B#mi2/ b Z = N (0, σ]- r?2`2 N Bb M ?TT Q7 BMi2MbBiv λ BM/2T2M/2Mi Q7 i?2 `M/QK p`B#H2 σ rBi? b2`pB+2 iBK2 /Bbi`B#miBQM GX h?2`27Q`2- {Xn }n≥0 Bb  /Bb+`2i2@iBK2 ?K+- +HH2/ i?2 2K#2//2/ +?BM i /2T`im`2 iBK2bX Aib i`MbBiBQM Ki`Bt Bb ⎞ ⎛ a0 a1 a2 a3 . . . ⎜ a0 a1 a2 a3 . . . ⎟ ⎟ ⎜ ⎟ ⎜ P = ⎜ 0 a0 a1 a2 . . . ⎟ , ⎜ 0 0 a0 a1 . . . ⎟ ⎠ ⎝ XX XX XX XX X X X X r?2`2

9Re

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: σ5

σ2

σ1

σ3 T3

T2

T1

σ6

σ4 T4

T5

T6

σ5

σ6

W (t) σ2

σ1

σ3 τ1

τ2

σ4

τ3

τ4

τ5

X(t)

``BpH T`Q+2bb- 7B7Q rQ`FHQ/- M/ +QM;2biBQM T`Q+2bb2b 



ai = P (Zn+1 = i) = P (N (0, σ] = i) =

P (N (0, t] = i) dG(t), 0

i?i Bb





ai =

e−λt

0

(λt)i dG(t). i!

URRXRdV

h?Bb +?BM Bb B``2/m+B#H2 M/  bm{+B2Mi +QM/BiBQM Q7 TQbBiBp2 `2+m``2M+2 Bb E[Z] < 1- r?2`2b B7 E[Z] > 1- Bi Bb i`MbB2MiXRR HbQ- BM i?2 TQbBiBp2 `2+m``2Mi +b2- i?2 ;2M2`iBM; 7mM+iBQM Q7 i?2 biiBQM`v /Bbi`B#miBQM Bb ∞ 

π(i)z i = (1 − E[Z])

i=0

AM pB2r Q7 URRXRdV



E [Z] = 0



∞ 





gZ (z) =

e 0

i!

i=0

M/ 

ie

−λt (λt)

−λt

∞   i=0

∞ 0

i

(z − 1)gZ (z) . z − gZ (z)







λ t dG(t) = λE [σ]

dG(t) =

(λt)i i z i!

URRXR3V

0



 dG(t) =



e−λt(1−z) dG(t).

0

AM i?2 Jf:AfRf∞f7B7Q [m2m2BM; bvbi2K- i?2 MmK#2` Q7 +mbiQK2`b Xn = X(τn ) H27i #2?BM/ #v i?2 n@i? +mbiQK2` r?2M ?2 H2p2b i?2 bvbi2K Bb 2t+iHv i?2 RR

a22 Mv i2ti QM J`FQp +?BMb- Q` rBi 7Q` i?2 KQ`2 ;2M2`H `2bmHib Q7 a2+iBQM RRXjX

RRXRX  _1oA1q P6 J_EPoAL Zl1l1AL: h>1P_u

9Rd

MmK#2` Q7 +mbiQK2`b ``BpBM; /m`BM; ?2` bQDQm`M- i?i Bb- BM i?2 BMi2`pH #2ir22M ?2` ``BpH iBK2 Tn M/ ?2` /2T`im`2 iBK2 τn = Tn + Vn - r?2`2 Vn Bb ?2` bQDQm`M iBK2, Xn = A(Tn , Tn + Vn ]. AMpQFBM; i?2 bi`QM; J`FQp T`QT2`iv Q7 ?TTb M/ MQiBM; i?i Vn /2T2M/b QMHv QM (σ1 , . . . , σn ) M/ i?2 Tbi Q7 i?2 ``BpH T`Q+2bb A i iBK2 Tn - Bi 7QHHQrb i?i i 2[mBHB#`BmK∞  π(i)z i = E z N ((0,Vn ]) , i=0

r?2`2 N Bb M ?TT Q7 BMi2MbBiv λ BM/2T2M/2Mi Q7 Vn X LQr ∞  ∞ N ((0,Vn ]) N ((0,v]) E z = /FVn (v) = E z eλv(z−1) /FVn (v), 0

0

r?2`2 FVn (v) Bb i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 Vn X h?Bb b?Qrb BM T`iB+mH` i?i i?2 /Bbi`B#miBQM Q7 Vn Bb BM/2T2M/2Mi Q7 n M/ i?i- /2MQiBM; #v V Mv `M/QK p`B#H2 rBi? i?2 bK2 /Bbi`B#miBQM b Vn ∞ 

 π(i)z i =

eλs(z−1) /FV (s).

URRXRNV

i=0

AM bmKK`v, i 2[mBHB#`BmK- Xn Bb /Bbi`B#mi2/ b X = N (V )- r?2`2 N Bb M ?TT Q7 BMi2MbBiv λ- M/ V Bb  `M/QK p`B#H2 BM/2T2M/2Mi Q7 N rBi? i?2 /Bbi`B#miBQM Q7 i?2 biiBQM`v bQDQm`M iBK2X 1tKTH2 RRXRXR8, :AfJfRf∞f7B7QX h?2 :AfJfRf∞f7B7Q [m2m2 Bb Q7 i?2 bK2 Mim`2 b i?2 JfJfRf∞f7B7Q [m2m2- 2t+2Ti 7Q` i?2 ``BpH T`Q+2bb- r?B+? Bb MQi SQBbbQMBM #mi `2M2rH, i?2 BMi2```BpH iBK2b 7Q`K M BB/ b2[m2M+2 rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM F (x) M/ K2M λ−1 X h?2 b2`pB+2 iBK2b `2KBM 2tTQM2MiBH rBi? K2M μ−1 X b BM i?2 Jf:AfRf∞f7B7Q [m2m2- i?2 +QM;2biBQM T`Q+2bb {X(t)}t≥0 Bb MQi J`FQpBM #mi M2p2`i?2H2bb K2M#H2 iQ J`FQpBM MH@ vbBbX AM/22/- B7 r2 H2i Xn := X(Tn −) #2 i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K b22M mTQM ``BpH #v i?2 ni? +mbiQK2`r2 ?p2 i?2 `2HiBQM Xn+1 = (Xn + 1 − Zn+1 )+ , r?2`2  ivTB+H Zn+1 Bb i?2 MmK#2` Q7 /2T`im`2b BM i?2 BMi2`pH (Tn , Tn+1 ]X h?2 b2[m2M+2 {Zn }n≥1 Bb BB/- BM/2T2M/2Mi Q7 X0 M/ /Bbi`B#mi2/ b Z = N (0, τ ]- r?2`2 τ M/ N `2 BM/2T2M/2Mi M/ N Bb M ?TT rBi? BMi2MbBiv μ M/ i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 τ Bb F X h?2`27Q`2 {Xn }n≥0 Bb M ?K+ M/ 



bk := P (Zn+1 = k) = 0

e−μt

(μt)k dF (t) . k!

h?2 ii? `Qr Q7 i?2 i`MbBiBQM Ki`Bt P Bb i?2`27Q`2

9R3

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: (1 −

i 

bk , bi , bi−1 , . . . , b0 , 0, 0, . . .).

k=0

aBM+2 bk > 0 7Q` HH k ≥ 0- P Bb B``2/m+B#H2 M/ T2`BQ/B+X P#b2`p2 i?i ∞ 

kbk = ρ−1 ,

k=0

r?2`2 ρ =

λ μ

Bb i?2 i`{+ BMi2MbBivX AM/22/∞ 

kbk =

k=0

 ∞  k

0 k=0 ∞ ∞

 =



0

e−μt

ke−μt

k=0

(μt)k dF (t) k!

(μt)k dF (t) = k!





μt dF (t) = 0

μ . λ

A7 ρ < 1- i?2 +?BM Bb TQbBiBp2 `2+m``2MiX hQ T`Qp2 i?Bb- Bi bm{+2b iQ T`Qp2 i?2 2tBbi2M+2 Q7  biiBQM`v /Bbi`B#miBQMX q2 KF2 i?2 2/m+i2/ ;m2bb i?i π ?b i?2 7Q`K π(i) = ξ i (1 − ξ) URRXkyV 7Q` bQK2 ξ ∈ (0, 1)X hQ p2`B7v i?2 +Q``2+iM2bb Q7 i?Bb ;m2bb- r2 Kmbi }M/ ξ ∈ (0, 1) bm+? i?i ∞  ξ j−i+1 bj−i+1 = ξ (i ≥ 1) () j=i−1

M/

∞ ∞   ( bk )ξ j = 1 ,

()

j=0 k=j+1

bBM+2 i?2b2 2[miBQMb `2 i?2 #HM+2 2[miBQMb r?2M π Bb ;Bp2M #v URRXkyVX 1[m@ iBQMb UV HH `2/m+2 iQ ξ = gZ (ξ), URRXkRV

∞ k r?2`2 gZ (ξ) = k=0 bk ξ Bb i?2 ;2M2`iBM; 7mM+iBQM Q7 Z1 X aBM+2 HH i?2 bk Ƕb `2 TQbBiBp2- i?2`2 Bb  mMB[m2 bQHmiBQM ξ0 Q7 URRXkRV BM (0, 1) B7 M/ QMHv B7 gZ (1) =

∞ 

kbk > 1 ,

k=1

i?i Bb- B7 M/ QMHv B7 ρ < 1X q2 Kmbi HbQ biBb7v UVX h?2 H27i@?M/ bB/2 2[mHb k−1 ∞   k=1

∞ 

1 − ξk 1 bk ξ = bk ( )= 1 − ξ 1 − ξ j=0 k=1 j

M/ BM pB2r Q7 URRXkRV- i?Bb 2[mHb

1 (1 1−ξ

 1 − b0 −

∞  k=1

− b0 − (ξ − b0 )) = 1X

 bk ξ

k

,

RRXkX SPAaaPL auah1Ja

9RN

1tKTH2 RRXRXRe, h?2 rBiBM; iBK2 /Bbi`B#miBQM Q7 :AfJfRf∞f7B7QX M ``BpBM; +mbiQK2` }M/b Xn = X(Tn− ) +mbiQK2`b BM 7`QMi Q7 ?BK- M/ i?2`27Q`2BM i?2 7B7Q /Bb+BTHBM2- ?Bb rBiBM; iBK2 Wn Ui?2 iBK2 #2ir22M ?Bb ``BpH iBK2 M/ i?2 iBK2 r?2M ?2 bi`ib `2+2BpBM; b2`pB+2V Bb i?2 iBK2 M22/2/ 7Q` i?2 b2`p2` iQ iF2 +`2 Q7 i?2 Xn +mbiQK2`b T`2b2Mi i iBK2 Tn −X AM T`iB+mH`Wn =

Xn 

Yj ,

j=1

r?2`2 i?2 Yj `2 BB/ 2tTQM2MiBHb Q7 K2M μ−1 M/ +QKKQM +?`+i2`BbiB+ 7mM+iBQM E[eiuY ] =

μ , μ − iu

M/ `2 BM/2T2M/2Mi Q7 Xn X HbQ- P (Xn = k) = (1 − ξ0 )ξ0k X h?2`27Q`2- BM bi2/v bii2E[eiuWn ] = E[ =

∞ 

eiu

k

j=1

Yj

1{Xn =k} ]

k=0 ∞ 

∞ 

k=0

k=0

E[eiuY ]k P (Xn = k) =

(

μ )k (1 − ξ0 )ξ0k . μ − iu

q2 }M/ 7Q` i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 i?2 biiBQM`v rBiBM; iBK2   μ(1 − ξ0 ) (1 − ξ0 ) + ξ0 . μ(1 − ξ0 ) + iu h?Bb Bb i?2 +?`+i2`BbiB+ 7mM+iBQM Q7  `M/QK p`B#H2 i?i Bb MmHH rBi? T`Q##BHBiv 1 − ξ0 M/ 2tTQM2MiBH Q7 K2M [μ(1 − ξ0 )]−1 rBi? T`Q##BHBiv ξ0 X

_2K`F RRXRXRd h?2 #Qp2 MHvbBb Q7 MQM@J`FQpBM [m2m2b ;Bp2b i?2 biiBQM@ `v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb 2Bi?2` Dmbi #27Q`2 i?2 ``BpH iBK2b UM/GI/1/∞V- Q` Dmbi 7i2` i?2 /2T`im`2 iBK2b UGI/M/1/∞VX q?i #Qmi i?2 /Bbi`B#miBQMb Q7 i?2 biiBQM`v T`Q+2bb Dmbi 7i2`  /2T`im`2 iBK2 7Q` i?2 M/GI/1/∞ [m2m2 Q` Dmbi #27Q`2 i?2 ``BpH iBK2b 7Q` i?2 GI/M/1/∞ [m2m2\ M/ 7Q` #Qi?- r?i #Qmi i?2 biiBQM`v /Bbi`B#miBQM i M `#Bi``v }t2/ iBK2\ h?Bb rBHH #2 Mbr2`2/ BM am#b2+iBQM RRX9 BM p2`v ;2M2`H i2`KbX

RRXk

SQBbbQM avbi2Kb

h?2 T`QQ7 Q7 J`FQpBMBiv M/ i?2 +QKTmiiBQM Q7 i?2 BM}MBi2bBKH ;2M2`iQ`b Q7 J`FQpBM [m2m2BM; bvbi2Kb- 2p2M i?2 bBKTH2bi QM2b- /Q2b MQi BKK2/Bi2Hv 7QHHQr 7`QK  dzMim`HǴ /2b+`BTiBQM BM i2`Kb Q7 MmK#2` Q7 b2`p2`b- b2`pB+2 /Bb+B@ THBM2b- T`BQ`Biv bbB;MK2Mib- 2i+X 6Q` i?2 [m2m2BM; bvbi2Kb +QMbB/2`2/ bQ 7` Bi rb /KBii2/ i?i i?2B` +QM;2biBQM T`Q+2bb2b r2`2 ?K+ M/  ?2m`BbiB+ /2`BpiBQM Q7

9ky

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

i?2B` BM}MBi2bBKH ;2M2`iQ`b rb ;Bp2MX q2 MQr ;Bp2 i?2 7Q`KH T`QQ7b BM  [mBi2 ;2M2`H 7`K2rQ`F i?i HbQ ++QKKQ/i2b [m2m2BM; M2irQ`Fb M/ KQ`2 +QKTH2t bvbi2KbXRk q2 bi`i rBi?  +QMbi`m+iBQM bHB;?iHv KQ`2 ;2M2`H i?M M22/2/ 7Q` J`FQpBM [m2m2BM; bvbi2Kb UBM T`iB+mH` i?2 b2`pB+2 iBK2b Q` BMi2```BpH iBK2b M22/ MQi #2 2tTQM2MiBH `M/QK p`B#H2bVX h?Bb Bb i?2 ;bKT U:2M2`HBx2/ a2KB@J`FQp S`Q+2bb2bVRj +QMbi`m+iBQMX h?2 ;bKT *QMbi`m+iBQM h?2 `iBQMH2 #2?BM/ i?Bb +QMbi`m+iBQM Bb i?i i?2`2 `2 bQm`+2b Q7 2p2Mib i?i +M #2 2Bi?2` +iBp2 Q` BM+iBp2 /2T2M/BM; QM i?2 bii2 Q7 i?2 T`Q+2bbX +iBp2 bQm`+2b i?2M +QKT2i2 iQ T`Q/m+2 M 2p2Mi i?i rBHH #2 i?2 +mb2 7Q`  bii2 i`MbBiBQMX h?2`2 `2 irQ BM;`2/B2Mib, UV M BMTmi- r?B+? +QMbBbib Q7 `M/QK p`B#H2b Q` TQBMi T`Q+2bb2b- M/ U#V  i`MbBiBQM K2+?MBbK- i?i i`Mb7Q`Kb i?2 BMTmi BMiQ i?2 bii2 T`Q+2bbX AM i?2 JfJfRf∞ [m2m2 7Q` BMbiM+2- i?2 BMTmi +QMbBbib Q7 i?2 BMi2```BpH iBK2b M/ i?2 b2`pB+2 iBK2b- i?2 i`MbBiBQM K2+?MBbK /2b+`B#2b i?2 7mM+iBQMBM; Q7 i?2 bvbi2K UBM+Hm/BM; T`BQ`BiB2b- b2`pB+2 /Bb+BTHBM2- M/ bQ QMV M/ i?2 bii2 T`Q+2bb Bb i?2 +QM;2biBQM T`Q+2bbX AM i?Bb bBKTH2 2tKTH2 i?2`2 `2 irQ bQm`+2b, QM2 ;2M2`i2b i?2 BMi2```BpH iBK2b- i?2 Qi?2` T`Q/m+2b i?2 b2`pB+2 iBK2bX q2 b?HH MQr T`Q+22/ rBi? i?2 ;2M2`H +b2- M/ }`bi /2b+`B#2 i?2 i`MbBiBQM K2+?MBbKX G2i S M/ E #2 irQ +QmMi#H2 b2ib- i?2 b2i Q7 2p2Mi bQm`+2b UbQm`+2b 7Q` b?Q`iV M/ i?2 b2i Q7 bii2b- `2bT2+iBp2HvX qBi? 2+? bQm`+2@bii2 TB` (s, i) Bb bbQ+Bi2/  MmK#2` c(s, i) ≥ 0 K2bm`BM; i?2 BMi2MbBiv Q7 +iBpBiv Q7 bQm`+2 s r?2M i?2 bii2 Bb i UBM [m2m2BM; bvbi2Kb i?Bb HHQrb mb iQ +QMbB/2`- 7Q` BMbiM+2- i?2 b2`pB+2 bT22/ Q7  b2`p2`VX M 2p2Mi bQm`+2 s Bb bB/ iQ #2 +iBp2 BM bii2 i B7 c(s, i) > 0X h?2 +QHH2+iBQM Q7 bQm`+2b s +iBp2 BM bii2 i rBHH #2 /2MQi2/ #v A(i)X qBi? 2+? bQm`+2@bii2 TB` (s, i) Bb bbQ+Bi2/  T`Q##BHBiv /Bbi`B#miBQM p(s, i, ·) QM EX h?Bb /Bbi`B#miBQM i2HHb mb ?Qr iQ b2H2+i i?2 M2ti bii2 r?2M M 2p2Mi Bb i`B;;2`2/ #v i?2 bQm`+2 s +iBp2 BM bii2 iX h?2 [m/`mTH2 (E, S, p(·, ·, ·), c(·, ·)) URRXkkV Bb +HH2/ i?2 i`MbBiBQM K2+?MBbKX h?2 +QHH2+iBQM T Q7 i`BTH2b (s, i, j) bm+? i?i c(s, i)p(s, i, j) > 0 Bb +HH2/ i?2 b2i Q7 HHQr2/ i`MbBiBQMbX A7 (s, i, j) ∈ T - QM2 bvb i?i  i`MbBiBQM i → j +M #2 i`B;;2`2/ #v M 2p2Mi T`Q/m+2/ #v bQm`+2 s i?i Bb +iBp2 BM bii2 iX q2 MQr /2b+`B#2 i?2 BMTmiX 1+? bQm`+2 s ∈ S +M ;2M2`i2 b KMv BM/2T2M@ /2Mi `M/QK p`B#H2b b M22/2/ rBi? i?2 bK2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Rk h?Bb r?QH2 b2+iBQM +M ?Qr2p2` #2 bFBTT2/ B7 QM2 Bb biBb}2/ #v i?2 ?2m`BbiB+ `;mK2Mib Q7 i?2 }`bi b2+iBQM Q7 i?2 +m``2Mi +?Ti2`X Rj (Jii?2b- RNek)X

RRXkX SPAaaPL auah1Ja

9kR

Gs (x)X AM //BiBQM- HH i?2 `M/QK p`B#H2b ;2M2`i2/ #v i?2 p`BQmb bQm`+2b `2 BM/2T2M/2MiX h?2 bii2 T`Q+2bb Bb Q#iBM2/ 7`QK i?2 BMTmi M/ i?2 i`MbBiBQM K2+?MBbK b 7QHHQrbX h?2 bii2 T`Q+2bb Bb BM 7+i  DmKT T`Q+2bb- iFBM; M2r pHm2b i i?2 bm++2bbBp2 i`MbBiBQM iBK2b τ˜n Un ≥ 0V r?2`2 τ˜0 = 0X i 2+? i`MbBiBQM iBK2 τ˜n i?2 bBimiBQM Q7 i?2 bvbi2K Bb /2b+`B#2/ #v i?2 bii2 ˜n ∈ E X (˜ τn ) = X M/  +QHH2+iBQM Q7 +QMbmK#H2 `M/QK p`B#H2b Ys (n)

(s ∈ S) .

h?2 M2ti i`MbBiBQM Q++m`b i iBK2 τ˜n+1 = τ˜n +

Y (n) . s inf ˜ ˜n s∈A(Xn ) c s, X

URRXkjV

  ˜ n i?2 bQm`+2 `2HBxBM; i?2 BM}KmK BM URRXkjV-R9 V i?2 M2ti .2MQiBM; #v s˜n ∈ A X ˜ n+1 Bb +?Qb2M ++Q`/BM; iQ bii2 X   ˜ n = i, s˜n = s = p (s, i, j) , ˜ n+1 = j | X URRXk9V P X ˜ n−1 , s˜n−1 ;Bp2M X ˜ n M/ s˜n X ˜ 0 , s˜0 , . . . , X M/ Bb BM/2T2M/2Mi Q7 X i i?2 Q`B;BM Q7 iBK2 τ˜0 = 0- i?2 b2i Ys (0) Us ∈ SV Bb M BM/2T2M/2Mi +QHH2+iBQM Q7 `M/QK p`B#H2b rBi? i?2 `2bT2+iBp2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Gs (x) Us ∈ SVX h?2 b2i Q7 +QMbmK#H2 `M/QK p`B#H2b Ys (n + 1) Us ∈ SV pBH#H2 i i`MbBiBQM iBK2 τ˜n+1 Bb `2+imHBx2/ b 7QHHQrb,   ˜n Ys (n + 1) = Ys (n) B7 s ∈ A X Ui?2 +QMbmK#H2 p`B#H2b Q7 M BM+iBp2 bQm`+2 `2 MQi +imHHv +QMbmK2/V M/     ˜ n (˜ ˜ n s = s˜n Ys (n + 1) = Ys (n) − c s, X τn+1 − τ˜n ) B7 s ∈ A X Ui?2 +QMbmK#H2 p`B#H2b Q7 M +iBp2 bQm`+2 s /2TH2i2 i i?2 `i2 c(s, i) r?2M i?2 bvbi2K Bb BM bii2 iV- M/ 7Q` s = s˜n - i?2 +QMbmK2/ `M/QK p`B#H2b Ys (n) Bb `2TH+2/ #v  M2r `M/QK p`B#H2 Ys (n + 1) rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Gs M/ BM/2T2M/2Mi Q7 Ys (k) Uk ≤ n, s ∈ SV Ui?2 +QMbmK#H2 p`B#H2 Q7 i?Bb T`iB+mH` bQm`+2 rb 2MiB`2Hv +QMbmK2/ M/ i?2`27Q`2 ?b iQ #2 `2TH+2/VX q2 b?HH MQr brBi+? iQ SQBbbQM bvbi2Kb- r?B+? `2 +HQb2Hv `2Hi2/ iQ ;bKTb r?2M i?2 BMTmi `M/QK p`B#H2b ?p2 M 2tTQM2MiBH /Bbi`B#miBQMX R9

bbmK2/ mMB[m2X h?Bb ?b iQ #2 p2`B}2/ BM 2+? +b2X Ai Bb i`m2 Q7 i?2 SQBbbQM bvbi2Kb #2HQrX

9kk

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

SQBbbQM avbi2Kb h?2 i`MbBiBQM K2+?MBbK Bb i?2 bK2 b BM i?2 ;bKT +QMbi`m+iBQMX h?2 BMTmi Bb 7Q`KHHv UMQi 2bb2MiBHHvV /Bz2`2Mi- BM i?i Bi +QMbBbib Q7 ?TTb- HbQ +HH2/ i?2 /`BpBM; ?TTbX qBi? 2+? bQm`+2 s ∈ S Bb bbQ+Bi2/ M BMi2MbBiv- Q` `i2- λs > 0- M/ i?2`2 Bb ;Bp2M  7KBHv Q7 BM/2T2M/2Mi ?TTb Ns,i,j U(s, i, j) ∈ T V rBi? i?2 `2bT2+iBp2 BMi2MbBiB2b as,i,j := λs c(s, i)p(s, i, j) (s, i, j) ∈ T ) . G2i aij :=



λs c(s, i)p(s, i, j) (i, j ∈ E)

URRXk8V

s∈S

M/ ai :=



aij

(i ∈ E)

URRXkeV

j∈E

M/ BMi`Q/m+2 i?2 bi#BHBiv bbmKTiBQM, ai < ∞ (i ∈ E) .

URRXkdV

h?2 +QMbi`m+iBQM Q7 i?2 bii2 T`Q+2bb 7`QK i?2 #bB+ /`BpBM; ?TTb Bb bBKBH` iQ i?2 QM2 mb2/ BM am#b2+iBQM kX8 7Q` DmKT ?K+b- QMHv rBi? KQ`2 /2iBHb +Q``2@ bTQM/BM; iQ i?2 dzMim`HǴ /2b+`BTiBQM Q7 i?2 bvbi2K +QMbB/2`2/X h?2 bii2 T`Q+2bb {X(t)}t≥0 iF2b Bib pHm2b BM E ∪{Δ}- r?2`2 Δ Bb M 2H2K2Mi QmibB/2 E- M/ Bb +QM@ bi`m+i2/ b 7QHHQrbX Uh?2 `2/2` b?QmH/ ?p2 BM KBM/ h?2Q`2K kXRX8 +QM+2`MBM; i?2 bmT2`TQbBiBQM Q7 BM/2T2M/2Mi ?TTb- h?2Q`2K kXRXe- i?2 bQ@+HH2/ +QKT2iBiBQM i?2@ Q`2K- M/ h?2Q`2K kXRXd- i?2 bi`QM; J`FQp T`QT2`iv Q7 ?TTbXV h?2 BMBiBH bii2 X(0) Bb  `M/QK p`B#H2 BM/2T2M/2Mi Q7 i?2 ?TTb Ns,i,j Ƕb bm+? i?i (s, i, j) ∈ T X amTTQb2 i?i i?2 bii2 i iBK2 t Bb X(t) = i ∈ EX h?2 +iBp2 bQm`+2b `2 i?2M HH i?2 bQm`+2b s ∈ A(i)X h?2 ?TTb +QKT2iBM; 7Q` i?2 /2i2`KBMiBQM Q7 i?2 M2ti i`MbBiBQM 7i2` t `2 i?2 Ns,i,j Ƕb bm+? i?i s ∈ A(i) M/ p(s, i, j) > 0X Ns0 ,i,j0 Bb i?2 ?TT T`Q/m+BM; i?2 }`bi 2p2Mi bi`B+iHv 7i2` t- bv i iBK2 t + τ˜X h?2 bii2 `2KBMb mM+?M;2/ BM [t, t + τ˜]- M/  i`MbBiBQM 7`QK bii2 i iQ bii2 j0 Q++m`b i iBK2 t + τ˜X h?2 bi#BHBiv +QM/BiBQM ai < ∞ 2Mbm`2b i?i τ˜ > 0- bBM+2 ai Bb i?2 bmK Q7 i?2 BMi2MbBiB2b Q7 i?2 Ns,i,j BM +QKT2iBiBQM r?2M i?2 bii2 Bb iX h?Bb `2+m`bBp2 T`Q+2/m`2 bi`i2/ 7`QK iBK2 0 ;Bp2b  b2[m2M+2 Q7 i`MbBiBQM ˜1, X ˜ 2 , . . .- r?2`2 X ˜ n = X(˜ iBK2b τ˜1 , τ˜2 , . . .X h?2 2K#2//2/ T`Q+2bb Bb X τn )X SQbbB#Hvi?2`2 Bb  `M/QK BM/2t K bm+? i?i τ˜K = ∞- BM r?B+? +b2 r2 H2i τ˜k = ∞ ˜k = X ˜ K 7Q` k ≥ KX h?Bb T`Q+2/m`2 /2}M2b X(t) 7Q` HH t ∈ [0, τ˜∞ )- r?2`2 M/ X τ˜∞ = limn↑∞ τ˜n Bb i?2 2tTHQbBQM iBK2X 6Q` t ≥ τ˜∞ - b2i X(t) = ΔX 6`QK MQr QM- r2 b?HH bbmK2 i?i τ˜∞ = ∞- M/ i?2`27Q`2 i?i {X(t)}t≥0 Bb  `2;mH` DmKT ?K+ QM EX UAM HH i?2 [m2m2BM; M2irQ`Fb i?i r2 b?HH +QMbB/2`i?Bb bbmKTiBQM ?QH/b i`m2- rBi? M 2bv T`QQ7XV Ai Bb i?2M +H2` i?i {X(t)}t≥0 Bb  `2;mH` J`FQp DmKT ?K+ rBi? bii2 bT+2 EX q2 MQr HQQF BM /2iBH i i?2 }`bi i`MbBiBQM- Q++m``BM; i iBK2 τ˜1 - r?B+? Bb ivT@ B+H Q7 HH Qi?2` i`MbBiBQMbX amTTQb2 i?i X(0) = iX ai`iBM; 7`QK i- i?2 +QKT2iBM; T`Q+2bb2b `2 i?2 Ns,i,j r?2`2 s ∈ A(i) M/ r?2`2 j biBb}2b p(s, i, j) > 0X A7 QM2 Bb

RRXkX SPAaaPL auah1Ja

9kj

MQi BMi2`2bi2/ BM i?2 T`iB+mH` bQm`+2 i`B;;2`BM; i?2 i`MbBiBQM- QM2 +M HmKT HH i?2 Ns,i,j Us ∈ A(i)V iQ 7Q`K  bBM;H2 ?TT Nij Q7 BMi2MbBiv aij X q2 `2 i?2M BM i?2 bK2 bBimiBQM b BM am#b2+iBQM kX8- rBi? QM2 2t+2TiBQM, aii Kv #2 bi`B+iHv TQbB@ ˜ n BMbi2/ Q7 Xn , aij BMbi2/ Q7 iBp2X h?Bb KQiBpi2b i?2 MQiiBQMb τ˜n BMbi2/ Q7 τn , X qij - 2i+X- r?B+? `2 i?2`2 iQ r`M mb i?i  Tb2m/Q@i`MbBiBQM +M Q++m`- i?i Bb-  dzi`MbBiBQMǴ 7`QK i iQ i?2 bK2 bii2 iX Pi?2`rBb2- i?2 bBimiBQM Bb bBKBH` iQ i?i ˜ 0 := X(0), Q7 am#b2+iBQM kX8- M/ r2 +M ;Bp2 i?2 7QHHQrBM; `2bmHi- H2iiBM; X ˜ n }n≥0 Bb  /Bb+`2i2@iBK2 ?K+ rBi? pHm2b BM E M/ i`MbBiBQM Ki`Bt (α) {X {˜ pij }i,j∈E ;Bp2M #v  a p˜ij = aiji B7 ai > 0 URRXk3V p˜ii = 1 B7 ai = 0. (β) 6Q` HH n ≥ 1- x ∈ R+ ˜n, X ˜ n−1 , · · · , X ˜ 0 , τ˜n , τ˜n−1 , · · · , τ˜1 ) = e−ai X˜n . P (˜ τn+1 − τ˜n ≥ x | X

URRXkNV

Sb2m/Q@i`MbBiBQMb aQ 7` i?2`2 Bb MQi?BM; 2bb2MiBHHv M2r rBi? `2bT2+i iQ i?2 +QMbi`m+iBQM Q7 bm#@ b2+iBQM kX8 2t+2Ti i?i r2 HHQr Tb2m/Q@i`MbBiBQMb r?B+? +M THv  `QH2 BM bQK2 bBimiBQMb- BM 722/#+F [m2m2b- 7Q` BMbiM+2 Ub22 1tKTH2 RRXkX9VX G2i {τn }n≥0 #2 i?2 b2[m2M+2 Q7 dzi`m2Ǵ i`MbBiBQM Ui?i Bb- 7`QK bQK2 bii2 iQ  /Bz2`2Mi bii2V iBK2b Q7 i?2 bii2 T`Q+2bb- M/ H2i {Xn }n≥0 #2 i?2 2K#2//2/ T`Q+2bb i i?2b2 i`m2 i`MbBiBQM iBK2bX A7 QM2 Bb BMi2`2bi2/ QMHv BM i`m2 i`MbBiBQMb- Bi bm{+2b iQ ;2i `B/ Q7 HH i?2 ?TTb Ns,i,i #2+mb2 i?2v /Q MQi z2+i i?2 bii2 T`Q+2bbX q2 i?2`27Q`2 `2@ i`B2p2 i?2 bBimiBQM Q7 bm#b2+iBQM kX8X h?2 bii2 T`Q+2bb Bb M ?K+ rBi? BM}MBi2bBKH ;2M2`iQ`  qij = λs c(s, i)p(s, i, j), URRXjyV s∈S

r?2`2 j = i- M/ qi =

 

λs c(s, i)p(s, i, j).

URRXjRV

j∈E,j=i s∈S

q2 MQr T`Q+22/ iQ KF2 i?2 +QMM2+iBQM rBi? i?2 dzMim`HǴ ;bKTX AM pB2r Q7 i?Bb- r2 }`bi ;Bp2 M Hi2`MiBp2 rv Q7 HQQFBM; i i?2 ;bKT +QMbi`m+iBQM- mbBM; K`F2`bX J`F2`b qBi? 2+? bQm`+2 s ∈ S Bb bbQ+Bi2/ M mM/2Hv2/ `2M2rH TQBMi T`Q+2bb Ns ≡ s s {Tns }n≥0 - /2}M2/ 7Q` n = 0 #v T0s = 0 M/ 7Q` n ≥ 0 #v Tn+1 − Tns := Sn+1 s r?2`2 {Sn }n≥1 Bb M BB/ b2[m2M+2 rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Gs (x)X h?2 7KBHv Q7 TQBMi T`Q+2bb2b {Ns }s∈S Bb BM/2T2M/2MiX 1+? TQBMi T`Q+2bb ?b  K`F2` i?i 2Bi?2` `2KBMb biBHH Q` KQp2b +QMiBMmQmbHv iQ i?2 `B;?i UrBi? p`B#H2 bT22/VX i iBK2 t- Bi ?b `2+?2/ TQbBiBQM  t   ˜ (u) du. Ms (t) = c s, X 0

9k9

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

AM Qi?2` rQ`/b- i?2 K`F2` QM Ns KQp2b i bT22/ c (s, i) r?2M i?2 ;bKT Bb BM bii2 iX i iBK2 τ˜n - i?2 b2i Q7 +QMbmK#H2 (Ys (n + 1) , s ∈ S) Bb ;Bp2M #v i?2 7Q`r`/ `2+m``2M+2 iBK2b Ub22 i?2 };m`2 #2HQrV Ys (n + 1) = inf {Tks ; Tks > Ms (τn )} − Ms (τn ) . k

URRXjkV

Ms1 (˜ τn ) Ys1 (n + 1) s1 τn ) Ms2 (˜ Ys2 (n + 1) = 0 s˜n = s2 Ms3 (˜ τn ) Ys3 (n + 1) s3 Ms4 (˜ τn ) Ys4 (n + 1) s4 t=0 J`F2`b M/ +QMbmK#H2 p`B#H2b i  ;Bp2M BMbiMi t- QMHv i?2 K`F2`b QM i?2 TQBMi Ns bm+? i?i    T`Q+2bb2b  ˜ ˜ s ∈ A X(t) `2 KQpBM; i i?2 `2bT2+iBp2 bT22/b c s, X(t) - M/ i?2 }`bi QM2 i?i K22ib i?2 M2ti 2p2Mi Q7 Bib TQBMi T`Q+2bb T`QpQF2b  i`MbBiBQMX h?2 iBK2 Q7 ˜ n+1 `2 ;Bp2M #v URRXkjV M/ URRXk9V- `2bT2+@ i`MbBiBQM τ˜n+1 M/ i?2 M2r bii2 X iBp2Hv- r?2`2 s˜n Bb i?2 bQm`+2 +Q``2bTQM/BM; iQ i?2 K`F2` i?i }`bi `2+?2/ M 2p2Mi Q7 Bib TQBMi T`Q+2bbX *QMM2+iBM; ;bKT M/ SQBbbQM avbi2Kb q?2M Gs (x) = 1 − e−λs x 7Q` HH s ∈ S- i?2 ;bKT +QMbi`m+iBQM ;Bp2b i?2 bK2 bii2 T`Q+2bb b i?2 Tm`2Hv SQBbbQMBM /2b+`BTiBQMX AM/22/- BM i?Bb +QMbi`m+iBQM- i iBK2 τ˜n - i?2`2 Bb  +QKT2iBiBQM KQM; i?2 SQBbbQM T`Q+2bb2b {Sτ˜n N(s,i,j) }(s,j)∈T (i) , r?2`2 T (i) = {(s, j) ; (s, i, j) ∈ T }X "v i?2 bi`QM; J`FQp T`QT2`iv 7Q` ?TTbi?2b2 ?TTb `2 BM/2T2M/2Mi M/ /Bbi`B#mi2/ b {N(s,i,j) }(s,j)∈T (i) ,

URRXjjV

M/ KQ`2Qp2`- i?2v `2 BM/2T2M/2Mi Q7 r?i ?TT2M2/ #27Q`2 iBK2 τ˜n X q2 +M i?2`27Q`2 `2bi`B+i Qm` ii2MiBQM iQ i?2 +b2 τ˜n = τ˜0 = 0 rBi?Qmi HQbb Q7 ;2M2`HBivX

RRXkX SPAaaPL auah1Ja

9k8

h?2 }`bi i`MbBiBQM iBK2 τ˜1 Bb Q#iBM2/ #v +QKT2iBiBQM KQM; i?2 ?TTb BM ˜ 0 = iX h?Bb +QKT2iBiBQM +M #2 Q`;MBx2/ BM irQ URRXjjV- r?2`2 r2 bmTTQb2 i?i X bi;2bX 6B`bi i?2`2 Bb  +QKT2iBiBQM KQM; i?2 {N(s,i) }s∈A(i) r?2`2 N(s,i) :=



N(s,i,j) ,

j ; (s,i,j)∈ T

r?B+? T`Q/m+2b i?2 i`MbBiBQM iBK2 τ˜1 = inf Z(s,i) (0), s∈A(i)

r?2`2 Z(s,i) (0) Bb i?2 }`bi TQBMi Q7 N(s,i) - M 2tTQM2MiBH `M/QK p`B#H2 rBi? K2M (λs c (s, i))−1 X PM2 +M r`Bi2 Z(s,i) (0) =

Ys (0) , c (s, i)

r?2`2 Ys (0) Bb M 2tTQM2MiBH `M/QK p`B#H2 rBi? K2M λ−1 s - M/ i?2`27Q`2 τ˜1 − τ˜0 =

inf ˜0 ) s∈A(X

Y (0) s , ˜0 c s, X

M/ r2 `2i`B2p2 URRXkjV Q7 i?2 ;bKT +QMbi`m+iBQMX G2i s˜0 #2 i?2 bQm`+2 ;BpBM; i?2 ˜ 1 Bb +?Qb2M #v +QKT2iBiBQM KQM; i?2 SQBbbQM T`Q+2bb2b BM}KmKX h?2 M2r bii2 X Ns˜0 ,i,j - j ∈ E, p(˜ s0 , i, j) > 0X h?2`27Q`2  ˜ 0 = i, s˜0 = s = p (s, i, j) , ˜1 = j | X P X M/ r2 `2i`B2p2 URRXk9V Q7 i?2 ;bKT +QMbi`m+iBQMX J`FQpBM Zm2m2b b SQBbbQM avbi2Kb q2 b?HH MQr `2pBbBi bQK2 2tKTH2b BM Q`/2` iQ b?Qr ?Qr i?2v +M #2 KQ/2HH2/ b SQBbbQM bvbi2KbX h?Bb rBHH BM T`iB+mH` T`Qp2 i?i i?2 [m2m2b i?2`2Q7 `2 ?K+ M/ T`QpB/2  Ki?2KiB+HHv `B;Q`Qmb /2`BpiBQM Q7 i?2B` BM}MBi2bBKH ;2M2`iQ`X 1tKTH2 RRXkXR, JfJfRf∞f7B7QX q2 iF2 S = {α, δ}- r?2`2 α biM/b 7Q` ``BpH M/ δ 7Q` /2T`im`2X h?2 bii2 bT+2 Bb E = N = {0, 1, 2, . . .}- i ∈ E `2T`2b2MiBM; i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K UBM i?2 rBiBM; HBM2 Q` #2BM; ii2M/2/ #v i?2 b2`p2`VX A7 i > 0- A(i) = {α, δ}- K2MBM; i?i r?2M i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K Bb bi`B+iHv TQbBiBp2- M ``BpH Q`  /2T`im`2 +QmH/ Q++m`X A7 i = 0- A(0) = {α}- K2MBM; i?i B7 i?2 bvbi2K Bb 2KTiv- QM2 +MMQi 2tT2+i  /2T`im`2- QMHv M ``BpHX HH i?2 +iBpBiv BMi2MbBiB2b i?i `2 MQi MmHH `2 iF2M 2[mH iQ 1X HbQ- λα = λ M/ λδ = μ- #2+mb2 r2 rMi i?2 BMi2```BpH iBK2b iQ #2 2tTQM2MiBH `M/QK p`B#H2b rBi? K2M λ−1 M/ i?2 b2`pB+2 `2[m2bib iQ #2 2tTQM2MiBH rBi? K2M μ−1 X

9ke

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

A7 BM bii2 i ≥ 0  i`MbBiBQM Bb i`B;;2`2/ #v M 2p2Mi Q7 bQm`+2 s0 = α ∈ A(i)i?2 M2ti bii2 Bb j = i + 1 UM ``BpH BM+`2b2b i?2 MmK#2` Q7 +mbiQK2`b #v 1Vc r?2`2b BM bii2 i > 0- B7 s0 = δ ∈ A(i)- i?2 M2ti bii2 Bb j = i − 1X h?mbp(α, i, i + 1) = 1 7Q` HH i ≥ 0- M/ p(δ, i, i − 1) = 1 7Q` HH i > 0X h?2 };m`2 #2HQr /2TB+ib  ivTB+H 2pQHmiBQM Q7 i?2 bii2 T`Q+2bb- bi`iBM; 7`QK bii2 i0 = 0X α U``BpHbV

δ X(t) 2 1 0 t=0

t ;bKT +QMbi`m+iBQM Q7 JfJfRf∞f7B7Q

6Q`KmH URRXjyV KF2b i?2 BM}MBi2bBKH ;2M2`iQ` `2/#H2 7`QK i?2 Mim`H /2b+`BTiBQM BM i2`Kb Q7 i?2 `i2b λs - i?2 bT22/b c(s, i)- M/ i?2 i`MbBiBQM T`Q#@ #BHBiB2b p(s, i, j)X 6Q` i?2 #Qp2 JfJfRf∞f7B7Q [m2m2- Bi +M #2 `2/BHv +?2+F2/ i?i qi,i+1 = λ, qi,i−1 = μ1{i≥0} .

PM2 b?QmH/ TQBMi Qmi i?i 2p2M 7Q` i?2 bBKTH2bi 2tKTH2ěi?2 JfJfRf∞ [m2m2ěi?2 SQBbbQM bvbi2K /2b+`BTiBQM /Bz2`b 7`QK i?2 mbmH `2;2M2`iBp2 /2b+`BT@ iBQM Q7 i?2 +Q``2bTQM/BM; J`FQp +?BMX h?2 SQBbbQM bvbi2K /2b+`BTiBQM ;Bp2b HH i?2 /2iBHb #Qmi i?2 K2+?MBbK Q7 ;2M2`iBQM Q7 i`MbBiBQMb- M/ Bib +QHHTb2 BMiQ i?2 mbmH `2;2M2`iBp2 /2b+`BTiBQM Bb ++QKTMB2/ #v  HQbb Q7 BM7Q`KiBQM +QM+2`MBM; i?2 }M2 /2iBHbX AM/22/- 7`QK i?2 TQBMi Q7 pB2r Q7 i?2B` /Bbi`B#miBQMb- QM2 +MMQi /BbiBM;mBb? #2ir22M M JfJfRf∞f7B7Q [m2m2 i?i ?b M BM}MBi2bBKH ;2M2`iQ` Q7 i?2 7Q`K A = (λ + μ)(K − I), r?2`2



0

⎜ μ ⎜ λ+μ K=⎜ 0 ⎝ XX X

1 0 μ λ+μ

XX X

0 λ λ+μ

0 XX X

0 0 λ λ+μ

XX X,

⎞ ··· · · ·⎟ ⎟ , · · ·⎟ ⎠

RRXkX SPAaaPL auah1Ja

9kd

M/ i?2 mMB7Q`K J`FQp +?BM +QMbi`m+i2/ 7`QK  SQBbbQM T`Q+2bb Q7 BMi2MbBiv λ + μ M/  `M/QK rHF rBi? `2~2+iBM; #``B2` 0 rBi? i?2 i`MbBiBQM Ki`Bt KX Ai Bb +imHHv MQi 2bv iQ b22 7`QK i?2 mMB7Q`K /2b+`BTiBQM i?i QM2 Bb /2HBM; rBi? M JfJfRf∞ [m2m2X h?Bb /B{+mHiv BM+`2b2b rBi? i?2 bBx2 M/ i?2 +QKTH2tBiv Q7 i?2 bvbi2K U[m2mBM; M2irQ`Fb- 7Q` BMbiM+2VX 1tKTH2 RRXkXk, JfJfEfy- Q` 1`HM; [m2m2X PM2 TQbbB#H2 i`MbBiBQM K2+?MBbK Bb i?2 7QHHQrBM;, S E A(i) c(s, i)

= = = =

{0, 1, . . . , K}, {HH bm#b2ib Q7 S +QMiBMBM; y}, {HH 2p2Mi bQm`+2b BM i}, 1 7Q` HH s ∈ A(i).

h?2 i`MbBiBQM T`Q##BHBiB2b p(·, ·, ·) rBHH #2 /2b+`B#2/ BM  72r HBM2bX h?2 SQBbbQM T`Q+2bb2b +Q``2bTQM/BM; iQ 2p2Mi bQm`+2b 1, 2, . . . , K HH ?p2 i?2 bK2 BMi2MbBiv μ > 0- r?2`2b i?2 SQBbbQM T`Q+2bb Q7 2p2Mi bQm`+2 0 ?b i?2 BMi2MbBiv λ > 0X aQm`+2 0 +Q``2bTQM/b- b BM i?2 T`2pBQmb 2tKTH2- iQ i?2 ``BpHbX 6Q` Mv bii2 i ∈ E, i − {0} Bb  bm#b2i Q7 {1, . . . , K}- `2T`2b2MiBM; i?2 b2`p2`b i?i `2 #mbv r?2M i?2 bvbi2K Bb BM i?i bii2X A7 X(t) = i- i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K Bb |i|−1- r?2`2 |i| /2MQi2b i?2 +`/BMHBiv Q7 i?2 b2i iX M 2p2Mi bQm`+2 s s U1 ≤ s ≤ KV +Q``2bTQM/b iQ b2`p2` s M/ i?2 b2[m2M+2 Sns = Tns − Tn−1 (n ≥ 1) Bb i?2 b2[m2M+2 Q7 bm++2bbBp2 b2`pB+2 iBK2b T`QpB/2/ #v i?Bb b2`p2`X q2 b?HH MQr ;Bp2 i?2 i`MbBiBQM T`Q##BHBiB2b p(·, ·, ·)X 6Q` i?Bb r2 M22/ iQ /BbiBM;mBb? irQ bii2b, i0 = {0} M/ iK = {0, 1, . . . , K} +Q``2bTQM/BM; `2bT2+iBp2Hv iQ M 2KTiv M/  7mHH bvbi2KX A7 i = i0 M/  i`MbBiBQM Bb i`B;;2`2/ QM 2p2Mi bQm`+2 s ∈ i r?2`2 s > 0- i?Bb K2Mb i?i b2`p2` s `2H2b2b  +mbiQK2`- M/ i?2 M2ti bii2 Bb i?2M i − {s}X A7 i = iK M/  i`MbBiBQM Bb i`B;;2`2/ QM 2p2Mi bQm`+2 0 Ur?B+? K2Mb i?i  M2r +mbiQK2` ``Bp2bV- i?2 M2ti bii2 Bb i + {r}- r?2`2 r Bb +?Qb2M i `M/QK BM S − i- i?2 b2i Q7 B/H2 b2`p2`b BM bii2 iX A7 i = iK M/  i`MbBiBQM Bb i`B;;2`2/ QM 2p2Mi bQm`+2 0 U M2r +mbiQK2` ``Bp2bV- i?2 bii2 /Q2b MQi +?M;2 Ui?2 M2r +mbiQK2` Bb `2D2+i2/ bBM+2 HH i?2 b2`p2`b `2 #mbvVX h?2 BM}MBi2bBKH ;2M2`iQ` +M #2 +QKTmi2/ 7`QK 7Q`KmH URRXjyV M/ i?Bb ;Bp2b  1 B7 i ∈ E, r ∈ {1, . . . , K}, r ∈ i, λ K+1−|i| qi,i+{r} = qi,i−{r} = μ B7 i ∈ E, r ∈ {1, . . . , K}, r ∈ i. .2}MBM; Q(t) = |X(t)| − 1 Ui?2 MmK#2` Q7 #mbv b2`p2`bV- r2 +M TTHv h?2Q`2K kX8Xe iQ T`Qp2 i?i i?Bb Bb  `2;mH` DmKT ?K+ rBi? bii2 bT+2 E˜ = {0, 1, . . . , K} ˜ ;Bp2M #v M/ BM}MBi2bBKH ;2M2`iQ` A q˜n,n+1 = λ B7 0 ≤ n ≤ K − 1 q˜n,n−1 = nμ B7 1 ≤ n ≤ K.

9k3

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

 ;Bp2M bii2 n ∈ E˜ Bb Q#iBM2/ #v ;`QmTBM; i?2 bii2b i ∈ E bm+? i?i |i| − 1 = nX hQ b?Qr i?i q˜n,n−1 = μn 7Q` 1 ≤ n ≤ K- r2 Kmbi +QMbB/2` i?2 i`MbBiBQMb Q7 {X(t)} 7`QK i bm+? i?i |i| − 1 = n iQ j bm+? i?i |j| − 1 = n − 1X 6BtBM; i bm+? i?i |i| − 1 = n- i?2`2 `2 2t+iHv n bii2b j bm+? i?i qij > 0 M/ |j| − 1 = n − 1MK2Hv HH bii2b Q7 i?2 7Q`K j = i − {r}- r?2`2 r ∈ {1, . . . , K} M/ r ∈ iX h?2 +Q``2bTQM/BM; bmK j qij = nμ M/ Bb BM/2T2M/2Mi Q7 i bm+? i?i |i| − 1 = nX h?mb +QM/BiBQM UkXR8V Q7 h?2Q`2K kX8Xe Bb biBb}2/ 7Q` α = n, β = n − 1X h?2 T`QQ7 Q7 q˜n,n+1 = λ Bb bBKBH`X 1tKTH2 RRXkXj, JfJfRf∞fHB7Q T`22KTiBp2 `2bmK2X h?2 +QMbi`m+iBQM b  SQBbbQM bvbi2K Bb b 7QHHQrb S = N, E = {HH }MBi2 bm#b2ib Q7 S +QMiBMBM; 0}, M/- /2MQiBM;  bii2 i ∈ E b i = {0, i1 , i2 , . . . , in(i) }A(i) = {0, in(i) }, c(s, i) = 1 B7 s ∈ A(i), p(0, i, i + {in(i) + 1}) = 1, p(in(i) , i, i − {in(i) }) = 1. aQm`+2 0 Bb i?2 ``BpH bQm`+2- b mbmHX h?2 Hbi T`2b+`BTiBQM Bb i?i r?2M  M2r +mbiQK2` b?Qrb mT Ui?i Bb-  i`MbBiBQM Bb i`B;;2`2/ QM bQm`+2 0)- i?2 bii2 #2BM; i M2r bQm`+2 in(i) + 1 Bb //2/ iQ i iQ 7Q`K i?2 M2ti bii2- M/ i?Bb bQm`+2 #2+QK2b BKK2/Bi2Hv +iBp2X q?2M bQm`+2 in(i) i`B;;2`b M 2p2Mi- i?Bb K2Mb  /2T`im`2 7`QK i?2 [m2m2- M/ i?2 bQm`+2 in(i) /BbTT2`b 7`QK bii2 i- M/ i?2 bQm`+2 in(i)−1 Bb `2+iBpi2/X h?2 ?TT {Tn0 }n≥1 ?b BMi2MbBiv λ > 0- M/ 7Q` HH k ≥ 1, {Tnk }n≥1 ?b BMi2MbBiv μ > 0X q?2M k ≥ 1- i?2 BMi2`2p2Mi iBK2b BM {Tnk }n≥1 +Q``2bTQM/ iQ b2`pB+2 iBK2bX q2 b22 i?i BM bii2 i- i?2`2 `2 n(i) +mbiQK2`b BM i?2 bvbi2KX h?mb i?2 +QM@ ;2biBQM T`Q+2bb i iBK2 t Bb |X(t)| − 1- r?2`2 |X(t)| Bb i?2 +`/BMHBiv Q7 i?2 b2i X(t)X 1tKTH2 RRXkXR rb +?Qb2M 7Q` Bib bBKTHB+BivX 1tKTH2 RRXkXk b?Qrb i?i i?2 bii2 bT+2 E b?QmH/ bQK2iBK2b #2 +?Qb2M H`;2` i?M i?2 bT+2 BM r?B+?  T`Q+2bb Q7 BMi2`2bi- ?2`2 i?2 +QM;2biBQM T`Q+2bb- iF2b Bib pHm2bc ?Qr2p2`- i?2 Hii2` b?QmH/ #2  7mM+iBQM Q7 i?2 bii2 T`Q+2bb- M/ i?Bb Bb r?v i ∈ E Bb bQK2iBK2b +HH2/  K+`Qbii2X 1tKTH2 RRXkXj +QMiBMb i?2 bK2 i2+?BM;b b 1tKTH2 9Xk- THmb i?2 72im`2 i?i HH bQm`+2b BM i?2 UK+`QV bii2 i M22/ MQi #2 +iBp2X 1tKTH2 RRXkX9, JfJfRf∞f7B7Q [m2m2 rBi? BMbiMiM2Qmb 722/#+FX Hi?Qm;? QM2 +M // M/ bmTT`2bb Tb2m/Q@i`MbBiBQMb i rBHH rBi?Qmi Hi2`BM; i?2 bii2 T`Q+2bb- Tb2m/Q@i`MbBiBQMb `2 MQi Hrvb K2MBM;H2bbX 6Q` BMbiM+2- +QMbB/2` M JfJfRf∞ [m2m2 rBi? BMbiMiM2Qmb 722/#+F- r?2`2  +mbiQK2` }MBb?BM; b2`@ pB+2 2Bi?2` H2p2b i?2 bvbi2K rBi? T`Q##BHBiv 1 − p Q` Bb BKK2/Bi2Hv `2+v+H2/

RRXjX h>1 :f:fRf∞ Zl1l1

9kN

rBi? T`Q##BHBiv p i i?2 2M/ Q7 i?2 rBiBM; HBM2 Q` i i?2 b2`pB+2 #QQi? B7 i?2`2 Bb M 2KTiv rBiBM; HBM2- rBi?  M2r BM/2T2M/2Mi 2tTQM2MiBH b2`pB+2 `2[m2bi Ub22 i?2 };m`2 #2HQrVX h?2M i?2 iBK2b Q7 b2`pB+2 +QKTH2iBQM Q7 `2+v+H2/ +mbiQK2`b /Q MQi +Q``2bTQM/ iQ  ;2MmBM2 i`MbBiBQM Q7 {X(t)}t≥0 - r?2`2 X(t) Bb i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 tX

RRXj

h?2 :f:fRf∞ Zm2m2

h?Bb Bb  p2`v ;2M2`H KQ/2H Q7  [m2m2 rBi? QM2 b2`p2` M/ BM}MBi2 +T+Biv i?i b2`p2b b  #bBb 7Q` 2tKTH2b Q7 TTHB+iBQMX G2i σ M/ τ #2 BMi2;`#H2 MQM@M2;iBp2 `M/QK p`B#H2b /2}M2/ QM i?2 T`Q#@ #BHBiv bT+2 (Ω, F, P 0 ) M/ H2i θ : (Ω, F) → (Ω, F) #2  K2bm`#H2 KT rBi? K2bm`#H2 BMp2`b2X bbmK2 i?i (P 0 , θ) Bb 2`;Q/B+X  GBM/H2v T`Q+2bbR8 bbQ+Bi2/ rBi? i?2b2 `M/QK p`B#H2b Bb  biQ+?biB+ T`Q+2bb {Wn }n∈T - r?2`2 T = N Q` ZbiBb7vBM; i?2 `2+m`bBQM 2[miBQM Wn+1 := (Wn + σn − τn )+ r?2`2

σn := σ ◦ θn ,

(n ∈ T) ,

URRXj9V

τn := τ ◦ θn .

h?Bb 2[miBQM ?b i?2 7QHHQrBM; BMi2`T`2iiBQM BM i2`Kb Q7 [m2m2BM;X .2}M2  TQBMi T`Q+2bb A QM R #v Bib 2p2Mi iBK2b b2[m2M+2 {Tn }n∈Z , T0 ≡ 0 M/ Tn+1 − Tn := τn

(n ∈ Z) .

AMi2`T`2i Tn b i?2 ``BpH iBK2 BM  [m2m2BM; bvbi2K Q7 +mbiQK2` n M/ σn b i?2 KQmMi Q7 b2`pB+2 UBM iBK2 mMBibV `2[mB`2/ #v i?Bb +mbiQK2`X .2}M2 i?2 i`{+ BMi2MbBiv E0 [σ] ρ := 0 . E [τ ] a2`pB+2 Bb T`QpB/2/ i mMBi `i2 r?2M2p2` i?2`2 `2KBMb i H2bi QM2 +mbiQK2` BM i?2 bvbi2KX Pi?2`rBb2 i?2`2 Bb MQ 7m`i?2` T`2b+`BTiBQM b iQ b2`pB+2 /Bb+BTHBM2- T`BQ`BiB2b M/ bQ QMX A7 Wn Bb i?2 iQiH b2`pB+2 `2KBMBM; iQ #2 /QM2 Dmbi #27Q`2 +mbiQK2` n ``Bp2b Ui?i Bb- i iBK2 Tn −V- i?2M GBM/H2vǶb `2+m``2M+2 URRXj9V Bb biBb}2/X AM i?2 [m2m2BM; BMi2`T`2iiBQM- i?2 GBM/H2v T`Q+2bb Bb mbmHHv +HH2/ i?2 rQ`FHQ/ T`Q+2bbX q?2M T = N- i?2 GBM/H2v T`Q+2bb Bb BKK2/Bi2Hv +H+mHi2/ 7`QK i?2 BMBiBH rQ`FHQ/ W0 X AM i?2 +b2 T = Z- r2 ?p2 MQr?2`2 iQ bi`i i?2 `2+m`bBQMX h?Bb +b2 +Q``2bTQM/b iQ i?2 bBimiBQM Q7  [m2m2BM; bvbi2K i?i ?b #22M QT2`iBM; 7`QK i?2 BM}MBi2 TbiX q2 Kv 2tT2+i i?i mM/2` +2`iBM +B`+mKbiM+2b UQ7 +Qm`b2 ;QQ/ ;m2bb Bb i?i ρ < 1 rBHH /QV i?2 rQ`FHQ/ T`Q+2bb ?b  biiBQM`v p2`bBQMX PM2 Bb i?2`27Q`2 H2/ iQ TQb2 i?2 T`Q#H2K BM i?2 7QHHQrBM; i2`Kb, 2t?B#Bi  }MBi2 MQM@M2;iBp2 `M/QK p`B#H2 W bm+? i?i- H2iiBM; Wn := W ◦ θ−n , R8

(GBM/H2v- RN8k)X

9jy

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

i?2 GBM/H2v `2+m`bBQM URRXj9V Bb biBb}2/ 7Q` HH n ∈ ZX 1[mBpH2MiHv, r2 i`v iQ }M/  }MBi2 MQM@M2;iBp2 `M/QK p`B#H2 W bm+? i?i W ◦ θ = (W + σ − τ )+ .

URRXj8V

h?Bb 2[miBQM Bb +HH2/ GQvM2bǶ 2[miBQMX h?2Q`2K RRXjXR URe V A7 ρ < 1- i?2`2 2tBbib  mMB[m2 }MBi2 MQM@M2;iBp2 bQHmiBQM W Q7 URRXj8VX S`QQ7X 6Q` n ≥ 0- /2}M2 Mn iQ #2 i?2 rQ`FHQ/ 7QmM/ #v +mbiQK2` 0 bmTTQbBM; i?i +mbiQK2`  − n 7QmM/ M 2KTiv [m2m2 mTQM ``BpHX PM2 +?2+Fb #v BM/m+iBQM i?i  + m  Mn = max (σ−i − τ−i ) . URRXjeV 1≤m≤n

i=1

AM T`iB+mH`- Mn Bb BMi2;`#H2 7Q` HH n ∈ NX h?2 GQvM2bǶ b2[m2M+2 {Mn }n≥0 biBb}2b i?2 `2+m``2M+2 `2HiBQM Mn+1 ◦ θ = (Mn + σ − τ )+

URRXjdV

M/ URRXjeV b?Qrb i?i Bi Bb MQM@/2+`2bBM;X .2}M2 i?2 GQvM2bǶ p`B#H2 UTQbbB#Hv iFBM; BM}MBi2 pHm2bV iQ #2  M∞ := lim ↑ Mn = n→∞

sup n≥1

n 

+ (σ−i − τ−i )

.

URRXj3V

i=1

G2iiBM; n ;Q iQ ∞ BM URRXjdV- r2 b22 i?i M∞ Bb  MQM@M2;iBp2 `M/QK p`B#H2 biBb7vBM; GQvM2bǶ 2[miBQM, M∞ ◦ θ = (M∞ + σ − τ )+ .

URRXjNV

lbBM; i?2 B/2MiBiv (a − b)+ = a − a ∧ b- 1[mHBiv URRXjdV #2+QK2b Mn+1 ◦ θ = Mn − Mn ∧ (τ − σ)

URRX9yV

M/ i?2`27Q`2- bBM+2 P 0 Bb θ@BMp`BMi M/ bBM+2 {Mn }n≥1 Bb MQM@/2+`2bBM; M/ BMi2;`#H2E0 [Mn ∧ (τ − σ)] = E0 [Mn − Mn+1 ◦ θ] = E0 [Mn − Mn+1 ] ≤ 0 UmbBM; G2KK dXjXe 7Q` i?2 b2+QM/ 2[mHBivVX Ai 7QHHQrb #v KQMQiQM2 +QMp2`;2M+2 i?i E0 [M∞ ∧ (τ − σ)] ≤ 0 . URRX9RV 1[mHBiv URRXjNV b?Qrb i?i i?2 2p2Mi {M∞ = ∞} Bb θ@BMp`BMi U`2+HH i?i σ M/ τ `2 }MBi2VX "v 2`;Q/B+Biv- P 0 (M∞ = ∞) Bb 2Bi?2` 0 Q` 1X AM pB2r Q7 URRX9RVRe

(GQvM2b- RNek)X

RRXjX h>1 :f:fRf∞ Zl1l1

9jR

P 0 (M∞ = ∞) = 1 BKTHB2b E0 [τ − σ] ≤ 0X h?2`27Q`2- i?2 +QM/BiBQM E0 [σ] < E0 [τ ] BKTHB2b i?i M∞ < ∞- P 0 @XbX GQvM2bǶ p`B#H2 M∞ Bb i?2 KBMBKH MQM@M2;iBp2 bQHmiBQM Q7 GQvM2bǶb 2[m@ iBQMX h?Bb Bb T`Qp2/ #v BM/m+iBQMX AM/22/- bi`iBM; rBi?  MQM@M2;iBp2 bQHmiBQM W Q7 URRXj8V- W ≥ Mn BKTHB2b W ≥ Mn+1 Ui?Bb Bb i`m2 #2+mb2 Mn+1 ◦ θ = (Mn + σ − τ )+ ≤ (W + σ − τ )+ = W ◦ θVX h?2 }`bi i2`K Q7 i?2 BM/m+iBQM Bb p2`B}2/ bBM+2 W ≥ 0 = M0 X Ai `2KBMb iQ T`Qp2 mMB[m2M2bb Q7  }MBi2 bQHmiBQM Q7 GQvM2bǶ 2[miBQMX G2i W #2 MQi?2` }MBi2 bQHmiBQM- i?2M σ − τ ≤ W ◦ θ − W ≤ σ, M/ BM T`iB+mH` W ◦ θ − W Bb BMi2;`#H2X h?2`27Q`2- #v G2KK dXjXeE0 [W ◦ θ − W ] = 0X q2 b22F iQ T`Qp2 i?i B7 ρ < 1- i?2M W = M∞ Bb i?2 mMB[m2 MQM@M2;iBp2 }MBi2 bQHmiBQM Q7 URRXj8VX q2 ?p2 b22M i?i M∞ Bb i?2 KBMBKH bQ@ HmiBQMX h?2`27Q`2 7Q` Mv MQM@M2;iBp2 bQHmiBQM W - {W = 0} ⊆ {W = M∞ }X h?2 Hii2` 2p2Mi Bb θ@+QMi`+iBM; bBM+2 #Qi? W M/ M∞ biBb7v URRXj8VX aBM+2 (P 0 , θ) Bb 2`;Q/B+- r2 Kmbi i?2M ?p2 P 0 (W = M∞ ) = 0 Q` 1X Ai Bb i?2`27Q`2 2MQm;? iQ b?Qr i?i P 0 (W = 0) > 0- r?B+? 7QHHQrb 7`QK i?2 M2ti H2KKX G2KK RRXjXk A7 P 0 (W = 0) = 0- 7Q` bQK2 }MBi2 bQHmiBQM W Q7 URRXj8V- i?2M ρ = 1X S`QQ7X AM/22/- B7 W ◦ θ > 0 P 0 @XbX- W ◦ θ = W + σ − τ - M/ E0 [W ◦ θ − W ] = 0 BM pB2r Q7 G2KK dXjXe- M/ i?Bb BKTHB2b E0 [σ] = E0 [τ ]X   _2+m``2M+2 iQ 0 h?2Q`2K RRXjXj h?2 bi#BHBiv +QM/BiBQM ρ < 1 Bb bbmK2/ iQ ?QH/X G2i W = M∞ #2 i?2 mMB[m2 MQM@M2;iBp2 bQHmiBQM Q7 URRXj8VX h?2`2 2tBbib M BM}MBiv Q7 M2;iBp2 U`2bTX TQbBiBp2V BM/B+2b n bm+? i?i W ◦ θn = 0X S`QQ7X AM pB2r Q7 i?2 2`;Q/B+Biv Q7 (P 0 , θ) Bi bm{+2b iQ b?Qr i?i P 0 [M∞ = 0] > 0.

URRX9kV

h?Bb 7QHHQrb 7`QK G2KK RRXjXk- BKTHvBM; i?i r2 +MMQi ?p2 P 0 [M∞ = 0] = 0 B7 ρ < 1X  h?2 7QHHQrBM; Bb  T`iBH +QMp2`b2 Q7 h?2Q`2K RRXjXRX h?2Q`2K RRXjX9 A7 E0 [σ] > E0 [τ ]- i?2M MQ }MBi2 bQHmiBQM Q7 URRXj8V 2tBbibX

9jk

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

S`QQ7X hQ T`Qp2 i?Bb- Bi Bb 2MQm;? iQ b?Qr i?i P 0 (M∞ = ∞) = 1 bBM+2 M∞ Bb i?2 KBMBKH MQM@M2;iBp2 bQHmiBQM Q7 URRXj8VX h?Bb 7QHHQrb 7`QK 1 (σ−i − τ−i ) = E0 [σ − τ ] > 0, n→∞ n i=1 n

lim

bBM+2 i?Bb BM im`M BKTHB2b  M∞ =

sup n

n 

+ (σ−i − τ−i )

= ∞.

i=1

 h?2 :f:fRf∞ Zm2m2 BM *QMiBMmQmb hBK2 aQ 7`- r2 ?p2 bim/B2/ i?2 :f:fRf∞ [m2m2 i i?2 ``BpH iBK2b Q`- BM Qi?2` rQ`/bmM/2` i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? i?2 ``BpH T`Q+2bbX q2 MQr bi;2 HH i?Bb BM i?2 UiBK2@V biiBQM`v 7`K2rQ`FX G2i UΩ, F, P ) #2  T`Q##BHBiv bT+2 rBi?  K2bm`#H2 ~Qr {θt }t∈R - bm+? i?i (P, θt ) Bb 2`;Q/B+X G2i A #2  bBKTH2 TQBMi T`Q+2bb /2}M2/ QM (Ω, F)- +QKTiB#H2 rBi? i?2 ~Qr M/ rBi? }MBi2 p2`;2 `i2, λ := E[A((0, 1])] < ∞ . Ai Bb +HH2/ i?2 ``BpH UTQBMiV T`Q+2bb M/ Bib n@i? TQBMi Tn Bb BMi2`T`2i2/ b #27Q`2 b i?2 ``BpH iBK2 Q7 +mbiQK2` n BM i?2 b2`pB+2 bvbi2KX _2+HH i?2 +QMp2MiBQM Tn < Tn+1 Un ∈ ZV M/ T0 ≤ 0 < T1 X h?2 BMi2`@``BpH iBK2 #2ir22M +mbiQK2`b n M/ (n + 1) Bb τn := Tn+1 − Tn . *mbiQK2` n +``B2b M KQmMi Q7 UMQM@M2;iBp2V `2[mB`2/ b2`pB+2 UQ` b2`pB+2 iBK2V /2MQi2/ #v σn - r?2`2 i?2 b2[m2M+2 {σn }n∈Z Bb bbmK2/ iQ #2  b2[m2M+2 Q7 K`Fb Q7 i?2 ``BpH T`Q+2bbX G2iiBM; PA0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? P M/ A- r2 /2}M2 i?2 i`{+ BMi2MbBiv ρ := λE0A [σ0 ] . h?2 b2[m2M+2 {(Tn , σn )}n∈Z /2b+`B#2b i?2 BMTmi BMiQ bQK2 [m2m2BM; bvbi2K U G/G BMTmi- BM E2M/HHǶb i2`KBMQHQ;vVX Ai Bb biiBQM`v BM irQ /BbiBM+i U#mi `2Hi2/V b2Mb2b, mM/2` i?2 SHK T`Q##BHBiv PA0 - i?2 b2[m2M+2 {(τn , σn )}n∈Z Bb biiBQM`v- M/ mM/2` i?2 biiBQM`v T`Q##BHBiv P - i?2 K`F2/ TQBMi T`Q+2bb (A, σ) Bb biiBQM`vX Ai rb T`Qp2/ BM i?2 T`2pBQmb bm#b2+iBQM i?i B7 i?2 i`{+ BMi2MbBiv ρ Bb bi`B+iHv bKHH2` i?M 1- i?2M i?2`2 2tBbib  mMB[m2 }MBi2 b2[m2M+2 {Wn }n∈Z bm+? i?i Wn = W0 ◦ θn M/ i?i Wn+1 = (Wn + σn − τn )+

(n ∈ Z),

PA0 @XbX

()

RRXjX h>1 :f:fRf∞ Zl1l1

9jj

"v h?2Q`2K dXeXRy- UV HbQ ?QH/b P @XbX G2i {W (t)}t∈R #2 i?2 U+QMiBMmQmb@iBK2V biQ+?biB+ T`Q+2bb /2}M2/ #v W (t) = (Wn + σn − (t − Tn ))+

(t ∈ [Tn , Tn+1 ))

URRX9jV

Ui?2 KQmMi Q7 rQ`F `2KBMBM; i iBK2 tVX LQi2 i?i Wn = W (Tn −)X h?Bb /2}M2b  θt @+QKTiB#H2 rQ`FHQ/ T`Q+2bb {W (t)}t∈R X LQi2 HbQ i?i bBM+2 Wn = 0 7Q` M BM}MBiv Q7 BM/B+2b n ∈ Z,

PA0 @XbX ,

#v h?2Q`2K dXeXRy ;BM- i?Bb `2KBMb i`m2 P @XbX _2+TBimHiBM;, lM/2` i?2 bi#BHBiv +QM/BiBQM ρ < 1- i?2`2 2tBbib  mMB[m2 }MBi2 MQM@M2;iBp2 θt @+QKTiB#H2 T`Q+2bb {W (t)}t∈R biBb7vBM; 2[miBQM URRX9jVX JQ`2Qp2`- i?2`2 `2 M BM}MBi2 MmK#2` Q7 M2;iBp2 BM/B+2b n M/ M BM}MBi2 MmK#2` Q7 TQbBiBp2 BM/B+2b n bm+? i?i W (Tn −) = 0X h?2 a2+QM/`v S`Q+2bb2b G2i mb bbmK2 i?i ρ < 1X h?2 TQBMi T`Q+2bb R /2}M2/ #v  1C (Tn )1{0} (W (Tn −)), R(C) =

URRX99V

n∈Z

+QmMiBM; i?2 Tn Ƕb i r?B+? M ``BpBM; +mbiQK2` }M/b M 2KTiv [m2m2- Bb θt @ +QKTiB#H2X G2i {Un }n∈Z #2 i?2 b2[m2M+2 Q7 TQBMib Q7 R- rBi? i?2 mbmH +QMp2MiBQM U0 ≤ 0 < U 1 . 6Q` 2+? n ∈ Z- H2i Vn+1 #2 i?2 }`bi iBK2 t 7i2` Un i r?B+? W (t) = 0X h?2 BMi2`pH [Un , Un+1 ) Bb +HH2/ i?2 n@i? +v+H2- [Un , Vn+1 ) Bb i?2 n@i? #mbv T2`BQ/ M/ [Vn+1 , Un+1 ) Bb i?2 n@i? B/H2 T2`BQ/X h?2 Q#D2+iBp2 Bb iQ +QMbi`m+i  `B;?i@+QMiBMmQmb +QM;2biBQM T`Q+2bb {X(t)}t∈R - r?2`2 X(t) Bb i?2 MmK#2` Q7 +mbiQK2`b BM i?2 [m2m2@ BM; bvbi2K i iBK2 t- #2BM; 2Bi?2` BM i?2 rBiBM; `QQK Q` i i?2 b2`pB+2 #QQi?X *HH {X(t)}t∈R i?2 bvbi2K +QM;2biBQM T`Q+2bb- iQ /BbiBM;mBb? Bi 7`QK i?2 rBiBM; `QQK +QM;2biBQM T`Q+2bb {Q(t)}t∈R +QmMiBM; i?2 +mbiQK2`b BM i?2 rBiBM; `QQKX AM  G/G/1/∞ [m2m2- i?2 bvbi2K +QM;2biBQM T`Q+2bb M/ i?2 rBiBM; `QQK +QM;2biBQM T`Q+2bb `2 `2Hi2/ #v Q(t) = (X(t) − 1)+ . URRX98V h?2 #Qp2 +QM;2biBQM T`Q+2bb /2T2M/b QM i?2 T`BQ`Biv `mH2b M/ i?2 b2`pB+2 /Bb+B@ THBM2X >2`2 r2 b?HH bmTTQb2 i?i i?2`2 2tBbib QMHv QM2 +Hbb Q7 +mbiQK2`b bQ i?i MQ [m2biBQM Q7 T`BQ`Biv `Bb2bX h?2 b2`pB+2 /Bb+BTHBM2 Bb 2K#Q/B2/ BM  `mH2 /2+B/BM; r?B+? Q7 i?2 +mbiQK2`b T`2b2Mi i iBK2 t b?QmH/ #2 b2`p2/X AKTQ`iMi /Bb+BTHBM2b #2bB/2b 7B7Q- HB7Q M/ `M/QK `2 Ç a?Q`i2bi S`Q+2bbBM; hBK2 UbTiVX h?2 b2`p2` iF2b i?2 +mbiQK2` BM HBM2 rBi? i?2 b?Q`i2bi `2[mB`2/ b2`pB+2 iBK2X Ç a?Q`i2bi _2KBMBM; S`Q+2bbBM; hBK2 Ub`TiVX h?Bb Bb  T`22KTiBp2 `2bmK2 /Bb+BTHBM2- r?2`2 i 2+? BMbiMi t i?2 b2`p2` ii2M/b i?2 +mbiQK2` rBi? i?2 b?Q`i2bi `2KBMBM; T`Q+2bbBM; iBK2X  T`22KTiBQM i?2`27Q`2 Q++m`b B7 M/ QMHv B7 i?2 M2rHv ``Bp2/ +mbiQK2` ?b  `2[mB`2/ b2`pB+2 bKHH2` i?M i?2 `2KBMBM; b2`pB+2 iBK2 Q7 i?2 +mbiQK2` #2BM; b2`p2/X

9j9

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

HH i?2 #Qp2 b2`pB+2 /Bb+BTHBM2b #2HQM; iQ i?2 +Hbb Q7 HQ+H /Bb+BTHBM2bX "v i?Bb r2 K2M i?i B7 t ∈ [Un , Un+1 )- i?2 +?QB+2 Q7 i?2 +mbiQK2` #2BM; b2`p2/ i iBK2 t /2T2M/b mTQM i?2 ?BbiQ`v Q7 i?2 bvbi2K QMHv i?`Qm;? r?i ?TT2M2/ BM i?2 iBK2 BMi2`pH [Un , t)X AM //BiBQM- QM2 +M /BbiBM;mBb? #2ir22M i?Qb2 /Bb+BTHBM2b r?B+? /2T2M/ mTQM i?2 b2`pB+2 iBK2b UHBF2 bTi Q` b`TiV- M/ i?Qb2 r?B+? /Q MQi UHBF2 7B7Q- 7B7Q Q` `M/QKVX *H2`Hv- 7Q` Mv HQ+H b2`pB+2 /Bb+BTHBM2- i?2 +QM;2biBQM T`Q+2bb {X(t)} Ut ∈ [Un , Vn+1 )V +M #2 +QMbi`m+i2/ 7`QK {W (t)} Ut ∈ [Un , Vn+1 )V M/ 7`QK i?2 FMQrH@ 2/;2 Q7 i?2 b2`pB+2 /Bb+BTHBM2 UKv#2 iQ;2i?2` rBi? i?2 ?2HT Q7  `M/QK 2tT2`B@ K2Mi- b BM i?2 `M/QK /Bb+BTHBM2VX JQ`2Qp2`- i?Bb +QMbi`m+iBQM `2bmHib BM  +QM@ ;2biBQM T`Q+2bb {X(t)}t∈R i?i Bb θt @+QKTiB#H2X 1tKTH2 RRXjX8, *QM;2biBQM T`Q+2bb 7Q` i?2 7B7Q /Bb+BTHBM2X "v +QM@ bi`m+iBQM X(t) = 1{Tn ≤t} 1{Tn +W (Tn −)+σn >t} , n∈Z

M/ i?2`27Q`2- bBM+2 i?2 b2[m2M+2b {W (Tn −)}n∈Z M/ {σn }n∈Z `2 b2[m2M+2b Q7 K`Fb Q7 A- {X(t)}t∈R Bb  θt @+QKTiB#H2 T`Q+2bbX 1tKTH2 RRXjXe, *QM;2biBQM T`Q+2bb 7Q` i?2 `M/QK /Bb+BTHBM2X >2`2 i?2 #bB+ BMTmi T`Q+2bb Kmbi #2 m;K2Mi2/ rBi? M BB/ b2[m2M+2 {Zn }n∈Z Q7 [0, 1]N @ pHm2/ `M/QK p`B#H2b BM/2T2M/2Mi Q7 {(Tn , σn )}n∈Z X h?2 +QQ`/BMi2b Zn1 , Zn2 , . . . Q7 Zn `2 BB/ M/ mMB7Q`KHv /Bbi`B#mi2/ BM i?2 BMi2`pH [0, 1]X q?2M i?2 n@i? b2`pB+2 Bb Dmbi +QKTH2i2/- B7 i?2`2 `2 k +mbiQK2`b UrBi? BM/B+2b (n1 , . . . , nk )V rBiBM; BM i?2 [m2m2- i?2 b2`p2` +?QQb2b +mbiQK2` ni B7 Zni < Znj - 7Q` HH 1 ≤ j ≤ k- j = iX EMQrH2/;2 Q7 {(Tn , σn , Zn )}n∈Z M/ Q7 i?2 b2[m2M+2 Q7 +QMbi`m+iBQM TQBMib {Un }n∈Z bm{+2b iQ +QMbi`m+i i?2 +QM;2biBQM T`Q+2bb2bX 6Q` i?2 bK2 BMTmi {(Tn , σn )}n∈Z - i?2 rQ`FHQ/ T`Q+2bb `2KBMb i?2 bK2 r?2`2b /Bz2`2Mi b2`pB+2 /Bb+BTHBM2b mbmHHv ;Bp2 /Bz2`2Mi +QM;2biBQM T`Q+2bb2bb i?2 `2/2` Kv p2`B7v rBi? bBKTH2 2tKTH2bX 1tKTH2 RRXjXd, G/G BMTmi bi`2K rBi? T`BQ`Biv +Hbb2bX *QMbB/2`  G/G BMTmi bi`2K {(Tn , σn , Un }n∈Z - r?2`2 σn Bb i?2 KQmMi Q7 b2`pB+2 `2[mB`2/ #v +mbiQK2` n M/ Un Bb Bib T`BQ`Biv +Hbb, Un ∈ {1, 2, . . . , M }X G2i σ ˜ (t) := σn B7 t ∈ [Tn , Tn+1 ) . G2i {Ti,n }n∈Z #2 i?2 b2[m2M+2 bbQ+Bi2/ rBi? i?2 ``BpH T`Q+2bb Ai Q7 +mbiQK2`b Q7 +Hbb i- rBi? i?2 mbmH +QMp2MiBQM Ti,0 ≤ 0 < Ti,1 X h?2 [mMiBiv σi,n := σ(Ti,n ) Bb i?2 KQmMi Q7 b2`pB+2 `2[mB`2/ #v +mbiQK2` n Q7 ivT2 i ``BpBM; i iBK2 Ti,n X bbmK2 i?i λ = E[A((0, 1])] < ∞ M/ BM T`iB+mH` λi = E[Ai ((0, 1])] < ∞ U1 ≤ i ≤ M VX h?2`27Q`2 QM2 Kv /2}M2 i?2 SHK T`Q##BHBiB2b PA0 M/ PA0 i X 6`QK 1tKTH2 dXkXRy-

RRX9X Sah- GAhhG1- 1h*X

9j8 λi = PA0 (U0 = i). λ

*QMbB/2` i?2 i`{+ BMi2MbBiB2b ρi = λi E0Ai [σ0 ] = λi E0Ai [σi,0 ]X _2+HH i?i PA0 i (Ti,0 = 0) = 1 M/ i?2`27Q`2 σi,0 = σ0 , PA0 i @XbX h?2 BM/BpB/mH i`{+ BMi2M@

bBiB2b M/ i?2 ;HQ#H i`{+ BMi2MbBiv Bb ρ = ki=1 ρi Ub22 1tKTH2 dXkXRyVX

RRX9 Tbi- GBiiH2- 2i+X h?2 Tbi S`QT2`iv h?Bb +HbbB+H `2bmHi Q7 [m2m2BM; i?2Q`v bii2b i?i B7 i?2 ``BpH TQBMi T`Q+2bb Bb  SQBbbQM T`Q+2bb- QT2`iBQMH +?`+i2`BbiB+b Q7 i?2 bvbi2K +QKTmi2/ Dmbi #27Q`2 ``BpH iBK2b M/ i `#Bi``v iBK2b `2 i?2 bK2 UTbi Bb i?2 ##`2pBiBQM Q7 dzSQBbbQM ``BpHb a22 hBK2 p2`;2bǴVX >2`2 Bb  T`2+Bb2 bii2K2MiX G2i #2 ;Bp2M  K2bm`#H2 ~Qr {θt }t∈R M/  θt @BMp`BMi T`Q##BHBiv P QM (Ω, F)X G2i A #2  θt @+QKTiB#H2 TQBMi T`Q+2bb rBi? }MBi2 BMi2MbBiv λ Ur2 mb2 i?2 MQiiBQM A #2+mb2 i?2 Tbi i?2Q`2K Bb KQbiHv TTHB2/ iQ ``BpH TQBMi T`Q+2bb2bVX amTTQb2 i?i 7Q` bQK2 ?BbiQ`v {Ft }t∈R - A Bb M Ft @SQBbbQM T`Q+2bb Q7 BMi2MbBiv λX G2i {Z(t)}t∈R #2  +Q`HQH θt @+QKTiB#H2 Ft @/Ti2/ T`Q+2bb rBi? pHm2b BM bQK2 iQTQHQ;B+H K2bm`#H2 bT+2 (K, K)X G2i f : K → R #2  MQM@M2;iBp2 K2bm`#H2 7mM+iBQMX h?2ME0A [f (Z(0−))] = E[f (Z(0))] .

URRX9eV

LQi2 i?i- #v i?2 +`Qbb@2`;Q/B+ i?2Q`2K Uh?2Q`2K dXeXRkV- B7 (P, θt ) Bb 2`;Q/B+URRX9eV BKTHB2b i?i P @HKQbi bm`2Hv M/ P @HKQbi bm`2Hv N 1  1 T f (Z(Tn −)) = lim f (Z(s))ds = E[f (Z(0))] . N →∞ N T →∞ T 0 n=1

E0A [f (Z(0−))] = lim

h?2 7QHHQrBM; mb27mH 2ti2MbBQM Q7 i?2 #Qp2 +HbbB+H Tbi T`QT2`iv Bb  /B`2+i +QMb2[m2M+2 Q7 i?2 /2}MBiBQMb Q7 biQ+?biB+ BMi2MbBiv M/ Q7 SHK T`Q##BHBivX h?2Q`2K RRX9XR G2i A- P - {θt }t∈R M/ {Ft }t∈R #2 b #Qp2- rBi? i?2 /Bz2`2M+2 i?i A Bb MQi M2+2bb`BHv  SQBbbQM T`Q+2bb #mi ?b  θt @+QKTiB#H2 biQ+?biB+ Ft @ BMi2MbBiv {λ(t)}t∈R X aiBHH bbmK2 i?i Bib p2`;2 BMi2MbBiv λ Bb }MBi2X G2i PA0 #2 i?2 SHK T`Q##BHBiv bbQ+Bi2/ rBi? (A, θt , P )X h?2M- 7Q` Mv θt @+QKTiB#H2 +Q`HQH biQ+?biB+ T`Q+2bb {Z(t)}t∈R iFBM; Bib pHm2b BM bQK2 K2bm`#H2 bT+2 (K, K)M/ Mv MQM@M2;iBp2 7mM+iBQM f : K → RλE0A [f (Z(0−))] = E [λ(0)f (Z(0))] .

URRX9dV

9je

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

S`QQ7X URd V 



f (Z(s−)) N (ds) = E f (Z(s−))λ(s) ds (0,1] (0,1]

  f (Z(s))λ(s) ds = E [f (Z(s))λ(s)] ds =E (0,1] (0,1]  E [f (Z(0))λ(0)] ds = E [f (Z(0))λ(0)] . =

λE0A [f (Z(0−))] = E

(0,1]

 AM i?2 SQBbbQM +b2 Uλ(t) = λV r2 `2+Qp2` i?2 +HbbB+H Tbi T`QT2`ivX >Qr@ 2p2` i?2 #Qp2 ;2M2`HBxiBQM ?b mb27mH TTHB+iBQMb b i?2 7QHHQrBM; 2tKTH2 rBHH /2KQMbi`i2X 1tKTH2 RRX9Xk, h?2 ǯDQ#@Q#b2`p2`ǰ T`QT2`ivXR3 *QMbB/2`  :Q`/QMĜL2r2HH M2irQ`F BM biiBQM`v `2;BK2X h?2 TQBMi T`Q+2bb Aij +QmMiBM; i?2 i`Mb72`b 7`QK biiBQM i iQ biiBQM j /KBib i?2 FtX @BMi2MbBiv λij (t) = μi rij 1{Xi (t)>0} . Aib Up2`;2V BMi2MbBiv Bb i?2`27Q`2 λij = μi rij P (Xi (0) > 0)X "v h?2Q`2K RRX9XRλij PA0 ij [X(0−) = n] = E[μi rij 1{Xi (0)>0} 1{X(0)=n} ] . h?2`27Q`2 PA0 ij (X(0−)

 P (X(0) = n) = n)P (Xi (0) > 0) = 0

B7 ni > 0c Qi?2`rBb2X

AM pB2r Q7 i?2 2tT`2bbBQM URRXRyV 7Q` i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 :Q`/QMĜ L2r2HH M2irQ`F  !K nl 1 αini −1 B7 ni > 0c α l=1, l=i l PA0 ij (X(0−) = n) = C 0 Qi?2`rBb2r?2`2 C Bb  +QMbiMi Q#iBM2/ #v MQ`KHBxiBQM, 

K  

αlnl



αini −1 = C.

n1 +···+nK =M,ni >0 l=1, l=i

7i2` i?2 +?M;2 Q7 bmKKiBQM p`B#H2 ni → ni − 1- i?2 H27i@?M/ bB/2 Q7 i?2 #Qp2 2[mHBiv #2+QK2b  n1 +···+nK =M −1 Rd R3

K 

 αlnl = G(K, M − 1).

l=1

("`ûKm/- RN3N)X (_2Bb2` M/ Gp2M#2`;- RN3y)- (a2p+BF M/ JBi`MB- RN3R)X

RRX9X Sah- GAhhG1- 1h*X

9jd

h?2`27Q`2- 7Q` n bm+? i?i ni > 0PA0 ij (X(0−)

1 = n) = G(K, M − 1)



K 

 αlnl

αini −1 .

l=1, l=i

q?2M  +mbiQK2` Bb i`Mb72``2/ 7`QK i iQ j i iBK2 t- i?2 bBimiBQM ?2 b22b /m`BM; ?Bb i`Mb72` Ur?2M ?2 ?b H27i i #mi MQi v2i `2+?2/ jV 7Q` i?2 `2bi Q7 i?2 M2irQ`F Bb MQi X(t−) #mi X(t−) − ei X h?2`27Q`2- i?2 bii2 Q7 i?2 M2irQ`F #v i?Bb !K nQ#b2`p2/ 1 l +mbiQK2` U2t+Hm/BM; ?BKV Bb n rBi? T`Q##BHBiv G(K,M α X Ai Bb i?2 bK2 l=1 l −1) b i?2 bii2 Q7 i?2 bK2 M2irQ`F rBi? M − 1 +mbiQK2`b Q#b2`p2/ i M `#Bi``v iBK2 #v M 2ti2`MH Q#b2`p2`X 1tKTH2 RRX9Xj, AMb2MbBiBpBiv Q7 Jf:AfRf∞ HB7Q T`22KTiBp2X dzAMb2M@ bBiBpBivǴ `272`b iQ bBimiBQMb r?2`2  ;Bp2M p2`;2 /2T2M/BM; QM i?2 T`Q##BHBiv /Bbi`B#miBQM Q7 QM2 Q` b2p2`H `M/QK p`B#H2b +imHHv /2T2M/b QM i?2b2 /Bbi`B@ #miBQMb QMHv i?`Qm;? i?2B` K2MbX h?Bb Bb i?2 +b2 7Q` BMbiM+2 7Q` i?2 Jf:Af∞ /2Hv bvbi2K Q7 1tKTH2 jXjXk BM biiBQM`v `2;BK2- 7Q` r?B+? i?2 p2`;2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K /2T2M/b QM i?2 /2Hv QMHv i?`Qm;? Bib K2MX h?2`2 `2 KMv Qi?2` BMbiM+2b Q7 i?Bb T?2MQK2MQM BM [m2m2BM; i?2Q`v Ui?2 /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb Q7 i?2 biiBQM`v 1`HM; [m2m2 ?b i?Bb T`QT2`ivVXRN *HH X(t) i?2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K i iBK2 t BM M Jf:AfRf∞ HB7Q T`22KTiBp2 [m2m2X 6Q` }t2/ k ≥ 1- /2MQi2 #v Nk i?2 TQBMi T`Q+2bb +QmMiBM; i?2 ``BpHb i?i KF2 i?2 +QM;2biBQM T`Q+2bb `2+? H2p2H k,  Nk (C) = 1C (Tn )1{X(Tn −)=k−1} . n∈Z (k)

(k)

(k)

G2i {Tn } #2 i?2 b2[m2M+2 Q7 TQBMib Nk - rBi? i?2 mbmH +QMp2MiBQM T0 ≤ 0 < T1 X (k) (k)  +mbiQK2` ``BpBM; i iBK2 Tn `2[mB`2b i?2 b2`pB+2 σn X h?2 7QHHQrBM; 7+i Bb (k) i`m2, {σn }n∈Z Bb BB/ rBi? i?2 bK2 /Bbi`B#miBQM b {σn }n∈Z Ui?2 T`QQ7 Bb H27i 7Q` (k) i?2 `2/2`VX "2+mb2 Q7 i?2 HB7Q T`22KTiBp2 `mH2- +mbiQK2` Tn `2+2Bp2b HH ?Bb b2`pB+2 r?2M i?2 [m2m2 Bb i H2p2H k Ub22 i?2 };m`2 #2HQrVX (k)

σn

k

(k)

Tn RN

("`#Qm`- RNde)- (EƺMB; M/ CMb2M- RNde)X

9j3

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

aBM+2 k Bb `2+m``2Mi UmM/2` i?2 bi#BHBiv ?vTQi?2bBbV- i?2 Hr Q7 H`;2 MmK#2`b ;Bp2b t 1 ds 0 {X(s)=k} lim  = μ−1 XbX t→∞ 1 N (ds) (0,t] {X(s−)=k−1} "v 2`;Q/B+Biv1 t→∞ t



t

1{X(s)=k} ds = π(k) ,

lim

0

r?2`2 π Bb i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bbX JQ`2Qp2`  1 N (t) 1 lim 1{X(s−)=k−1} N (ds) = lim 1{X(s−)=k−1} N (ds). t→∞ t (0,t] t→∞ t N (t) (0,t] "mi limt→∞

N (t) t

1 lim t→∞ N (t)



= λ- M/ BM pB2r Q7 i?2 Tbi T`QT2`iv 1 1{X(s−)=k−1} N (ds) = lim t→∞ t (0,t]

bQ i?i

h?Bb H2/b iQ

π(k) =ρ π(k − 1) π(k) = (1 − ρ)ρk



t

1{X(s)=k−1} ds = π(k − 1) , 0

(k ≥ 1) . (k ≥ 0) ,

r?B+? b?Qrb BM T`iB+mH` i?2 BMb2MbBiBpBiv Q7 i?2 M/GI/1/∞ HB7Q T`22KTiBp2 [m2m2- i?i Bb- π /2T2M/b QM G QMHv i?`Qm;? Bib K2MX h?2 T`QQ7 Q7 BMb2MbBiBpBiv +M #2 2ti2M/2/ iQ i?2 bBimiBQM r?2`2 i?2 BMTmi T`Q+2bb N /KBib i?2 biQ+?biB+ (P, Ft )@BMi2MbBiv {λX(t) }t∈R - r?2`2 {Ft }t∈R Bb i?2 ?BbiQ`v `2+Q`/BM; i?2 Tbi Q7 N i iBK2 t M/ i?2 r?QH2 b2[m2M+2 {σn }n∈N X LQrr2 ?p2 iQ mb2 i?2 2ti2M/2/ p2`bBQM Q7 Tbi  t  1 t 1 t 1{X(s−)=k−1} N (ds) = lim 1{X(s)=k−1} λk−1 ds lim t→∞ N (t) 0 t→∞ N (t) t 0 λk−1 = π(k − 1) , λ r?B+? T`Qp2b BMb2MbBiBpBivX

Zm2m2 G2M;i? i .2T`im`2b Q` ``BpHb AM i?2 Jf:fRf∞ Q` :fJfRf∞ bvbi2Kb- r2 r2`2 #H2 iQ +QKTmi2 i?2 biiBQM`v [m2m2 /Bbi`B#miBQMb i /2T`im`2 iBK2b M/ Dmbi #27Q`2 M ``BpH iBK2 `2bT2+iBp2HvX 6Q` i?2 ;2M2`H :f:fRf∞ [m2m2- i?2`2 Bb  p2`v ;2M2`H `2HiBQM #2ir22M i?2 biiBQM`v /Bbi`B#miBQMb i /2T`im`2 iBK2b M/ Dmbi #27Q`2 M ``BpH iBK2X h?2 bBimiBQM Bb i?2 7QHHQrBM;, q2 ?p2  K2bm`#H2 ~Qr {θt }t∈R - irQ bBKTH2 TQBMi T`Q+2bb2b A M/ D rBi?Qmi +QKKQM TQBMib M/  biQ+?biB+ T`Q+2bb {X(t)}t∈R - HH θt @+QKTiB#H2 M/ bm+? i?i 7Q` HH t ≥ 0

RRX9X Sah- GAhhG1- 1h*X

9jN

X(t) = X(0) + A((0, t]) − D((0, t]) ≥ 0 . h?mb {X(t)}t∈R Bb i?2 +QM;2biBQM T`Q+2bb U[m2m2 H2M;i?V Q7  [m2m2- A M/ D #2BM; `2bT2+iBp2Hv i?2 ``BpH M/ /2T`im`2 T`Q+2bb2bX HbQ bbmK2 i?i A ?b  }MBi2 BMi2MbBiv λA = λ M/ i?i X(0) UM/ i?2`27Q`2 HbQ X(t)V ?b }MBi2 2tT2+i@ iBQMX h?2M- iFBM; 2tT2+iiBQM BM i?2 #Qp2 2pQHmiBQM 2[miBQM 7Q` i?2 +QM;2biBQM T`Q+2bb- r2 }M/ i?i E[A((0, t]] = E[D((0, t])] M/ i?2`27Q`2 i?2 BMi2MbBiv Q7 i?2 /2T`im`2 T`Q+2bb Bb λD = λX G2i MQr f : N → R #2  #QmM/2/ 7mM+iBQMX 6Q` HH t ≥ 0  f (X(t)) = f (X(0)) + (0,t]

{f (X(s)) − f (X(s−))}A(ds)  {f (X(s)) − f (X(s−))}D(ds) . + (0,t]

i M 2p2Mi iBK2 Q7 A- f (X(t)) = f (X(t−) + 1) M/ i M 2p2Mi iBK2 Q7 Df (X(t−)) = f (X(t) + 1)X h?2`27Q`2  f (X(t)) = f (X(0)) + (0,t]

{f (X(s−) + 1) − f (X(s−))}A(ds)  {f (X(s−) − 1) − f (X(s−))}D(ds) . + (0,t]

.BpB/BM; #v λt- iFBM; 2tT2+iiBQMb rBi? `2bT2+i iQ P - Q#b2`pBM; i?i E[f (X(t))] = E[f (X(0))]- r2 }M/ i?i E0A [f (X(0−) + 1) − f (X(0−))] + E0D [f (X(0)) − f (X(0) + 1)] = 0 . qBi? f (x) = 1{n} (x)- r?2`2 n ≥ 1- M/ H2iiBM; 0 0 (n) := PA0 (X(0−) = n) , πD (n) := PD0 (X(0) = n) , πA− 0 0 0 0 (n − 1) − πA− (n) = πD (n) − πD (n + 1)X 6Q` n = 0- Q#b2`pBM; i?i r2 Q#iBM πA− 0 0 X(0−) ≥ 0 M/ X(0−) > 0 B7 y Bb  /2T`im`2 iBK2- πA− (0) = πD (1)X "v bmKKBM; i?2 Hbi irQ 2[mHBiB2b 7Q` HH i ≥ 0, 0 0 πA− (i) = πD (i + 1) .

AM rQ`/b, i?2 biiBQM`v +QM;2biBQM T`Q+2bb2b Q#b2`p2/ Dmbi #27Q`2 M ``BpH M/ Dmbi 7i2`  /2T`im`2 ?p2 i?2 bK2 /Bbi`B#miBQMX 1tKTH2 RRX9X9, *b2 Q7 M Jf:AfRf∞ [m2m2X h?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 MmK#2` Q7 +mbiQK2`b Dmbi 7i2` i?2 /2T`im`2 iBK2b Q7  bi#H2 Jf:fRf∞ [m2m2 Bb 2[mH iQ i?2 biiBQM`v /Bbi`B#miBQMX h?Bb Bb M BKK2/Bi2 +QMb2[m2M+2 Q7 i?2 #Qp2 `2bmHi M/ Q7 i?2 Tbi T`QT2`ivX

99y

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

GBiiH2Ƕb 6Q`KmH G2i (Ω, F, P ) #2  T`Q##BHBiv bT+2 2M/Qr2/ rBi?  K2bm`#H2 ~Qr {θt }t∈R X bbmK2 i?i (P, θt ) Bb 2`;Q/B+X *QMbB/2`  TQBMi T`Q+2bb N +QKTiB#H2 rBi? i?2 ~Qr M/ i?2`27Q`2 biiBQM`vX bbmK2 Bib BMi2MbBiv λ Bb }MBi2X SQBMi Tn Bb BMi2`T`2i2/ b i?2 ``BpH iBK2 BMiQ bQK2 dzbvbi2KǴ Q` dz#H+F #QtǴ Ubv-  b2`pB+2 bvbi2K Q`  T`Q+2bbQ`V Q7  dz+mbiQK2`Ǵ Q` dzDQ#Ǵ- i?i `2KBMb BM i?2 bvbi2K 7Q` i?2 `M/QK iBK2 Wn X h?2 b2[m2M+2 {Wn }n∈Z Bb bbmK2/ θt @+QKTiB#H2 UM/ i?2`27Q`2 biiBQM`v mM/2` i?2 SHK T`Q##BHBiv PN0 VX

λ

X(t) Wn

λ

h?2 MmK#2` X(t) Q7 +mbiQK2`b- Q` DQ#b- BM i?2 bvbi2K i iBK2 t Bb X(t) =



1(−∞,t] (Tn )1(t,−∞) (Tn + Wn ).

n∈Z

h?2 T`Q+2bb {X(t)}t∈R Bb #v +QMbi`m+iBQM θt @+QKTiB#H2 M/ i?2`27Q`2 biiBQM`vX G2iiBM; t = 0 M/ iFBM; 2tT2+iiBQM ;Bp2b U#v J2+F2Ƕb 7Q`KmH M/ MQiBM; i?i W n = W 0 ◦ θTn V    E [X(0)] = E 1(−∞,0] (Tn )1(0,−∞) (Tn + W0 ◦ θTn ) n∈Z



=

λE0N 



0 −∞

1(0,−∞) (s + W0 )ds

0

=λ −∞ ∞ =λ

PN0 (s + W0 > 0)ds PN0 (W0 > u)du = λE0N [W0 ] .

0

q2 ?p2 Dmbi Q#iBM2/ i?2 +2H2#`i2/ GBiiH2Ƕb 7Q`KmH,ky E [X(0)] = λE0N [W0 ] . A7 (P, θt ) Bb 2`;Q/B+- i?2 +`Qbb@2`;Q/B+ i?2Q`2K Uh?2Q`2K dXeXRkV HHQrb mb iQ r`Bi2 i?2 #Qp2 2[mHBiv b 1 Wk n↑∞ n k=1 n

E [X(0)] = λ lim

P @XbX

h?Bb 7Q`KmH TT2`b BM i?2 TTHB2/ HBi2`im`2 mM/2` i?2 7Q`K Q = λW X Ai Bb  `i?2` p2`biBH2 7Q`KmH- i?2 +?QB+2 Q7 i?2 dz#H+F #QtǴ /2HBKBiBM; i?2 bvbi2K #2BM; H27i i QM2Ƕb BK;BMiBQMX ky

(GBiiH2- RNeR)X

RRX9X Sah- GAhhG1- 1h*X

99R

1tKTH2 RRX9X8, 1KTiBM2bb T`Q##BHBiv Q7 i?2 G/G/1/∞ [m2m2X *QM@ bB/2` i?2 biiBQM`v G/G/1/∞ [m2m2 Q7 a2+iBQM RRXj rBi?  b2`pB+2 /Bb+BTHBM2 rBi?Qmi T`22KTiBQM U b2`pB+2 QM+2 bi`i2/ +MMQi #2 BMi2``mTi2/VX .2MQi2 #v X(t) i?2 MmK#2` Q7 +mbiQK2`b BM i?2 r?QH2 bvbi2K UrBiBM; `QQK THmb b2`pB+2 #QQi?VX hF2 7Q` i?2 #H+F #Qt i?2 b2`pB+2 #QQi?- M/ H2i Wn #2 i?2 bQDQm`M iBK2 Q7 +mbiQK2` n UiBK2 bT2Mi BM rBiBM; THmb iBK2 bT2Mi BM b2`pB+2VX 1

h?2 MmK#2` Q7 T2QTH2 BM i?2 #QQi? Bb i KQbi R- +imHHv- 2[mH iQ 1{X(t)=1} i?2 2tT2+iiBQM Q7 r?B+? Bb 1 − π(0)- r?2`2 π(0) Bb i?2 biiBQM`v T`Q##BHBiv Q7 }M/BM; i?2 bvbi2K 2KTiv i Mv ;Bp2M iBK2 tX h?2 bQDQm`M iBK2 Q7 +mbiQK2` n BM i?2 #QQi? Bb σn X h?2 θt @+QKTiB#H2 TQBMi T`Q+2bb Q7 ``BpHb BM i?2 #QQi? ?b i?2 p2`;2 BMi2MbBiv λX "v GBiiH2Ƕb 7Q`KmH π(0) = 1 − ρ , r?2`2 i?2 i`{+ BMi2MbBiv ρ Bb ;Bp2M #v i?2 7Q`KmH 1 σk . n↑∞ n k=1 n

ρ = λ lim

h?Bb Bb  mMBp2`bH 7Q`KmH i?i r2 ?p2 H`2/v 2M+QmMi2`2/ BM p`BQmb T`iB+mH` +b2bX 1tKTH2 RRX9Xe, J2M@pHm2 MHvbBb Q7 +HQb2/ C+FbQM M2irQ`FbX *HQb2/ C+FbQM M2irQ`F KQ/2Hb `Bb2 7Q` BMbiM+2 BM i?2 bBimiBQM r?2`2 M QT2M M2irQ`F Bb QT2`i2/ rBi?  #HQ+FBM; /KBbbBQM TQHB+v, B7 i?2`2 `2 H`2/v N +mb@ iQK2`b BM i?2 M2irQ`F- i?2 M2r+QK2`b rBi i  ;i2 UBM  [m2m2V mMiBH bQK2 +mbiQK2` Bb `2H2b2/ 7`QK i?2 M2irQ`F- i r?B+? iBK2 QM2 KQM; i?2 #HQ+F2/ +mb@ iQK2`b- B7 Mv- Bb /KBii2/X AM i?2 M2irQ`F- i?2`2 `2 i KQbi N +mbiQK2`bX i dzbim`iBQMǴ- i?2`2 Bb Hrvb QM2 +mbiQK2` `2/v iQ `2TH+2  /2T`iBM; +mbiQK2`#v /2}MBiBQM Q7 bim`iBQMX h?2`27Q`2- i bim`iBQM- Q` 7Q` HH T`+iB+H Tm`TQb2b M2` bim`iBQM- 2p2`vi?BM; HQQFb b B7  /2T`iBM; +mbiQK2` rb #2BM; BKK2/Bi2Hv `2+v+H2/X a

b

≤N ∞ ≥N

a

b

99k

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: N a

b

Ai Bb BKTQ`iMi BM T`+iB+2 iQ #2 #H2 iQ +QKTmi2 i?2 p2`;2 MmK#2` Q7 +mb@ iQK2`b d(N ) TbbBM; i?`Qm;? i?2 2Mi`M+2 TQBMi a T2` mMBi Q7 iBK2X h?Bb [mMiBiv Bb i?2 KtBKmK i?`Qm;?Tmi M/ Bb i?2`27Q`2 `2Hi2/ iQ i?2 2{+B2M+v Q7 i?2 bvbi2K 7`QK i?2 TQBMi Q7 pB2r Q7 i?2 QT2`iQ` Ur?Q KF2b KQM2v rBi? +mbiQK2`bVX 6`QK i?2 TQBMi Q7 pB2r Q7 i?2 [mHBiv Q7 b2`pB+2- M BKTQ`iMi T`K2i2` Bb W (N )- i?2 p2`;2 iBK2 bT2Mi #v  +mbiQK2` #2ir22M a M/ bX b  Kii2` Q7 7+i- N - W (N )M/ d(N ) `2 `2Hi2/ #v d(N )W (N ) = N , ()  T`iB+mH` +b2 Q7 GBiiH2Ƕb 7Q`KmHX h?2 7QHHQrBM; H;Q`Bi?K- +HH2/ K2M pHm2 MHvbBb-kR KF2b mb2 Q7 GBiiH2Ƕb 7Q`KmH M/ Q7 i?2 DQ#@Q#b2`p2` T`QT2`iv 7Q` +QKTmiBM; d(N )X G2i vi /2MQi2 i?2 p2`;2 MmK#2` Q7 pBbBib i?i  +mbiQK2` 2Mi2`BM; BM a KF2 BM biiBQM i #27Q`2 H2pBM; i?2 M2irQ`F pB bX G2i Li (N ) #2 i?2 biiBQM`v p2`;2 MmK#2` Q7 +mbiQK2`b BM biiBQM i- H2i Wi (N ) #2 i?2 +mbiQK2`@p2`;2 bQDQm`M iBK2 BM biiBQM i M/ H2i λi (N ) #2 i?2 p2`;2 MmK#2` Q7 +mbiQK2`b b2`p2/ #v biiBQM i T2` mMBi Q7 iBK2X q2 ?p2 λi (N ) = d(N )vi M/ 7`QK GBiiH2Ƕb 7Q`KmH Li (N ) = λi (N )Wi (N ) = d(N )vi Wi (N ). aBM+2 i?2 iQiH MmK#2` Q7 +mbiQK2`b Bb N - r2 ?p2 N=

K 

Li (N ) = d(N )

i=1

N 

vi Wi (N ).

i=1

G2i Yi (N ) /2MQi2 i?2 p2`;2 MmK#2` Q7 +mbiQK2`b BM biiBQM i b22M #v  +mbiQK2` ``BpBM; i biiBQM iX "v i?2 DQ#@Q#b2`p2` T`QT2`iv- Yi (N ) = Li (N − 1)X HbQ- #v qH/Ƕb B/2MiBivYi (N ) + 1 Wi (N ) = , μi r?2`2 μi Bb i?2 b2`pB+2 `i2 i biiBQM iX *QK#BMBM; i?2 #Qp2 `2HiBQMb- r2 Q#iBM i?2 7QHHQrBM; `2+m``2M+2 `2HiBQMb, 1 (1 + Li (N − 1)) , μi N , T (N ) = N v i=1 i Wi (N ) Li (N ) = vi T (N )Wi (N ) .

Wi (N ) =

kR

(_2Bb2` M/ Gp2M#2`;- RN3y)- (a2p+BF M/ JBi`MB- RN3R)X

URRX93V

RRX9X Sah- GAhhG1- 1h*X

99j

PM2 +M i?2`27Q`2 +QKTmi2 d(N ) M/ i?2M W (N ) pB UV- M/ i?2M +?QQb2 M QT2`iBM; TQBMi N i?i T`QpB/2b i?2 `2[mB`2/ #HM+2 #2ir22M i?2 QT2`iQ`Ƕb T`Q}i M/ i?2 +mbiQK2`Ƕb +QK7Q`iX

6mM+iBQMH 6Q`K Q7 J2+F2Ƕb 6Q`KmH AM J2+F2Ƕb 7Q`KmH QM2 +M iF2 /pMi;2 Q7 i?2 TQbbB#BHBiv 7Q` Zn iQ iF2 pH@ m2b BM M `#Bi``v K2bm`#H2 bT+2 (K, K)- 7Q` BMbiM+2 K = D(R; R)- i?2 b2i Q7 +Q`HQH 7mM+iBQMb 7`QK R BMiQ R- rBi? Mv bmBi#H2 iQTQHQ;v M/ K i?2 "Q`2H }2H/ bbQ+Bi2/ rBi? i?Bb iQTQHQ;vX M 2H2K2Mi Q7 i?Bb bT+2 Bb  7mM+iBQM z := (z(t), t ∈ R)X G2i Zn := (Zn (t, ω), t ∈ R)X aBM+2 Bi Bb  θt @+QKTiB#H2 K`FZn = Z0 ◦ θTn := {Z0 (t − Tn }t∈R X J2+F2Ƕb 7Q`KmH i?2M iF2b i?2 7Q`K  

   E Zn (Tn ) = λ E0N [Z0 (t)] dt = λE0N Z0 (t)dt . URRX9NV R

n∈Z

R

h?Bb 2ti2MbBQM Q7 GBiiH2Ƕb 7Q`KmH Bb +HH2/ i?2 H = λG 7Q`KmHXkk 1tKTH2 RRX9Xd, h?2 SQHH+x2FėE?BMi+?BM2 7Q`KmHX *QMbB/2`  G/G/s/c [m2m2 BM biiBQM`v `2;BK2- rBi?  MQM@T`22KTiBp2 b2`pB+2 /Bb+BTHBM2, QM+2 bi`i2/i?2 b2`pB+2 Q7  +mbiQK2` +MMQi #2 BMi2``mTi2/X h?2 +QMi`B#miBQM Q7 +mbiQK2` n Un ≤ 0V iQ i?2 rQ`FHQ/ W (0) i iBK2 0 Bb Zn (0 − Tn )- r?2`2  + Zn (t) = σn 1{Tn ≤t≤Tn +V˜n } + σn − (t − Tn − V˜n ) 1{t≥Tn +V˜n } M/ r?2`2 V˜n Bb i?2 rBiBM; iBK2 Q7 +mbiQK2` nX Zn (t)

Zn (t-Tn )

σn

Zn (-Tn ) 0

V˜n

V˜n +σn

t

Tn

Tn +V˜n

Tn +V˜n +σn

h?2 rQ`FHQ/ i iBK2 y +M #2 2tT`2bb2/ b  Zn (0 − Tn )1(−∞,0] (Tn ) W (0) = n≤0

M/ i?2`27Q`2- #v i?2 7mM+iBQM bT+2 J2+F2 7Q`KmH 

0 Z0 (t)dt . E[W (0)] = λEA R

AM pB2r Q7 i?2 b?T2 Q7 Z0 (t) M/ mbBM; i?2 7+i i?i PA0 (T0 = 0) = 1kk

("`mK2HH2- RNdR)X

t

999

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:  E0A

1 Z0 (t)dt = E0A [σ0 V˜0 + σ02 ]. 2 R

h?Bb `2bmHib BM i?2 7Q`KmH, 1 E[W (0)] = λE0A [σ0 V˜0 ] + λE0A [σ02 ]. 2 q?2M σ0 M/ V˜0 `2 BM/2T2M/2Mi- 7Q` BMbiM+2 BM  GI/GI/1/∞ 7B7Q Q` HB7Q MQM@T`22KTiBp2 [m2m2 U#mi MQi 7Q` bTi 7Q` BMbiM+2V 1 E[W (0)] = λE0A [σ0 ]E0A [V˜0 ] + λE0A [σ02 ] . 2

()

q2 MQr bT2+BHBx2 i?2 #Qp2 iQ  M/GI/1/∞ 7B7Q [m2m2X "2+mb2 Q7 i?2 7B7Q /Bb+BTHBM2 M/ bBM+2 i?2`2 Bb QMHv QM2 b2`p2`- i?2 rBiBM; iBK2 V˜n Q7 +mbiQK2` n Bb 2[mH iQ i?2 rQ`FHQ/ b?2 b22b BM 7`QMi Q7 ?2` mTQM ``BpH- i?i Bb- W (Tn− )X "v i?2 Tbi T`QT2`ivE0A [W (Tn −)] = E[W (0)] M/ i?2`27Q`2- mbBM; UV M/ i?2 T`2pBQmb `2K`Fb E0A [V˜0 ] =

1 λE0A [σ02 ] 2

URRX8yV

1−ρ

Q` 2[mBpH2MiHv E[W (0)] =

1 λE[σ02 ] 2

1−ρ

.

"v GBiiH2Ƕb 7Q`KmH- λE0A [V˜0 ] = E[Q(0)]X HbQ- E[Q(0)] = E[(X(0) − 1)+ ] = E[X(0)] − 1 + P (X(0) = 0). h?2`27Q`2 BM pB2r Q7 URRX98V- E[Q(0)] = E[X(0)] − ρX *QK#BMBM; HH i?2 #Qp2 `2K`Fb- r2 }M/ i?2 2tT2+i2/ MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2K M/GI/1/∞ 7B7Q, E[X(0)] = ρ +

1 2 λ E[σ02 ] 2

1−ρ

.

q`BiBM; λ2 E[σ02 ] = λ2 (Var(σ0 ) + (E[σ0 ])2 ) = λ2 Var(σ0 ) + ρ2 BM i?2 T`2pBQmb 2[mHBiv- r2 Q#iBM i?2 7Q`KmHkj E[X(0)] = ρ +

kj

(SQHH+x2F- RNjy)- (E?BMi+?BM- RNjk)X

1 λ2 1 ρ2 + Var(σ0 ). 21−ρ 21−ρ

RRX9X Sah- GAhhG1- 1h*X

998

1tKTH2 RRX9X3, EH2BM`Q+Fǰb +QMb2`piBQM HrXk9 h?Bb 2tKTH2 72im`2b i?2 G/G/s/∞ [m2m2BM; bvbi2K rBi? M T`BQ`Biv +Hbb2b Q7 1tKTH2 RRXjXdX A7 r2 TTHv i?2 SQHH+x2FĜE?BM+?BM K2M pHm2 7Q`KmH iQ 2+? Q7 i?2 T`BQ`Biv +Hbb2br2 Q#iBM- rBi? i?2 MQiiBQM BMi`Q/m+2/ BM 1tKTH2 RRXjXd1 2 E[Wi (0)] = λi E0Ai [σi,0 V˜i,0 ] + λi E0Ai [σi,0 ]. 2

URRX8RV

h?Bb 7Q`KmH TTHB2b r?2M i?2 b2`pB+2 Q7  +mbiQK2` Q7 Mv ivT2 +MMQi #2 BMi2`@ `mTi2/ M/ i?2 b2`pB+2 `i2 Bb 1X h?2 QMHv `2[mB`2K2Mi 7Q` i?2 b2`pB+2 /Bb+BTHBM2 KQM; +mbiQK2`b Q7 i?2 bK2 +Hbb M/ 7Q` i?2 T`BQ`Biv bbB;MK2Mi #2ir22M i?2 M +Hbb2b Bb i?i HH i?2 Q#D2+ib +QMbB/2`2/ }i i?2 θt @7`K2rQ`FX h?2 bBKTH2bi +b2 rQmH/ #2  7B7Q /Bb+BTHBM2 KQM; i?2 K2K#2`b Q7 i?2 bK2 +Hbb- M/  MQM@ T`22KTiBp2 T`BQ`Biv `mH2 rBi? ;Bp2M Q`/2` Q7 T`BQ`Biv- bv 1 ( 2 ( · · · ( M - r?2`2 i ( j K2Mb i?i

i ?b T`BQ`Biv Qp2` jX amKKBM; mT i?2 M 2[miBQMb URRX8RV M/ Q#b2`pBM; i?i M i=1 Wi (t) Bb i?2 iQiH rQ`FHQ/- M/ i?i 1 1 2 λi E0Ai [σi,0 ] = λE0A [σ02 ], 2 i=1 2 M

2 ] = λ λλi E0Ai [σ02 ] = λPA0 (U0 = i)E0A [σ02 | U0 = i]V- r2 Q#iBM UbBM+2 λi E0Ai [σi,0 M  i=1

1 λi E0Ai [σi,0 V˜i,0 ] = E[W (0)] − λE0A [σ02 ] . 2

()

h?2 `B;?i@?M/ bB/2 /Q2b MQi /2T2M/ mTQM i?2 b2`pB+2 /Bb+BTHBM2b KQM; K2K#2`b Q7 i?2 bK2 +Hbb Q` i?2 T`BQ`Biv bbB;MK2Mi b HQM; b i?2 +QM/BiBQMb bii2/ Dmbi 7i2` 7Q`KmH URRX8RV `2 BM 7Q`+2X h?Bb ;Bp2b i?2 +QMb2`piBQM Hr, M 

λi E0Ai [σi,0 V˜i,0 ] = +QMbiMi.

i=1

P7 +Qm`b2- B7 7Q` 2+? +Hbb i- σi,0 M/ V˜i,0 `2 BM/2T2M/2Mi UmM/2` P Q` PA0 i - i?Bb Bb 2[mBpH2MiV- UV iF2b i?2 7Q`K M  i=1

1 ρi E0Ai [V˜i,0 ] = E[W (0)] − λE0A [σ02 ]. 2

h?Bb 7Q`KmH ?QH/b 7Q` BMbiM+2 BM M M/GI/1/∞ bvbi2K rBi? T`BQ`Biv +Hbb2b- MQ T`22KTiBQM- M/ 7B7Q /Bb+BTHBM2 KQM; i?2 K2K#2`b Q7  ;Bp2M +HbbX k9

(EH2BM`Q+F- RNe8)X

99e

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

RRX8 a2H2+i2/ TTHB+iBQMb *?`+i2`BbiB+ 6mM+iBQM Q7 i?2 qQ`FHQ/ G2i {W (t)}t∈R #2 i?2 biiBQM`v rQ`FHQ/ T`Q+2bb Q7  biiBQM`v G/G/1/∞ [m2m2 UrBi? i`{+ BMi2MbBiv ρ < 1VX "v HQQFBM; b2T`i2Hv i r?i ?TT2Mb i i?2 DmKT iBK2b M/ r?i ?TT2Mb #2ir22M i?2 DmKTb- QM2 +M p2`B7v i?2 7QHHQrBM; 2pQHmiBQM 2[miBQM,  t eiuW (t) = eiuW (0) − iu eiuW (s) 1{W (s)>0} ds 0  + {eiuW (Tn ) − eiuW (Tn −) }1{0≤Tn ≤t} . n∈Z

hFBM; 2tT2+iiBQMb M/ Q#b2`pBM; i?i E[eiuW (t) ] = E[eiuW (0) ] M/ eiuW (s) 1{W (s)>0} = eiuW (s) − 1{X(s)=0} −iuE[eiuW (0) ] + iuP (X(0) = 0) + λE0A [eiuW (0) − eiuW (0−) ] = 0 . amTTQb2 i?i σn Bb BM/2T2M/2Mi Q7 W (Tn −) Ur?B+? Bb i?2 +b2 BM  GI/GI/1/∞ [m2m2 7Q` BMbiM+2VX h?2M E0A [eiuW (0) − eiuW (0−) ] = E0A [eiuW (0−) (eiuσ0 − 1)] = E0A [eiuW (0−) ](E0A [eiuσ0 ] − 1) . AM  GI/GI/1/∞ [m2m2- E0A [eiuσ0 ] = E[eiuσ0 ]- M/ i?2`27Q`2- iFBM; BMiQ ++QmMi P (X(0) = 0) = 1 − ρiuE[eiuW (0) ] = λE0A [eiuW (0−) ](E[eiuσ0 ] − 1) + iu(1 − ρ) . AM i?2 bT2+BH +b2 Q7  SQBbbQM ``BpH T`Q+2bb (M/GI/1/∞)- i?2 Tbi T`QT2`iv ;Bp2b E0A [eiuW (0−) ] = E[eiuW (0) ]- M/ i?2`27Q`2 E[eiuW (0) ] =

iu(1 − ρ) , iu − λ(Ψσ (u) − 1)

r?2`2 Ψσ (u) Bb i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 σ0 X JmHiBTH2 o+iBQMb G2i MQr {W (t)}t∈R #2 i?2 rQ`FHQ/ T`Q+2bb Q7  M/GI/1/∞ biiBQM`v [m2m2 rBi? KmHiBTH2 b2`p2` p+iBQMb, b bQQM b i?2 [m2m2 #2+QK2b 2KTiv- i?2 b2`p2` iF2b  p+iBQMX h?2 b2[m2M+2 Q7 p+iBQM iBK2b {Vk }k∈Z Bb bbmK2/ BB/ M/ BM/2T2M/2Mi Q7 i?2 ``BpH T`Q+2bbX Ai Bb bbmK2/ i?i i i?2 2M/ Q7  p+iBQM- i?2 b2`p2` }M/BM; i?2 [m2m2 biBHH 2KTiv iF2b MQi?2` p+iBQMc M/ bQ QM- b HQM; b i?2 [m2m2 `2KBMb 2KTivX *HHBM; tk i?2 bi`iBM; iBK2b Q7 i?2 p+iBQMb  σn 1{Tn ≤t} + Vk 1{tk ≤t} − t . () W (t) = W (0) + n≥1

k≥1

RRX8X a1G1*h1. SSGA*hAPLa

99d

A7 2[mBHB#`BmK Bb bbmK2/- E[W (t)] = E[W (0)]- M/ i?2`27Q`2- /2MQiBM; #v λV i?2 BMi2MbBiv Q7 i?2 p+iBQM T`Q+2bb NV - r2 ?p2 ρt + λV tE[V0 ] − t = 0- r?B+? ;Bp2b λV =

1−ρ . E[V0 ]

URRX8kV

6`QK UV 7QHHQrb i?2 2pQHmiBQM 2[miBQM,  exp(iu W (t)) = exp(iu W (0)) + exp(iu W (Tn −))(eiu σn − 1)1{Tn ≤t} n≥1

+



exp(iu W (tk −))(e

iu Vk



− 1)1{tk ≤t} − iu

k≥1

t

eiu W (s) 1{W (s)>0} ds . 0

P#b2`pBM; i?i W (tk −) = 0 M/ i?i i?2 G2#2b;m2 K2bm`2 Q7 {t; W (t) = 0} Bb XbX MmHH- i?2 7QHHQrBM; 7Q`KmH Bb Q#iBM2/ 7i2` +QKTmiiBQMb bBKBH` iQ i?Qb2 T2`7Q`K2/ BM i?2 #Qp2 /2`BpiBQM Q7 i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 i?2 rQ`FHQ/bbmKBM; i?2 2tBbi2M+2 Q7  biiBQM`v bii2, E[exp(iu W (0))] =

λV (ΨV (u) − 1) , iu − λ(Ψσ (u) − 1)

r?2`2 ΨV (u) M/ Ψσ (u) `2 i?2 +?`+i2`BbiB+ 7mM+iBQMb Q7 V0 M/ σ0 `2bT2+iBp2HvX h?2`27Q`2- mbBM; 2tT`2bbBQM URRX8kV 7Q` i?2 BMi2MbBiv Q7 i?2 p+iBQM T`Q+2bbE[exp(iu W (0))] =

ΨV (u) − 1 1−ρ × . iu − λ(Ψσ (u) − 1) E[V0 ]

URRX8jV

(u)−1 P#b2`p2 i?i ΨVE[V Bb i?2 +?`+i2`BbiB+ 7mM+iBQM Q7 i?2 7Q`r`/ UQ` 2[mBp@ 0] H2MiHv- Q7 i?2 #+Fr`/V `2+m``2M+2 iBK2 Q7  biiBQM`v U/2Hv2/V `2M2rH T`Q+2bb rBi? BMi2`@`2M2rH /Bbi`B#miBQM FV - i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 V0 X 6`QK 2tT`2bbBQM URRX8jV M/ i?Bb `2K`F- M/ HbQ Q#b2`pBM; i?i BM i?2 p+iBQM n Bb i?2 rBiBM; iBK2 #27Q`2 bvbi2K b r2HH b BM i?2 Q`B;BMH bvbi2K- W (Tn −) = V b2`pB+2 r?2M  7B7Q /Bb+BTHBM2 Bb bbmK2/- QM2 ?b- rBi? Q#pBQmb MQiiBQM-

ΨV˜0 (u)|F IF O,

V AC

= ΨV 0 (u)|F IF O,

N OV AC

× ΨY (u),

r?2`2 Y Bb  `M/QK p`B#H2 ?pBM; i?2 bK2 /Bbi`B#miBQM b i?2 7Q`r`/ `2@ +m``2M+2 iBK2 Q7  biiBQM`v `2M2rH T`Q+2bb +Q``2bTQM/BM; iQ i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM FV X AM rQ`/b-k8 h?2Q`2K RRX8XR h?2 rBiBM; iBK2 Q7  M/GI/1/∞ 7B7Q bvbi2K rBi? KmHiBTH2 BM/2T2M/2Mi p+iBQMb Q7 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM FV Bb /Bbi`B#mi2/ b i?2 bmK Q7 irQ BM/2T2M/2Mi `M/QK p`B#H2b X M/ Y - r?2`2, Ç X Bb /Bbi`B#mi2/ b i?2 rBiBM; iBK2 BM i?2 bK2 M/GI/1/∞ 7B7Q bvbi2K rBi?Qmi p+iBQMbc Ç Y Bb /Bbi`B#mi2/ b i?2 7Q`r`/ `2+m``2M+2 iBK2 Q7  biiBQM`v `2M2rH T`Q+2bb +Q``2bTQM/BM; iQ i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM FV X k8

(6m?`KMM- RN39)- (6m?`KMM M/ *QQT2`- RN38)X

993

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

S`BQ`BiB2b BM Jf:AfRf∞ h?2 p2`;2 rBiBM; iBK2 Q7  +mbiQK2` Q7  ;Bp2M ivT2 BM  Jf:AfRf∞ bvbi2K rBi? M T`BQ`Biv +Hbb2b rBHH MQr #2 +QKTmi2/X JQ`2 T`2+Bb2Hv- i?2 bBimiBQM Bb i?i Q7 1tKTH2 RRXjXd- rBi? i?2 //BiBQMH 72im`2 i?i i?2 M BMTmi ~Qrb +Q``2bTQM/BM; iQ i?2 M +Hbb2b Q7 T`BQ`Biv `2 BM/2T2M/2Mi M/ Q7 i?2 M/GI ivT2X h?2`2 Bb QM2 b2`p2` QT2`iBM; i mMBi bT22/- i?2 b2`pB+2 /Bb+BTHBM2 Bb MQM@T`22KTiBp2- 7B7Q 7Q` i?2 +mbiQK2`b Q7 i?2 bK2 +Hbb- M/ T`BQ`Biv Bb ;Bp2M iQ +mbiQK2`b Q7 +Hbb i Qp2` +mbiQK2`b Q7 +Hbb j B7 i < jX 1[mBHB#`BmK Bb bbmK2/- M/ BM T`iB+mH` ρ=

M 

ρi < 1 .

i=1

_2+HH i?2 /2}MBiBQM σi (t) = σi,n - B7 t ∈ [Ti,n , Ti,n+1 )X .2}M2 i?2 ?BbiQ`v {Fi,t } #v   M 3 σi ,Ai σk ,Ak Fi,t = Ft . URRX89V ∨ F∞ k=1, k=i

h?2 K`F2/ TQBMi T`Q+2bb- (Ai , {σi,n }) /KBib i?2 Fi,t @BMi2MbBiv F2`M2H λi Gi (dσ)i?i Bb iQ bv       E H(t, σ)Ni (dt × dσ) = E H(t, σ)λi Gi (dσ)dt , R

R+

7Q` HH MQM@M2;iBp2 P(Fi,t )


Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL: E0Ai [V˜i,0 ] = E0A [V˜0 | +mbiQK2` ``BpBM; i T0 = 0 Bb Q7 ivT2 i],

URRX8NV

r?2`2 V˜n Bb i?2 rBiBM; iBK2 Q7  +mbiQK2` ``BpBM; i iBK2 Tn X _2K`F, AM Q`/2` iQ b?Qr i?i E[V˜l (0)] = E0Al [V˜l,0 ], mbBM; i?2 Tbi T`QT2`ivbQK2 +`2 Kmbi #2 2t2`+Bb2/X AM/22/ Al Bb M Fl,t @SQBbbQM T`Q+2bb rBi? BMi2MbBiv λl M/ V˜l,0 = V˜l (Tl,0 ) = V˜l (0) PA0 l @a.s. >Qr2p2`- {V˜l (t)} Bb MQi M Fl,t @T`2/B+i#H2 T`Q+2bb- r?2M Fl,t Bb /2}M2/ #v URRX89VX h?Bb T`Q#H2K Bb 2bBHv +B`+mKp2Mi2/ #v BMi`Q/m+BM; i?2 σ@}2H/ σl Gl,t = Fl,t ∨ F∞ Ur2 // iQ Fl,t i?2 7mim`2 Q7 i?2 b2`pB+2 T`Q+2bbVX h?2 TQBMi T`Q+2bb Bb  Gl,t @SQBbbQM T`Q+2bb Q7 BMi2MbBiv λl - M/ i?Bb iBK2 {V˜l (t)} Bb  H27i@+QMiBMmQmb T`Q+2bb /Ti2/ iQ Gl,t - M/ i?2`27Q`2 Gl,t @T`2/B+i#H2X h?Bb }MBb?2b i?2 T`QQ7 Q7 *Q#?KǶb 7Q`KmHX 1tKTH2 RRX8Xk, Jf:AfRf∞ bTi MQM@T`22KTiBp2X AM i?Bb /Bb+BTHBM2 i?2 +mb@ iQK2` rBi? bKHH2bi `2[mB`2/ b2`pB+2 UbKHH2bi T`Q+2bbBM; iBK2V ?b MQM@T`22KTiBp2 T`BQ`Biv Qp2` i?2 Qi?2`bX G2iiBM; G #2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 i?2 `2[mB`2/ b2`pB+2 iBK2  E0A V˜0 | σ0 ∈ (x − h, x] = 

1−λ

1 λE[σ 2 ] 2  0

 (0,x−h]

ydG(y)

1−λ

.

 (x−h,x]

URRXeyV

ydG(y)

S`QQ7X *QMbB/2` i?`22 T`BQ`Biv +Hbb2b, BM +Hbb R- Tmi HH +mbiQK2`b rBi? `2[mB`2/ b2`pB+2 bKHH2` i?M Q` 2[mH iQ x − h- BM +Hbb k i?Qb2 rBi? `2[mB`2/ b2`pB+2 BM (x − h, x]- M/ BM +Hbb j i?2 `2bi Q7 i?2 +mbiQK2`bX h?mb UB;MQ`BM; +Hbb jV   ρ1 = λ ydG(y), ρ2 = λ ydG(y) . (0,x−h]

(x−h,x]

TTHvBM; URRX83VĜURRX8NV- r2 Q#iBM URRXeyVX AM i?2 +b2 r?2M x Bb MQi  /Bb+QMiB@ MmBiv TQBMi Q7 G(x)- H2iiBM; h ;Q iQ 0 BM URRXeyV- r2 Q#iBM E0A [V˜0 | σ0 = x] =

1 λE[σ02 ] 2 . (1 − λ (0,x] ydG(y))2



AM i?2 T`22KTiBp2 +b2- i?2 pB`imH bQDQm`M iBK2 Q7  +mbiQK2` Q7 +Hbb i ``BpBM; i iBK2 t Bb Vi (t) = Fi (t) + Gi (t) , URRXeRV r?2`2 Fi (t) Bb i?2 iBK2 #2ir22M t M/ i?2 }`bi iBK2 i?2 +mbiQK2` `2+2Bp2b ii2MiBQM 7`QK i?2 b2`p2`- M/ Gi (t) Bb i?2 bmK Q7 i?2 b2`pB+2 σ ˜i (t) M/ Q7 i?2 b2`pB+2 Q7 HH +mbiQK2`b Q7 +Hbb 1, . . . , i − 1 ``BpBM; BM i?2 bvbi2K BM i?2 BMi2`pH (t + Fi (t), t + Fi (t) + Gi (t)]X *H2`Hv BM i?2 +QKTmiiBQM Q7 E[Fi (t)]- i?2 +mbiQK2`b Q7 +Hbb i +

RRX8X a1G1*h1. SSGA*hAPLa

98R

1, . . . , K /Q MQi THv  `QH2- M/ i?2`27Q`2 E[Fi (t)] Bb i?2 p2`;2 rBiBM; iBK2 Q7  +Hbb i pB`imH +mbiQK2` ``BpBM; i iBK2 t BM  bvbi2K rBi? QMHv +mbiQK2`b Q7 ivT2 1, . . . , i- BM i?2 MQM@T`22KTiBp2 +b2ke

i 1 2 k=1 λk E[σk ] 2 . E[Fi (t)] = 

i−1  

i 1 − k=1 ρk 1 − k=1 ρk h?2 i2`K Gi (t) +M #2 2tT`2bb2/ b ˜i (t) + Gi (t) = σ

i−1   k=1

 R×R+

σ1(t+Fi (t),t+Fi (t)+Gi (t)] (u)Nk (du × dσ)

M/ `;mBM; b BM i?2 MQM@T`22KTiBp2 +b2- r2 Q#iBM σi (t)] + E[Gi (t)] = E[˜

i−1 

λk E[σk ]E[Gi (t)] .

k=1

h?2`27Q`2- Q#b2`pBM; i?i E[˜ σi (t)] = E[σi ] UTbiV- r2 Q#iBM E[Gi (t)] =

1−

E[σi ]

i−1 k=1

ρk

.

URRXekV

*QK#BMBM; URRXeRVĜURRXekV- r2 Q#iBM

i 1 2 E[σi ] k=1 λk E[σk ] 2 + . E[Vi (t)] = 

i−1

i−1  

i 1 − 1 − k=1 ρk 1 − k=1 ρk k=1 ρk >2`2 HbQ- BM pB2r Q7 TbiE[Vi (t)] = E0Ai [Vi,0 ] = E0A [V0 | i?2 +mbiQK2` ``BpBM; i T0 = 0 Bb Q7 ivT2 i] , r?2`2 Vi,n Bb i?2 bQDQm`M iBK2 Q7 i?2 +mbiQK2` Q7 ivT2 i ``BpBM; i iBK2 Ti,n M/ Vn Bb i?2 bQDQm`M iBK2 Q7 i?2 +mbiQK2` UQ7 Mv ivT2V ``BpBM; i iBK2 Tn X 1tKTH2 RRX8Xj, h?2 c/ρ `mH2X *QMbB/2` i?2 M/GI/1/∞ bvbi2K rBi? M T`B@ Q`Biv +Hbb2b rBi? MQM@T`22KTiBp2 T`BQ`Biv b /2b+`B#2/ i i?2 #2;BMMBM; Q7 i?2 T`2b2Mi bm#b2+iBQMX 6Q` bBKTHB+Biv- /2MQi2 E[V˜i (0)] #v V˜i X h?2 Q#D2+iBp2 BM i?Bb 2tKTH2 Bb iQ KBMBKBx2 i?2 7mM+iBQMH C=

M 

ci V˜i ,

i=1

r?2`2 ci > 0- KQM; HH TQbbB#H2 `MFBM;b UBM i2`Kb Q7 T`BQ`BivV Q7 i?2 M +Hbb2bX Ai im`Mb Qmikd i?i i?2 +Hbb2b b?QmH/ #2 `MF2/ ++Q`/BM; iQ i?2 pHm2 Q7 c/ρ- i?i ke kd

(*Q#?K- RN89)X (a2p+BF- RNd9)X

98k

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

Bb- MQM@T`22KTiBp2 T`BQ`Biv Bb ;Bp2M iQ +Hbb i Qp2` +Hbb j B7 M/ QMHv B7 ρcii > c i?2 +b2 Q7 2[mHBiv ρcii = ρjj - ;Bp2 ?B;?2bi T`BQ`Biv iQ i Q` j BM/Bz2`2MiHvVX

cj ρj

UBM

hQ T`Qp2 i?Bb `2bmHi- +QKT`2 irQ `MFBM;b A M/ B /Bz2`BM; QMHv BM irQ /D@ +2Mi +Hbb2bX JQ`2 T`2+Bb2Hv- bmTTQb2 U7Q` +QMp2MB2M+2V i?i `MFBM; A +Q``2bTQM/b iQ 1 ( 2 ( · · · ( M, r?2`2 i ( j K2Mb i?i +Hbb i ?b UMQM@T`22KTiBp2V T`BQ`Biv Qp2` +Hbb j- M/ bmTTQb2 i?i `MFBM; B 2t+?M;2b +Hbb2b i M/ i + 1 1 ( 2 ( · · · ( i − 1 ( i + 1 ( i ( i + 2 ( · · · ( M. *HH V˜jA M/ V˜jB i?2 p2`;2 rBiBM; iBK2 7Q`  +mbiQK2` Q7 +Hbb j mM/2` i?2 `MFBM;b A M/ B `2bT2+iBp2HvX HbQ H2i CA =

M 

cj V˜jA ,

CB =

j=1

M 

cj V˜jB .

j=1

6`QK i?2 2tT`2bbBQM URRX83V- r2 b22 i?i V˜kA = V˜kB - 7Q` HH k, 1 ≤ k ≤ M, k = i, k = i + 1- bQ i?i A B C A − C B = ci (V˜iA − V˜iB ) + ci+1 (V˜i+1 − V˜i+1 ).

URRXejV

"v EH2BM`Q+FǶb +QMb2`piBQM Hr Ua2+iBQM RRX9X3VM 

ρj V˜jA =

M 

j=1

ρj V˜jB ,

j=1

i?i Bb- BM pB2r Q7 i?2 T`2pBQmb `2K`F A B ρi (V˜iA − V˜iB ) = −ρi+1 (V˜i+1 − V˜i+1 ).

URRXe9V

6`QK URRXejV M/ URRXe9V- r2 Q#iBM  C − C = ρi (V˜iA − V˜iB ) A

B

ci ci+1 − ρi ρi+1

 .

LQr V˜iA − V˜iB < 0 Ui?BMF- Q` mb2 6Q`KmH URRX83VV M/ i?2`27Q`2 C A − C B < 0 B7 M/ QMHv BM ρcii > ρci+1 X i+1 h?Bb T`Qp2b i?2 `2bmHi bBM+2- 7Q` Mv `MFBM; MQi biBb7vBM; i?2 ρc +QM/BiBQM- r2 +M }M/ irQ /D+2Mi +Hbb2b i M/ i + 1 pBQHiBM; i?2 +QM/BiBQM- M/ 2t+?M;BM; i?2b2 irQ +Hbb2b rQmH/ ;Bp2  #2ii2` `MFBM;X 1tKTH2 RRX8X9, h?2 μc `mH2X amTTQb2 r2 rBb? iQ KBMBKBx2 C=

M  i=1

ci Qi ,

RRX8X a1G1*h1. SSGA*hAPLa

98j

r?2`2 Qi = E[Qi (t)]- i?2 p2`;2 MmK#2` Q7 +mbiQK2`b Q7 +Hbb i BM i?2 rBiBM; `QQK i iBK2 tX 6`QK GBiiH2Ƕb 7Q`KmHC=

M 

ci λi V˜i

i=1

M/ i?2`27Q`2- 7`QK i?2 T`2pBQmb `2bmHib- r2 b22 i?i T`BQ`Biv b?QmH/ #2 ;Bp2M iQ +Hbb i Qp2` +Hbb j B7 μi ci > μj cj X 1tKTH2 RRX8X8, PTiBKHBiv Q7 bTi MQM@T`22KTiBp2X 6Q`  [m2m2BM; bvb@ i2K Q7 i?2 M/GI/1/∞ ivT2- KQM; HH i?2 MQM@T`22KTiBp2 b2`pB+2 /Bb+BTHBM2b #b2/ QM i?2 `2[mB`2/ b2`pB+2 iBK2- bTi MQM@T`22KTiBp2 KBMBKBx2b i?2 p2`;2 MmK#2` Q7 +mbiQK2`b BM i?2 bvbi2KX h?Bb `2bmHi 7QHHQrb 7`QK i?2 μc `mH2 Q7 1t@ KTH2 RRX8X9 rBi? ci ≡ 1 UKBMBKBxBM; i?2 p2`;2 MmK#2` Q7 +mbiQK2`b E[Q(t)] BM i?2 rBiBM; `QQK Bb i?2 bK2 b KBMBKBxBM; i?2 p2`;2 MmK#2` E[X(t)] Q7 +mbiQK2`b BM i?2 bvbi2K- r?2M i?2`2 Bb QM2 b2`p2`VX h?mb BM 1tKTH2 RRX8X9- BM Q`/2` iQ KBMBKBx2 E[Q(t)]- r2 Kmbi ;Bp2 ?B;?2bi T`BQ`Biv iQ i?2 +Hbb rBi? bKHH2bi p2`;2 b2`pB+2 iBK2X  HBKBiBM; `;mK2Mi UrBi? +Hbb i 7Q`K2/ Q7 i?Qb2 +mbiQK2`b rBi? `2[mB`2/ b2`pB+2 #2ir22M ih M/ (i + 1)hV ;Bp2b i?2 QTiBKHBiv Q7 bTi MQM@ T`22KTiBp2X 1tKTH2 RRX8Xe, iiBM2/ b2`pB+2 BM i?2 G/GI/1/∞ T`Q+2bbQ` b?`BM; [m2m2X AM i?Bb 2tKTH2- i?2 ``BpH T`Q+2bb Bb Q7  ;2M2`H Mim`2 M/ i?2 b2`pB+2 iBK2 b2[m2M+2 Bb BB/- BM/2T2M/2Mi Q7 i?2 ``BpH T`Q+2bbX h?2 [m2m2BM; /Bb+BTHBM2 Bb T`Q+2bbQ` b?`BM; UTbV, 2+? +mbiQK2` T`2b2Mi BM i?2 bvbi2K `2+2Bp2b b2`pB+2 i  `i2 BMp2`b2Hv T`QTQ`iBQMH iQ i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2KX h?mb  +mbiQK2` ``BpBM; i iBK2 Tn < 0 M/ `2[mB`BM; b2`pB+2 σn rBHH #2 T`2b2Mi i iBK2 y M/ rBHH ?p2 iiBM2/ b2`pB+2 MQi ;`2i2` i?M a B7 M/ QMHv B7  0 1 ds ≤ σn ∧ a , Tn X(s) r?2`2 X(t) /2MQi2b i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi i iBK2 tX .2MQiBM; #v Xa (t) i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi i iBK2 t M/ rBi? iiBM2/ b2`pB+2 MQi ;`2i2` i?M a,  Xa (0) = 1Tn ≤0 1{ 0 1 ds≤σn ∧a} . Tn X(s) n∈Z

J2+F2Ƕb 7Q`KmH ;Bp2b  E[Xa (0)] =





λE0A

t

0

1{

1 0 X(s) ds≤σ0 ∧a

} dt .

G2i X ∗ (t) = X(t) B7 t ≤ V0 UV0 Bb i?2 bQDQm`M iBK2 Q7 +mbiQK2` yV M/ X ∗ (t) = X(t) + 1 B7 t > V0 - M/ 7Q` HH u > 0- H2i r(u) #2 /2}M2/ #v  r(u) 1 ds = u . ∗ (s) X 0

989

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

h?2M r(σ0 ) = V0

PA0 @XbX

M/ r(u) Bb i?2 bQDQm`M iBK2 Q7 +mbiQK2` y ;Bp2M Bib `2[mB`2/ b2`pB+2 Bb uX h?2 T`Q+2bb {r(u)}u≥0 Bb +HH2/ i?2 `2bTQMb2 iBK2 T`Q+2bb Q7 +mbiQK2` yX HbQ PA0 @XbX ∞  ∞  t 1{ 1 ds≤σ0 ∧a} dt = 1{t≤r(σ0 ∧a)} dt = r(σ0 ∧ a). 0

0 X(s)

0

LQr- i?2 b2`pB+2 b2[m2M+2 Bb BB/ M/ BM/2T2M/2Mi Q7 i?2 ``BpH T`Q+2bb- M/ r(t) /2T2M/b QM i?2 BMTmi T`Q+2bb 2t+2Ti σ0 X h?2`27Q`2 B7 G /2MQi2b i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 σ0   r(x ∧ a)G(dx) = r¯(x)G(dx) + r¯(a)(1 − G(a)), E0A [r(σ0 ∧ a)] = E0A [0,∞)

[0,a]

r?2`2 r¯(x) = E0A [r(x)] Bb i?2 p2`;2 `2bTQMb2 iBK2 iQ `2[mB`2/ b2`pB+2 tX "mi   r¯(x)G(dx) + r¯(a)(1 − G(a)) = (1 − G(x))¯ r(dx), [0,a]

[0,a]

bQ i?i }MHHv

 (1 − G(x))¯ r(dx).

E[Xa (0)] = λ

URRXe8V

[0,a]

1tKTH2 RRX8Xd, _2bTQMb2 iBK2 Q7 i?2 M/GI/1/∞ T`Q+2bbQ` b?`BM; [m2m2X G2i i?2 bii2 i iBK2 t Q7  M/GI/1/∞ T`Q+2bbQ` b?`BM; [m2m2 +QMbBbi Q7 URV i?2 MmK#2` X(t) Q7 +mbiQK2`b BM i?2 bvbi2K- M/ UkV i?2 2HTb2/ M/ `2bB/mH b2`pB+2 iBK2b Q7 i?2b2 +mbiQK2`bX h?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 bii2 mM/2` P UQ7 +Qm`b2 r2 bbmK2 ρ < 1V +M #2 +QKTmi2/ #v ;2M2`H K2i?Q/b Q7 SHK i?2Q`vX Ai Bb Q#iBM2/ b 7QHHQrb, Ç 6B`bi b2H2+i i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K iQ #2 n rBi? T`Q##BHBiv ρn (1 − ρ)X Ç h?2M b2H2+i n BM/2T2M/2Mi TB`b Q7 2HTb2/@`2bB/mH b2`pB+2 iBK2b /Bbi`B#mi2/ b  TB` Q7 #+Fr`/@7Q`r`/ `2+m``2M+2 iBK2b Q7  `2M2rH T`Q+2bb rBi? +X/X7X G- i?2 +X/X7X Q7 i?2 b2`pB+2 iBK2bX AM T`iB+mH` i?2 `2bB/mH b2`pB+2 iBK2b M/ i?2  ∞ 2HTb2/ b2`pB+2 iBK2b ?p2 i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Ge (x) = μ x (1 − G(x))dx- r?2`2 μ−1 Bb i?2 K2M b2`pB+2 iBK2X q2 +M mb2 i?Bb `2bmHi iQ Q#iBM i?2 p2`;2 `2bTQMb2 iBK2X AM/22/ i?2 T`Q##BHBiv 7Q`  ;Bp2M +mbiQK2` iQ ?p2 iiBM2/ b2`pB+2 MQ ;`2i2` i?M a Bb Ge (a)- M/ i?2`27Q`2 ρ Ge (a) . E[Xa (0)] = E[X(0)]Ge (a) = 1−ρ

RRXeX 1s1_*Aa1a

988

*QKT`BM; rBi? URRXe8V ;Bp2b  a  a λ (1 − G(t))dt = λ (1 − G(t))d¯ r(t) 1−ρ 0 0 M/ i?2`27Q`2-k3 bBM+2 r¯(0) = 0r¯(t) =

RRXe

t . 1−ρ

1t2`+Bb2b

1t2`+Bb2 RRXeXRX _2;mH`Biv Q7 [m2m2b M/ i?2B` M2irQ`Fb a?Qr i?i HH i?2 ?K+b `BbBM; 7`QK i?2 [m2m2BM; KQ/2Hb Q7 a2+iBQM RRXR `2 `2;mH` DmKT T`Q+2bb2bX 1t2`+Bb2 RRXeXkX h?2 `2p2`bH i2bi G2i A = {qij }i,j∈E #2  bi#H2 M/ +QMb2`piBp2 ;2M2`iQ` QM i?2 +QmMi#H2 bii2 bT+2 E- M/ H2i π #2  bi`B+iHv TQbBiBp2 T`Q##BHBiv /Bbi`B#miBQM QM EX .2}M2 ˜ = {˜ A qij }i,j∈E #v π(i)˜ qij = π(j)qji . S`Qp2 i?i B7 7Q` HH i ∈ E-



q˜ij = qi ,

j∈E,j=i

i?2M π T A = πX 1t2`+Bb2 RRXeXjX JfJfRf∞fSa AM i?Bb JfJfRf∞ [m2m2- 2+? Q7 i?2 +mbiQK2`b T`2b2Mi i iBK2 t `2+2Bp2b b2`pB+2 1 i bT22/ X(t) X :Bp2  ;bKT /2b+`BTiBQM Q7 i?Bb T`Q+2bbX q?i `2 i?2 BM}MBi2b@ BKH ;2M2`iQ`- i?2 2`;Q/B+Biv +QM/BiBQM- M/ i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb\ 1t2`+Bb2 RRXeX9X C+FbQMǰb M2irQ`F rBi? bii2@/2T2M/2Mi b2`pB+2 bT22/b h?2 7QHHQrBM; KQ/B}+iBQM Q7 i?2 #bB+ C+FbQM M2irQ`F Bb +QMbB/2`2/X 6Q` HH i1 ≤ i ≤ K- i?2 b2`p2` i biiBQM i ?b  bT22/ Q7 b2`pB+2 φi (ni ) r?2M i?2`2 `2 ni +mbiQK2`b T`2b2Mi BM biiBQM i- r?2`2 φi (k) > 0 7Q` HH k ≥ 1 M/ φi (0) = 0X h?2 M2r BM}MBi2bBKH ;2M2`iQ` Bb Q#iBM2/ 7`QK i?2 biM/`/ QM2 #v `2TH+BM; μi #v μi φ(ni )X *?2+F i?Bb M/ b?Qr i?i B7 7Q` HH i- 1 ≤ i ≤ KAi := 1 +

∞   Ai =1

k3

(aFi- LQ;m+?B M/ PBxmKB- RNeN)X

ρni !ni i k=1 φ(k)

 < ∞,

98e

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

r?2`2 ρi = λi /μi M/ λi Bb i?2 bQHmiBQM Q7 i?2 i`{+ 2[miBQM- i?2M i?2 M2irQ`F Bb 2`;Q/B+ rBi? biiBQM`v /Bbi`B#miBQM π(n) =

K 

πi (ni ),

i=1

r?2`2 πi (ni ) =

ρn i 1 !ni i . Ai k=1 φi (k)

1t2`+Bb2 RRXeX8X *E *QMbB/2` i?2 +HQb2/ C+FbQM M2irQ`F #2HQr r?2`2 HH b2`pB+2 iBK2b i /Bz2`2Mi [m2m2b ?p2 i?2 bK2 U2tTQM2MiBHV /Bbi`B#miBQM Q7 K2M 15 X *QKTmi2 7Q` N = 5 i?2 p2`;2 iBK2 bT2Mi #v  +mbiQK2` iQ ;Q 7`QK i?2 H27iKQbi TQBMi A iQ i?2 `B;?iKQbi TQBMi B- M/ i?2 p2`;2 MmK#2` Q7 +mbiQK2`b TbbBM; #v A T2` mMBi iBK2X

1t2`+Bb2 RRXeXeX *HQb2/ C+FbQM rBi?  bBM;H2 +mbiQK2` *QMbB/2` i?2 +HQb2/ C+FbQM M2irQ`F Q7 i?2 i?2Q`v- rBi? N = 1 +mbiQK2`X G2i {Y (t)}t≥0 #2 i?2 T`Q+2bb ;BpBM; i?2 TQbBiBQM Q7 i?Bb +mbiQK2`- i?i Bb- Y (t) = i B7 b?2 Bb BM biiBQM i i iBK2 tX a?Qr i?i {Y (t)}t≥0 Bb  `2;mH` DmKT ?K+- B``2/m+B#H2M/ ;Bp2 Bib biiBQM`v /Bbi`B#miBQMX 1t2`+Bb2 RRXeXdX AK#2//2/ ?K+ Q7 M Jf:AfRf∞f7B7Q [m2m2 a?Qr i?i i?2 BK#2//2/ ?K+ Q7 M Jf:AfRf∞f7B7Q Bb B``2/m+B#H2 Ub HQM; b i?2 b2`pB+2 iBK2b `2 MQi B/2MiB+HHv MmHHVX 1t2`+Bb2 RRXeX3X Z = N (0, τ ] S`Qp2 i?2 bii2K2Mi BM 1tKTH2 RRXRXR8 +QM+2`MBM; i?2 b2[m2M+2 {Zn }n≥1 - MK2Hvi?i Bi Bb BB/ M/ BM/2T2M/2Mi Q7 X0 - M/ Bb /Bbi`B#mi2/ b Z = N (0, τ ]- r?2`2 τ M/ N `2 BM/2T2M/2Mi- N Bb M ?TT rBi? BMi2MbBiv μ M/ i?2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM Q7 τ Bb F X 1t2`+Bb2 RRXeXNX h`MbB2Mi :AfJfRf∞ [m2m2 a?Qr i?i i?2 :AfJfRf∞ [m2m2 Q7 1tKTH2 RRXRXR8 Bb i`MbB2Mi B7 ρ > 1X 1t2`+Bb2 RRXeXRyX *QM7BM2/ ?K+ G2i {X(t)}t≥0 #2 M B``2/m+B#H2 TQbBiBp2 `2+m``2Mi ?K+ rBi? BM}MBi2bBKH ;2M2`iQ` A M/ biiBQM`v /Bbi`B#miBQM πX amTTQb2 i?i (A, π) Bb `2p2`bB#H2X G2i S #2  bm#b2i ˜ #v Q7 i?2 bii2 bT+2 EX .2}M2 i?2 BM}MBi2bBKH ;2M2`iQ` A

RRXeX 1s1_*Aa1a

98d  q˜ij =

αqij B7 i ∈ S, j ∈ E − S qij Qi?2`rBb2

r?2M i = jX h?2 +Q``2bTQM/BM; ?K+ Bb B``2/m+B#H2 B7 α > 0X A7 α = 0- i?2 bii2 bT+2 rBHH #2 `2/m+2/ iQ S- iQ KBMiBM B``2/m+B#BHBivX a?Qr i?i i?2 +QMiBMmQmb ˜ /KBib i?2 biiBQM`v /Bbi`B#miBQM π iBK2 ?K+ bbQ+Bi2/ iQ A ˜ ;Bp2M #v  αC × π(i) B7 i ∈ S, π ˜i = C × απ(i) B7 i ∈ E − S, rBi? i?2 Q#pBQmb KQ/B}+iBQM r?2M α = 0X 1t2`+Bb2 RRXeXRRX p2`;2 i`77B+ BM C+FbQMǰb M2irQ`F a?Qr i?i BM 2[mBHB#`BmK- i?2 i`{+ 2[miBQMb 7Q` i?2 C+FbQM M2irQ`F `2+2Bp2 i?2 7QHHQrBM; BMi2`T`2iiBQM, λi = E[Ai (0, 1]],

λi ri j = E[Ai,j (0, 1]]

r?2`2 Ai Bb i?2 TQBMi T`Q+2bb +QmMiBM; i?2 ``BpHb U2ti2`MH M/ BMi2`MHV BMiQ biiBQM i- M/ Ai,j Bb i?2 TQBMi T`Q+2bb +QmMiBM; i?2 i`Mb72`b 7`QK biiBQM i iQ biiBQM jX 1t2`+Bb2 RRXeXRkX E2HHvǰb M2irQ`F *QMbB/2`  i2H2+QKKmMB+iBQMb M2irQ`F rBi? K `2HvbX M BM+QKBM; +HH +?QQb2b  dz`Qmi2Ǵ r KQM;  b2i RX h?2 M2irQ`F i?2M `2b2`p2b i?2 b2i Q7 `2Hvb r1 , . . . , rk(r) +Q``2bTQM/BM; iQ i?Bb `Qmi2- M/ T`Q+2bb2b i?2 +HH iQ /2biBMiBQM BM M 2tTQM2MiBH iBK2 Q7 K2M μ−1 r X h?2 BM+QKBM; +HHb rBi? `Qmi2 r 7Q`K  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb Q7 BMi2MbBiv λr X Ai Bb bbmK2/ i?i HH i?2 `2Hvb `2 mb27mH- BM i?i i?2v `2 T`i Q7 i H2bi QM2 `Qmi2 BM RX kN

URV h?2 +T+Biv Q7 i?2 bvbi2K Bb 7Q` i?2 iBK2 #2BM; bbmK2/ iQ #2 BM}MBi2- i?i Bbi?2 MmK#2` Xr (t) Q7 +HHb QM `Qmi2 r i iBK2 t +M iF2 Mv BMi2;2` pHm2X HH i?2 mbmH BM/2T2M/2M+2 ?vTQi?2b2b `2 K/2, i?2 T`Q+2bbBM; iBK2b M/ i?2 SQBbbQM T`Q+2bb2b `2 BM/2T2M/2MiX :Bp2 i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 +QMiBMmQmb iBK2 ?K+ {X(t)}t≥0 - r?2`2 X(t) = (Xr (t), r ∈ R)X UkV h?2 +T+Biv Q7 i?2 bvbi2K Bb MQr `2bi`B+i2/ b 7QHHQrbX *QMbB/2`  ;Bp2M TB` (a, b) Q7 `2HvbX Ai `2T`2b2Mib  dzHBMFǴ BM i?2 M2irQ`FX h?Bb HBMF ?b }MBi2 +T+Biv Cab X h?Bb K2Mb i?i i?2 iQiH MmK#2` Q7 +HHb mbBM; i?Bb HBMF +MMQi 2t+22/ i?Bb +T+Biv- rBi? i?2 +QMb2[m2M+2 i?i M BM+QKBM; +HH `2[mB`BM;  `Qmi2 TbbBM; i?`Qm;? i?Bb HBMF rBHH #2 HQbi B7 i?2 HBMF Bb bim`i2/ r?2M Bi ``Bp2bX h?2 T`Q+2bb {X(t)}t≥0 i?2`27Q`2 ?b 7Q` bii2 bT+2  ˜ = {n = (nr ; r ∈ R); E nr ≤ Cab 7Q` HH HBMFb (a, b)}. r∈R,(a,b)∈r

q?i Bb i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 +?BM {X(t)}t≥0 \ kN

(E2HHv- RNdN)X

983

*>Sh1_ RRX SPALh S_P*1aa1a AL Zl1l1AL:

1t2`+Bb2 RRXeXRjX o+iBQM [m2m2b a?Qr i?i BM i?2 M/GI/1/∞ 6A6P bvbi2K rBi? KmHiBTH2 BM/2T2M/2Mi p+iBQMbi?2 p2`;2 rBiBM; iBK2 Bb ;Bp2M #v i?2 7Q`KmH λ E[σ 2 ] 1 E[V 2 ] + . 2 1 − ρ 2 E[V ] 1t2`+Bb2 RRXeXR9X qQ`FHQ/ Q7 JfJfRf∞ a?Qr i?i i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 rQ`FHQ/ T`Q+2bb Q7 M JfJfRf∞ [m2m2 rBi? ``BpH `i2 λ M/ K2M b2`pB+2 iBK2 μ−1 bm+? i?i ρ = μλ < 1 Bb FW (x) = 1 − ρ (1 − exp{−(μ − λ)x}) .

1t2`+Bb2 RRXeXR8X hrQ 2tT`2bbBQMb 7Q` i?2 i`77B+ BMi2MbBiv o2`B7v i?2 7QHHQrBM; irQ 2tT`2bbBQMb 7Q` i?2 i`{+ BMi2MbBivX 6B`bi BM i2`Kb Q7 P , N 1  σk , N →∞ N k=1

ρ = λ lim M/ i?2M BM i2`Kb Q7 PA0 ,

ρ=

P @XbX ,

E0A [σ0 ] . E0A [τ0 ]

1t2`+Bb2 RRXeXReX h?2 JfJfRf∞ [m2m2 G2i A M/ D #2 irQ BM/2T2M/2Mi ?TTb rBi? `2bT2+iBp2 BMi2MbBiB2b λ M/ μ- M/ H2i X(0) #2 M BMi2;2`@pHm2/ `M/QK p`B#H2 BM/2T2M/2Mi Q7 i?2 #Qp2 TQBMi T`Q+2bb2bX .2}M2 7Q` HH t ≥ 0  X(t) = X(0) + A(0, t] − 1{X(s−)>0} dD(s) . (0,t]

a?Qr i?i {X(t)}t≥0 Bb i?2 +QM;2biBQM T`Q+2bb Q7 M JfJfRf∞ [m2m2 rBi? ``BpH `i2 λ M/ b2`pB+2 iBK2b rBi? 2tTQM2MiBH /Bbi`B#miBQM Q7 K2M μX 1t2`+Bb2 RRXeXRdX _2bB/mH b2`pB+2 iBK2 P#iBM i?2 `2bB/mH b2`pB+2 iBK2 7Q`KmH 1 E[R(0)] = λE0A [σ02 ], 2

URRXeeV

pHB/ 7Q`  G/G/1/∞ [m2m2 rBi?  MQM@T`22KTiBp2 /Bb+BTHBM2X 1t2`+Bb2 RRXeXR3X *QMbiMi b2`pB+2 iBK2 KBMBKBx2b +QM;2biBQM a?Qr i?i 7Q`  }t2/ i`{+ BMi2MbBiv ρ- +QMbiMi b2`pB+2 iBK2b KBMBKBx2 p2`;2 +QM;2biBQM BM i?2 M/GI/1/∞ 7B7Q [m2m2X

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1t2`+Bb2 RRXeXRNX qBiBM; iBK2 Q7 i?2 G/G/1/∞ [m2m2 AM  biiBQM`v G/G/1/∞ [m2m2 UλE0A [σ0 ] < 1V- H2i Vn #2 i?2 rBiBM; iBK2 UbT2Mi BM i?2 rBiBM; `QQKV Q7 i?2 n@i? +mbiQK2`X :Bp2 i?2 2tT`2bbBQM Q7 1 Vk n↑∞ n k=1 n

lim

BM i2`Kb Q7 λ- π(0) M/ E [X(0)]X 1t2`+Bb2 RRXeXkyX h?2 biiBQM`v Jf:fRf∞ [m2m2 h?2 M/GI/1/∞ [m2m2 Bb  bT2+BH +b2 Q7 i?2 G/G/1/∞ [m2m2X h?2 b2`pB+2 iBK2 b2[m2M+2 {σn }n∈Z Bb BB/ rBi? +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM G(x) M/ K2M μ−1 M/ i?2 BMTmi T`Q+2bb N Bb SQBbbQM rBi? BMi2MbBiv λX G2i π /2MQi2 i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 MmK#2` Q7 +mbiQK2`b Dmbi 7i2` i?2 /2T`im`2 iBK2b Q7  bi#H2 Jf:fRf∞ [m2m2X a?Qr i?i i?2 biiBQM`v /Bbi`B#miBQM Q7 i?2 +QM;2biBQM T`Q+2bb Bb 2[mH iQ π, P (X(0) = i) = π(i)- 7Q` HH iX

*?Ti2` Rk >rF2b SQBMi S`Q+2bb2b >rF2b T`Q+2bb2b Kv #2 `2;`/2/ b i?2 biQ+?biB+ 7Q`K Q7 i?2 GQiFĜoQHi2`` KQ/2H Q7 1tKTH2 8XRXke Q` b  T`iB+mH` 7Q`K Q7  #`M+?BM; TQBMi T`Q+2bb- BM im`M  T`iB+mH` +b2 Q7  +Hmbi2` TQBMi T`Q+2bbXR h?2v +M KQ/2HBx2 i?2 ;`Qri? Q7 M 2TB/2Kv Q` b2BbKB+ +iBpBiv- BM r?B+? +b2 Bi Bb BKTQ`iMi iQ ?p2 2biBKi2b Q7 i?2 T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 iBK2 iQ biiBQM`Biv U7Q` BMbiM+2- i?2 iBK2 iQ 2t@ iBM+iBQMVX M BMi2`2biBM; bT2+i Q7 i?2b2 TQBMi T`Q+2bb2b Bb i?2 2tBbi2M+2 Q7  T?b2 i`MbBiBQM T?2MQK2MQM U#`M+?BM; T`Q+2bb2b dzrBi?Qmi M+2biQ`bǴ- a2+iBQM RkX8VX  bQK2r?i `2Hi2/ +Hbb Q7 TQBMi T`Q+2bb2b Bb i?2 bQ@+HH2/ MQM@HBM2` >rF2b T`Q+2bb2b i?i rBHH #2 bim/B2/ BM i?Bb +?Ti2` BM i2`Kb Q7  biQ+?biB+ /Bz2`2MiBH 2[miBQM /`Bp2M #v  irQ@/BK2MbBQMH SQBbbQM T`Q+2bbX

RkXR

b  "`M+?BM; SQBMi S`Q+2bb

.2}MBiBQM RkXRXR  TQBMi T`Q+2bb Z Bb +HH2/  #`M+?BM; TQBMi T`Q+2bb rBi? bBM;H2 M+2biQ` TQBMi i 0 B7 Z = z 0 + z1 + z2 + · · · r?2`2 z0 := ε0 Ui?2 TQBMi T`Q+2bb rBi?  bBM;H2 TQBMi i 0V M/ 7Q` HH n ≥ 0- zn+1 Bb i?2 +Hmbi2` TQBMi T`Q+2bb rBi? ;2`K TQBMi T`Q+2bb zn M/ +Hmbi2` αn -  bBKTH2 }MBi2 TQBMi T`Q+2bb bm+? i?i αn ({0}) = 0- M/ r?2`2 i?2 b2[m2M+2 {αn }n≥1 Bb BB/X AM i?2 MQiiBQM BMi`Q/m+2/ BM am#b2+iBQM RX8- 1tKTH2 RX8Xk i?2`2BM, zn+1 = zn ∗ αn+1 X

h?2 b2[m2M+2 { nk=0 zk (E)}n≥0 Bb i?2`27Q`2  :HiQMĜqibQM T`Q+2bb rBi?  bBM;H2 M+2biQ` M/ ivTB+H T`Q;2Mv bBx2 α(E) U `M/QK p`B#H2 rBi? i?2 +QKKQM T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 αn (E)VX AM T`iB+mH`- B7 E [α(E)] < 1- Z Bb  }MBi2 TQBMi T`Q+2bb Q7 p2`;2 bBx2 E [Z(E)] = R

1 < ∞. 1 − E [α(E)]

(>rF2b- RNdR)- (>rF2b M/ PF2b- RNd9)X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9_12

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.2}MBiBQM RkXRXk  ;2M2`H #`M+?BM; T`Q+2bb Bb  +Hmbi2` TQBMi T`Q+2bb rBi? i?2 7QHHQrBM; bT2+B}+BiB2b, UV E = Rm U#V i?2 BMi2MbBiv K2bm`2 Q7 i?2 ;2`K TQBMi T`Q+2bb Bb ν0 - M/ U+V Z := Z1 Bb i?2 #`M+?BM; TQBMi T`Q+2bb rBi? M+2biQ` TQBMi i 0 Dmbi /2b+`B#2/r?2`2 Bi Bb bbmK2/ i?i E [α(Rm )] < 1X h?2 ;2M2`B+ +Hmbi2` α Bb +HH2/ i?2 ;2M2`B+ T`Q;2Mv Q7 i?2 #`M+?BM; +Hmbi2` TQBMi T`Q+2bbX h?Bb TQBMi T`Q+2bb Bb HbQ +HH2/ M Bi2`i2/ +Hmbi2` T`Q+2bb bBM+2 Bi +QMbBbib Q7  bm++2bbBQM Q7 ;2M2`iBQMb- N0 , N1 , N2 , . . .- r?2`2 7Q` k ≥ 1- Nk Bb Q#iBM2/ #v α@+Hmbi2`BM; Q7 Nk−1 - i?i Bb- Nk = Nk−1 ∗ αk X h?2 }MH TQBMi T`Q+2bb Bb  N= Nk . k≥0

h?2 BMi2MbBiv K2bm`2b Q7 i?2 bm++2bbBp2 ;2M2`iBQMb `2 ν0 , ν1 , ν2 , . . .- r?2`2 νk = νk−1 ∗ να X AM T`iB+mH`- 7Q` HH k ≥ 1 νk (C) = νk−1 (C − x) να (dx) . Rm

AM i?2 bT2+BH +b2 r?2`2 ν0 = λ0 m - r2 ?p2 i?i νk = λk m r?2`2 λk = λ0 να (E)k X h?2M- bBM+2 να (E) < 1- i?2 BMi2MbBiv K2bm`2 Q7 N Bb ν(dx) =

λ0

m (dx) . 1 − να (E)

1tKTH2 RkXRXj, h?2 P`B;BMH >rF2b S`Q+2bbX AM i?Bb  +b2- i?2 ;2M2`B+ α Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv 7mM+iBQM h(t) bm+? i?i R+ h(t) dt < ∞X 1tKTH2 RkXRX9, TT`QtBKi2 aKTH2b Q7  "`M+?BM; SQBMi S`Q+2bbX AM i?Bb 2tKTH2- r2 +QMbB/2` i?2 T`Q#H2K Q7 bKTHBM; i?2 /Bbi`B#miBQM Q7 i?2 ;2M2`H #`M+?BM; T`Q+2bb Q7 .2}MBiBQM RkXRXk BM  +QKT+i dzrBM/QrǴ W ∈ B(Rm )X h?Bb `2[mB`2b mb iQ +QMbi`m+i i?2 #`M+?BM; T`Q+2bb2b ii+?2/ iQ HH i?2 TQBMib Q7 i?2 ;2`K T`Q+2bb- r?Qb2 MmK#2` Bb TQbbB#Hv BM}MBi2X h?Bb Bb BM ;2M2`H Qmi Q7 `2+? U+b2b r?2`2 i?Bb Bb 72bB#H2 rBHH #2 +QMbB/2`2/ Hi2`VX 6Q` i?2 iBK2 #2BM;- bmTTQb2 i?i BMbi2/ Q7 N - QM2 bm++22/b BM bKTHBM; Bib TT`QtBKiBQM N (n) =

n 

Nk ,

k=0

r?Qb2 BMi2MbBiv K2bm`2 Bb- bbmKBM; i?i i?2 BMi2MbBiv K2bm`2 Q7 i?2 ;2`K TQBMi m n α (R ) m T`Q+2bb Bb λ0 m (dx)- λ0 1−ν1−|ν

(dx)X AM T`iB+mH`| α E (N − N (n) )(W ) = |να |n

λ0

m (W ) , 1 − |να |

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M/ i?2`27Q`2- bBM+2 7Q` Mv BMi2;2`@pHm2/ `M/QK p`B#H2 Y - P (Y > 0) ≤ E [Y ]P ((N − N (n) )(W ) > 0) ≤ γ|να |n , r?2`2 γ=

λ0

m (W ) . 1 − |να |

h?Bb bvb i?i i?2 T`Q##BHBiv i?i N (n) ≡ N QM W Bb bKHH2` i?M γνα (Rm )n X AM biBHH Qi?2` rQ`/b- /2MQiBM; #v NW i?2 `2bi`B+iBQM Q7 N iQ W (n)

dV (NW , NW ) ≤ γ|να |n , r?2`2 dV Bb i?2 p`BiBQM /BbiM+2X AM/22/, (n)

|P (NW ∈ Γ) − P (NW ∈ Γ)| (n)

(n)

(n)

(n)

=|P (NW ∈ Γ, NW ≡ NW ) + P (NW ∈ Γ, NW ≡ NW ) (n)

(n)

− P (NW ∈ Γ, NW ≡ NW ) − P (NW ∈ Γ, NW ≡ NW )| (n)

(n)

(n)

= |P (NW ∈ Γ, NW ≡ NW ) + P (NW ∈ Γ, NW ≡ NW ) (n)

(n)

− P (NW ∈ Γ, NW ≡ NW ) − P (NW ∈ Γ, NW ≡ NW )| (n)

(n)

(n)

(n)

= |P (NW ∈ Γ, NW ≡ NW ) − P (NW ∈ Γ, NW ≡ NW )| ≤ P (NW ≡ NW ) . >Qr2p2`- i?2 /B{+mHiv K2MiBQM2/ i i?2 #2;BMMBM; Q7 i?2 +m``2Mi 2tKTH2 `2KBMbX h?2 bKTHBM; Q7 N (n) `2[mB`2b iFBM; BMiQ ++QmMi TQBMib Q7 i?2 ;2`K TQBMi T`Q+2bb r?Qb2 MmK#2` +M BM T`BM+BTH2 #2 BM}MBi2X >Qr2p2`- B7 i?2 bmTTQ`i Q7 α Bb }MBi2i?i Bb- B7 7Q` bQK2 R < ∞- P (α({x ∈ Rm ; ||x|| ≥ R}) = 0) = 1-  HBiiH2 i?Qm;?i b?QmH/ +QMpBM+2 QM2 i?i BM Q`/2` iQ Q#iBM N (n) QM W - Bi bm{+2b iQ +QMbB/2` QMHv i?2 #`M+?BM; T`Q+2bb2b ii+?2/ iQ i?2 ;2`K TQBMib i  /BbiM+2 H2bb i?i nR 7`QK W X aBM+2 i?2B` MmK#2` Bb }MBi2- r2 i?2`27Q`2 Q#iBM M TT`QtBKiBQM Q7 i?2 bKTH2 r2 HQQF2/ 7Q`- #mi i?2 [mHBiv Q7 i?Bb TT`QtBKiBQM BM i2`Kb Q7 i?2 p`BiBQM /BbiM+2 +M #2 +QMi`QHH2/ M/ K/2 b ;QQ/ b /2bB`2/ #v  T`QT2` +?QB+2 Q7 nX

h?2 >rF2b SQBMi S`Q+2bb >rF2b TQBMi T`Q+2bb2b ?p2 #22M T`QTQb2/ b KQ/2Hb Q7 b2BbKB+ +iBpBiv Q` Q7 T`QT;iBQM Q7 2TB/2KB2b- 7Q` BMbiM+2X 1tKTH2 RkXRX8, 7i2`b?Q+FbX amTTQb2 i?i i?2`2 Bb Dmbi QM2 M+2biQ` i iBK2 0- i?i Bb- N0 ≡ ε0 X h?2 7QHHQrBM; BMi2`T`2iiBQM Bb QM2 KQM; Qi?2`b,k i?2`2 Bb M 2`i?[mF2 i iBK2 0 Ui?2 dzTQBMi T`Q+2bbǴ ε0 V r?B+? ;2M2`i2b 7i2`b?Q+Fb 7Q`KBM; i?2 TQBMi T`Q+2bb Q7 7i2`b?Q+Fb N − ε0 X k

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h?Bb ivT2 Q7 KQ/2H 7Q` b2BbKB+ +iBpBiv ?b `2+2Bp2/ +QMbB/2`#H2 ii2MiBQM- M/ Bb FMQrM b M 1ha U1TB/2KB+ hvT2 7i2`@a?Q+FV KQ/2HXj 1tKTH2 RkXRXe, 1TB/2KB2bX MQi?2` BMi2`T`2iiBQM Bb i?2 T`QT;iBQM Q7  bKHH 2TB/2Kv bi`iBM; 7`QK  bBM;H2 BM72+i2/ BM/BpB/mH i iBK2 0X U"v dzbKHHǴ- Bi Bb mM/2`biQQ/ i?i i?2 BM72+i2/ TQTmHiBQM `2KBMb M2;HB;B#H2 rBi? `2bT2+i iQ i?2 TQTmHiBQM Q7 bmb+2TiB#H2b- bQ i?i Mv BM/BpB/mH +M QMHv #2 BM72+i2/ QM+2XV >2`2 ;BM- QM2 b22Fb 2biBKi2b Q7 i?2 iBK2 Q7 2tiBM+iBQM Q7 i?2 2TB/2KvX M Bbbm2 Q7 BMi2`2bi BM #Qi? BMi2`T`2iiBQMb ;Bp2M BM i?2 #Qp2 2tKTH2b Bb iQ 2biBKi2 i?2 2tiBM+iBQM iBK2- i?i Bb- i?2 /m`iBQM Q7- `2bT2+iBp2Hv- i?2 7i2`b?Q+F b2[m2M+2 M/ i?2 2TB/2KvX h?2 2tiBM+iBQM Bbbm2 rBHH #2 /2Hi rBi? BM a2+iBQM RkXkX AM M 2TB/2Kv Ub22 1tKTH2 RkXRXeV- Bi Bb BKTQ`iMi iQ iF2 BMiQ ++QmMi i?2 /Bz2`2M+2 #2ir22M i?2 TQTmHiBQMb Q7 bmb+2TiB#H2b- #v BMi`Q/m+BM; ivT2b- 2+? ivT2 +Q``2bTQM/BM; iQ  b2i Q7 72im`2b- bm+? b p++BMiBQM biimb- bQ+BH T`+iB+2 M/ ;2M2iB+ #+F;`QmM/X h?Bb KQiBpi2b i?2 BMi`Q/m+iBQM Q7 i?2 KmHiBp`Bi2 >rF2b T`Q+2bb-  KmHiBivT2 #`M+?BM; TQBMi T`Q+2bb r?Qb2 +QMbi`m+iBQM Bb bBKBH` iQ i?2 mMBp`Bi2 >rF2b T`Q+2bbX Ai +QMbBbib Q7  7KBHv {N (i) }1≤i≤M r?2`2 2+? N (i) Bb  TQBMi T`Q+2bb QM Rm +QMbi`m+i2/ b  bmT2`TQbBiBQM Q7 TQBMi T`Q+2bb2b,  (i) Nk , N (i) := k≥0 (i)

r?2`2 N0 U1 ≤ i ≤ M V `2 bBKTH2 HQ+HHv }MBi2 TQBMi T`Q+2bb2b rBi? `2bT2+iBp2 (i) BMi2MbBiv K2bm`2 ν0 U1 ≤ i ≤ M V +HH2/ i?2 M+2biQ`b TQBMi T`Q+2bb Q7 ivT2 i M/ (i) Nk Uk ≥ 1V,Bb i?2-k@i? ;2M2`iBQM TQBMi T`Q+2bb Q7 ivT2 i /2}M2/ `2+m`bBp2Hv b 7QHHQrbX G2i

(i,j)

Zn,k

n∈N,k∈N

U1 ≤ i, j ≤ M V #2  7KBHv Q7 BB/ `M/QK p`B#H2b-

BM/2T2M/2Mi Q7 N0 - rBi? pHm2b BM  K2bm`#H2 bT+2 (K, K) M/ rBi? +QKKQM (i) (i) /Bbi`B#miBQM Qij X h?2 TQBMi T`Q+2bb2b Nk ≡ {Xn,k }n∈N `2 i?2 #b2 TQBMi T`Q+2bb , (i,j) (i,j) Q7 i?2 K`F2/ TQBMi T`Q+2bb2b N k QM Rm × K rBi? i?2 BB/ K`Fb Zn,k n∈N

i?i Bb(i,j)

N k (C × L) =



(i)

(i,j)

1C (Xn,k )1L (Zn,k )

(C ⊆ B(Rm ), L ∈ K) .

n∈N

G2i #2 ;Bp2M MQM@M2;iBp2 72`iBHBiv `i2 7mM+iBQMb hij : Rm × K → R U1 ≤ i, j ≤ M V M/ /2}M2 

t

hij (Z (i,j) , t) dt

aij := E

(1 ≤ i, j ≤ M ) ,

0

r?2`2 Z (i,j) Bb  K@pHm2/ `M/QK p`B#H2 rBi? /Bbi`B#miBQM Qij X Ai Bb bbmK2/ i?i i?2 Ki`Bt A := {aij }M i,j=1 j (P;i- RN33 M/ RNN3)- b22 (.H2vĜo2`2@CQM2b- kyyj)- 1tKTH2 eX9 U/V 7Q`  /2iBH2/ /2b+`BT@ iBQMX

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λk (t) :=

M   j=1

 Rm

(j)

hji (t − s, z)N k−1 (ds × dz) = K



(j)

(j,i)

hji (t − Xn,k−1 , Zn,k−1 ) .

n∈N (i)

(i)

h?2 BMi2`T`2iiBQM Bb i?2 7QHHQrBM;, 2+? TQBMi Xn,k−1 ∈ Nk−1 Q7 ;2M2`iBQM k − 1 ;Bp2b #B`i? iQ /2b+2M/Mib Q7 ivT2 j 7Q` i?2 M2ti ;2M2`iBQM ++Q`/BM; iQ  *Qt (i) (i,j) TQBMi T`Q+2bb Q7 BMi2MbBiv hij (t − Xn,k−1 , Zn,k−1 )X h?2`27Q`2 2+? M+2biQ` U TQBMi (i)

(i)

Q7 N0 7Q` bQK2 1 ≤ i ≤ M V M/ 2+? /2b+2M/2Mi U TQBMi Q7 Nk 7Q` bQK2 k ≥ 1 M/ bQK2 1 ≤ i ≤ M V ?p2 aij /B`2+i /2b+2M/Mib Q7 ivT2 j QM i?2 p2`;2X G2i    N := Nk , N := Nk . k≥0

k≥0

A7 i?2 M+2biQ`b TQBMi T`Q+2bb2b `2 DQBMiHv biiBQM`v- rBi? `2bT2+iBp2 p2`;2 (i) BMi2MbBiv λ0 - i?2 bm#b2[m2Mi ;2M2`iBQMb `2 biiBQM`v- M/ 7`QK i?2 #Qp2 2t@ (i) T`2bbBQM Q7 λk (t) M/ *KT#2HHǶb 7Q`KmH(i)

λk =

M  j=1

 (j)

λk−1

Rm

M  (j) E h(t, Z (j,i) ) dt = λk−1 aij ,

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j=1

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r?2`2 h : R+ → R+ Bb  MQM@M2;iBp2 7mM+iBQM bm+? i?i q2 b?HH +QMbB/2` i?2 KQ/2H λ(t) = ϕ(X(t)) + Y (t), r?2`2 X(t) = X0 + ct −

N (t)  n=1

Zn

∞ 0

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M/ Y (t) = Y0 e−αt + k



e−α(t−s) dN (s) ,

(0,t]

r?2`2 ϕ Bb  TQbBiBp2 7mM+iBQMX h?Bb Bb  KQ/B}+iBQM Q7 i?2 T`2pBQmb KQ/2H- r?2`2 ϕ `2TH+2b i?2 2tTQM2MiBH- M/ h Bb MQr `2bi`B+i2/ iQ #2 M 2tTQM2MiBHX h?2 T`Q#H2K Bb iQ }M/  M2+2bb`v +QM/BiBQM 7Q` i?2 2tBbi2M+2 M/ mMB[m2M2bb Q7 i?2 +Q``2bTQM/BM; biiBQM`v T`Q+2bb M/- 7Q` Mv BMBiBH /Bbi`B#miBQM Q7 X0 M/ Y0 - Q7 i?2 +QMp2`;2M+2 iQ i?i /Bbi`B#miBQM- M/ iQ T`Qp2 7Q`KHHv i?i Bi Bb HbQ bm{+B2Mi UmM/2`  7m`i?2` bKQQi?M2bb +QM/BiBQM QM i?2 /Bbi`B#miBQM Q7 Zi VX h?2 2tBbi2M+2 Q7 M 2`;Q/B+ bQHmiBQM Bb T`Qp2/9 BM i?2 +b2 k < 1. α

URkXkV

h?Bb Bb BM/22/  Mim`H UBMimBiBp2V +QM/BiBQM M/ KQ`2Qp2`- Bi Bb M2+2bb`v B7 r2 b22F QMHv i?Qb2 bQHmiBQMb 7Q` r?B+? i?2 bi2/v@bii2 p2`;2 BMi2MbBiv λ := E[λ(t)] biBb}2b 0 < λ < ∞X h?2 T`Q+2bb (X(t), Y (t)), t ≥ 0 Bb  iBK2@?QKQ;2M2Qmb J`FQp T`Q+2bb rBi? BMBiBH pHm2 (X0 , Y0 )- M/ λ(t) = ϕ(X(t)) + Y (t). 6m`i?2`- 2`;Q/B+Biv K2Mb- BM T`iB+mH`- i?i i?2`2 2tBbib  biiBQM`v p2`bBQM Q7 i?2 T`Q+2bb (X(t), Y (t))X 6Q` bm+?  biiBQM`v p2`bBQM- H2i λ1 = E[ϕ(X(t))] M/ λ2 = EY (t)X h?2M λ = λ1 + λ2 - bQ i?2 }MBi2M2bb Q7 λ BKTHB2b i?i Q7 λ1 M/ Q7 λ2 X "v *KT#2HHǶb 7Q`KmH E[Y (t)] = E[Y (0)]e−αt + kE[ e−α(t−s) N (ds)] = E[Y (0)]e−αt + k



(0,t] t

e−α(t−s) λds]

0

k = E[Y (0)]e−αt + λ (1 − e−αt ) . α h?2`27Q`2- #v biiBQM`Biv- E[Y (0)] = λ αk = E[Y (t)]X h?2M λ = λ1 + λ

k ≡ λ1 + λ2 . α

h?2 bmT2`+`BiB+H +b2X amTTQb2- BM pB2r Q7 +QMi`/B+iBQM- i?i αk > 1X h?2 Hbi 2[mHBiv i?2M BKTHB2b i?i λ = ∞- r?B+? r2 2t+Hm/2/- Q` i?i λ = 0- M/ i?2M λ1 = E[ϕ(X(t))] = 0X aBM+2 ϕ(X(t)) ≥ 0- i?Bb BKTHB2b P (ϕ(X(t)) = 0) = 1- i?i Bb P (X(t) = −∞) = 1X aBKBH`Hv P (Y (t) = 0) = 1X h?2 +`BiB+H +b2X amTTQb2 MQr- ;BM BM pB2r Q7 +QMi`/B+iBQM- i?i αk = 1X h?2 Hbi /BbTHv2/ 2[mHBiv BKTHB2b i?2M i?i λ = ∞ U2t+Hm/2/V Q` λ1 = 0 M/ i?2`27Q`2 P (ϕ(X(t)) = 0) = 1X h?2M 9

("`ûKm/ M/ 6Qbb- kyRy)X

RkXRX a  "_L*>AL: SPALh S_P*1aa λ(t) = Y (0)e

−αt



9ed

e−α(t−s) N (ds).

+k (0,t]

q2 b?Qr i?i Mv TQBMi T`Q+2bb N rBi? i?Bb biQ+?biB+ BMi2MbBiv M/ rBi? }MBi2 p2`;2 BMi2MbBiv Bb M2+2bb`BHv MmHH UrBi? BMi2MbBiv 2[mH iQ 0VX amTTQb2 i?i λ > 0X *H2`HvP (N (R+ ) = 0) = E[P (N (R+ ) = 0|Y (0))] = E[e−

∞ 0

Y (0)e−αt dt

1 −E[Y (0)] α

≥e

=e

]

1 −λ α

>0

M/ i?2`27Q`2- bBM+2 r2 bbmK2/ λ < ∞- r2 ?p2 i?i P (N (R+ ) = 0) > 0X LQr{N (R+ ) = 0} ⊆ θt {N (R+ ) = 0} = {N ([t, ∞)) = 0}X h?i Bb- {N (R+ ) = 0} Bb 2tTM/2/ #v i?2 U2`;Q/B+V b?B7i- M/ i?2`27Q`2 Bi ?b T`Q##BHBiv 0 Q` 1X "v i?2 #Qp2- i?Bb T`Q##BHBiv Kmbi #2 1X q2 +QM+Hm/2 i?i λ = 0-  +QMi`/B+iBQMX h?2`27Q`2 BM i?2 +`BiB+H +b2 i?2`2 Bb MQ bQHmiBQM 2t+2Ti i?2 i`BpBH QM2 UMQ 2`i?[mF2bVX h?2`2 Bb M BMi2`2biBM; 72im`2 Q7 i?2 KQ/2HX q2 bbmK2 ?2`2 ;BM 2`;Q/B+Biv M/ i?2 +QM/BiBQM 0 < λ < ∞X q2 +QMiBMm2 iQ +QMbB/2` i?2 KQ/2H BM i?2 biiBQM`v `2;BK2X q`BiBM; ⎛ ⎛ ⎞⎞ N (t)  N (t) 1 ϕ(X(t)) = ϕ ⎝X(0) + t ⎝c − Zn ⎠⎠ t N (t) n=1 = ϕ(X(0) + t(c − λE[Z1 ] + ε(t))), r?2`2 limt↑∞ ε(t) = 0 XbX G2i Δ := c − λE[Z1 ]X G2i τ #2 i?2 UXbX }MBi2V `M/QK iBK2 bm+? i?i t ≥ τ BKTHB2b |ε(t)| ≤ 12 |Δ|X amTTQb2 i?i c − λE[Z1 ] > 0X q2 ?p2 1 E[ϕ(X(t))] ≥ E[ϕ(X(t))1{t≥τ } ] ≥ E[ϕ(X(0) + t Δ)1{t≥τ } ] . 2 "mi ϕ(X(0) + 12 Δ)t1{t≥τ } ↑ ∞ b t → ∞ M/ i?2`27Q`2 λ1 = E[ϕ(X(t))] → ∞ , BKTHvBM; i?i λ = ∞- r?B+? Bb 2t+Hm/2/X amTTQb2 i?i c − λE[Z1 ] < 0X q2 b?Qr i?i i?Bb Bb BKTQbbB#H2X >2`2 r2 ?p2 limt↑∞ ϕ(X(t)) = 0 #v  bBKBH` `;mK2MiX q2 T`Qp2 i?i limt↑∞ E[ϕ(X(t))] = 0λ1 = 0 M/ i?2`27Q`2 λ = 0- r?B+? Bb BKTQbbB#H2X 6Q` i?2 T`QQ7 i?i limt↑∞ E[ϕ(X(t))] = 0 r2 KF2 mb2 Q7 i?2 7QHHQrBM; H2KK UiFBM; +`2 Q7 #Qi? bBimiBQMb r?2M c − λE[Z1 ] = 0VX G2KK RkXRX3 A7 i?2 biiBQM`v biQ+?biB+ T`Q+2bb {Z(t)}t≥0 Bb bm+? i?i Bi i2M/b HKQbi bm`2Hv iQ  /2i2`KBMBbiB+ +QMbiMi c b t ↑ ∞- i?2M Bi Bb HKQbi bm`2Hv 2[mH iQ i?Bb +QMbiMiX

9e3

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

S`QQ7X 6Bt ε > 0- M/ +QMbB/2` i?2 b2i C = {ω; Z(t, ω) ∈ [c − ε, c + ε] 7Q` HH t ≥ 0}. h?2M 7Q` HH a > 0θa C = {ω; Z(t, ω) ∈ [c − ε, c + ε] 7Q` HH t ≥ a}. "mi θa C ↑ Ω- M/ i?2`27Q`2 P (C) = P (θa C) ↑ 1X aQ i?i P (C) = 1X aBM+2 i?Bb Bb i`m2 7Q` HH ε > 0P {Z(t) = c} = 1, 7Q` HH t ≥ 0. h?2`27Q`2- M2+2bb`BHv λ=

c . E[Z1 ] 

h?2`27Q`2- BM i?Bb KQ/2H- i?2 `i2 Q7 Q++m``2M+2b Q7 2`i?[mF2b Bb ;Bp2M #v i?2 T?vbB+b Q7 bi`2bb #mBH/ mT Ui?2 +QMbiMi cV M/ bi`2bb `2H2b2 UE[Z1 ]VX Ai Bb BM/2T2M@ /2Mi Q7 i?2 2tBbi2M+2 Q7 7i2`b?Q+Fb5 h?2 ;HQ#H `i2 Bb b?`2/ KQM; T`BK`v M/ b2+QM/`v 2`i?[mF2b ++Q`/BM; iQ i?2 T?vbB+b Q7 i?2 7i2`b?Q+Fb Uα M/ kVX h?2 7i2`b?Q+Fb /2+B/2 ?Qr iQ b?`2 i?2 `i25

RkXk _i2b Q7 1tiBM+iBQM M/ Q7 AMbiHHiBQM AM i?Bb b2+iBQM- r2 +QMbB/2` 2biBKi2b `2Hi2/ iQ i?2 `i2 Q7 +QMp2`;2M+2 iQ 2[mB@ HB#`BmK Q7 K`F2/ >rF2b T`Q+2bb2bX8 6B`biHv- i?2 biiBQM`v T`Q+2bb Bb i?2 2KTiv T`Q+2bb- BM r?B+? +b2 r2 bT2F Q7 i?2 `i2 Q7 2tiBM+iBQMX a2+QM/Hv- i?2 biiBQM`v T`Q+2bb Bb i?2 mMB[m2 biiBQM`v M/ MQM@i`BpBH K`F2/ >rF2b T`Q+2bb rBi? ;Bp2M 72`iBHBiv `i2- BM r?B+? +b2 r2 bT2F Q7 i?2 `i2 Q7 BMbiHHiBQMX _2+HH i?2 7QHHQrBM; bBKTH2 dz2TB/2KB+Ǵ KQ/2HX h?2 iBK2b Q7 TT2`M+2 Q7 M BM72+i2/ BM/BpB/mH 7Q`K  TQBMi T`Q+2bb N QM R rBi? bbQ+Bi2/ b2[m2M+2 Q7 2p2Mib {Tn }n∈Z X qBi? 2+? Tn Bb bbQ+Bi2/  K`F- i?i Bb-  `M/QK p`B#H2 Zn rBi? pH@ m2b BM bQK2 K2bm`#H2 bT+2 (K, K) `2T`2b2MiBM; bQK2 ii`B#mi2 Q7 i?2 BM/BpB/mH r?B+? #2;BMb iQ #2 BM72+i2/ i iBK2 Tn U;2M/2`- r2B;?i- p++BMiBQM biimbVX h?2 b2[m2M+2 {Zn }n∈Z Bb bbmK2/ BB/- rBi? /Bbi`B#miBQM QX qBi? i?2 /Qm#H2 b2[m2M+2 {(Tn , Zn )}n∈Z Bb bbQ+Bi2/ i?2 TQBMi T`Q+2bb NZ QM (R × K, B(R) ⊗ K) /2}M2/ #v  NZ (A) = 1A (Tn , Zn ) (A ∈ B(R) ⊗ K) . n∈Z

G2i {Ft }t∈R #2  ?BbiQ`v Q7 NZ X h?2 /vMKB+b Q7 i?2 2TB/2Kv +Q``2bTQM/b iQ i?2 Ft @BMi2MbBiv F2`M2H λ(t) dt × Q(dz) r?2`2  λ(t) = ν(t) + h(t − s, z)NZ (ds × dz) , URkXjV (−∞,t)×K 8

(hQ``BbB- kyyy M/ kyyk)- ("`ûKm/- LTTQ M/ hQ``BbB- kyyk)X

RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL

9eN

r?2`2 {ν(t)}t∈R Bb  HQ+HHv BMi2;`#H2 MQM@M2;iBp2 biQ+?biB+ T`Q+2bb M/ h : R × K → R+ Bb  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM bm+? i?i (t, z) ∈ (−∞, 0] × L =⇒ h(t, z) = 0 .

URkX9V

h?2 T`iB+mH` /vMKB+b /2b+`B#2/ #v URkXjV `2T`2b2Mi i?2 bBimiBQM r?2`2 i?2`2 Bb M dzBKTQ`i2/Ǵ `i2 Q7 BM72+iBQM ν(t) MQi /m2 iQ +QMi;BQMX h?2 T`i  h(t − s, z)NZ (ds × dz) (−∞,t)×K

+Q``2bTQM/b iQ BMi2`MH +QMi;BQMX q2 +QMbB/2` TQBMi T`Q+2bb2b N i?i `2 2KTiv QM (−∞, 0)X lM/2` +QM/BiBQMb  ∞ E[h(t, Z1 )]dt < 1 URkX8V 0

M/





ν(t)dt < ∞,

URkXeV

0

i?2 b?B7i2/ T`Q+2bb St N +QMp2`;2b BM /Bbi`B#miBQM iQ i?2 2KTiv T`Q+2bb- M/ B7 KQ`2Qp2`  ∞ tE[h(t, Z1 )]dt < ∞ , URkXdV 0

i?2 +QMp2`;2M+2 Bb BM p`BiBQM Ui?Bb rBHH #2  +QMb2[m2M+2- BM 7+i  T`iB+mH` +b2- Q7 i?2 KQ`2 ;2M2`H h?2Q`2K RkXeXN QM r?B+? r2 MiB+BTi2VX _i2 Q7 1tiBM+iBQM, i?2 GB;?i@iBH *b2 q2 +QMbB/2` i?2 +b2 r?2M i?2`2 rb MQ +b2 Q7 BM72+iBQM `2+Q`/2/ BMbB/2 i?2 dz+QHQMvǴ Ui?2 TQTmHiBQM Q7 BMi2`2biV #27Q`2 U 0 rBi?  ∞ eβt E[h(t, Z1 )]dt = 1. URkX3V 0

Ai Bb HbQ bbmK2/ i?i HKQbi bm`2Hv t → E[h(t, Z1 )] Bb HQ+HHv #QmM/2/ QM R+ .

URkXNV

9dy

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

h?2Q`2K RkXkXR lM/2` bbmKTiBQMb URkX3V- URkXNV- M/ B7 KQ`2Qp2` i?2 7mM+iBQM eβt ν(t) Bb /B`2+iHv _B2KMM BMi2;`#H2 QM R+ - i?2M 7Q` HH K bm+? i?i  ∞ βt e ν(t)dt URkXRyV K >  ∞ 0 βt β 0 te E[h(t, Z1 )]dt i?2`2 2tBbib  t0 = t0 (K) bm+? i?i 7Q` HH t ≥ t0 P (T > t) ≤ Ke−βt .

URkXRRV

AM i?2 #Qp2 BM2[mHBiv (12.10) i?2 `B;?i@?M/ bB/2 Bb /2}M2/ iQ #2 0 B7 i?2 /2MQK@ BMiQ` Bb BM}MBi2X S`QQ7X 6Q` HH t > 0- r2 ?p2



P (T > t) = P (N (t, ∞) ≥ 1) ≤ E[N (t, ∞)] = E



 λ(u)du =

t



λ(u)du . t

HbQ- #v i?2 bKQQi?BM; 7Q`KmH U`2+HHBM; i?i N Bb 2KTiv QM (−∞, 0]V 

λ(t) = ν(t) + E h(t − s, z)λ(s)Q(dz)ds , (0,t)×L

M/ i?2`27Q`2- λ(t) biBb}2b i?2 /272+iBp2 `2M2rH 2[miBQM  t λ(t) = ν(t) + E[h(t − s, Z1 )]λ(s)ds,

URkXRkV

0

rBi? i?2 HB;?i@iBH +QM/BiBQM URkX3VX AM T`iB+mH`- Q#b2`pBM; i?i λ(t) Bb HQ+HHv #QmM/2/ QM R+ - ν(t) Bb #QmM/2/ M/ λ(t) Bb BMi2;`#H2 QM R+  ∞ βt e ν(t)dt βt  , lim e λ(t) = ∞ 0βt t→∞ te E[h(t, Z1 )]dt 0 

7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX

lM/2` bbmKTiBQMb URkX8V M/ URkXeV- λ(t) Bb BMi2;`#H2 QM R+ M/ i?2`27Q`2i?2  ∞ 2tT2+i2/ MmK#2` Q7 BM72+i2/ T2QTH2 BM i?2 2TB/2Kv Bb }MBi2 bBM+2 E[N (0, ∞)] = λ(s)dsX JQ`2Qp2`0 ∞ ν(t)dt  ∞0 . E[N (0, ∞)] = 1 − 0 E[h(t, Z1 )]dt S`QQ7X AM 7+i- #v 1[X URkXRkV r2 ?p2  t  t  t  λ(s)ds = ν(s)ds + 0

t

0

0

t−u

 λ(s)ds E[h(u, Z1 )]du .

0

h?2`27Q`2- 0 λ(s)ds biBb}2b  `2M2rH 2[miBQM Q7 i?2 /272+iBp2 ivT2- M/ i?2`27Q`2 Uam#b2+iBQM 9Xk- h?2Q`2K 9XRXR8V

RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL  t→∞

∞

t

λ(s)ds =

lim

9dR

0

1−

 ∞0 0

ν(t)dt . E[h(t, Z1 )]dt 

1tKTH2 RkXkXk, h?2 2772+i Q7 p++BMiBQM M/ +QM7BM2K2MiX *QMbB/2` i?2 2TB/2KB+b KQ/2H Q7 1tKTH2 RkXRXe rBi?  MQM@`M/QK HQ+HHv #QmM/2/ HB;?i@ iBH 72`iBHBiv `i2 h- bQ i?i i?2 2tBbi2M+2 Q7 β > 0 biBb7vBM;  ∞ eβt h(t)dt = 1 0

Bb ;m`Mi22/ M/ i?2 +QM+HmbBQMb Q7 h?2Q`2K RkXkXR TTHvX q2 MQr BMi`Q/m+2 p++BMiBQM BM i?2 KQ/2H #v `2TH+BM; h(t) #v Z1 h(t)- r?2`2 Z1 Bb i?2 p++BMiBQM biimb Q7 i?2 ivTB+H BM/BpB/mH- rBi? E[Z1 ] = p ∈ (0, 1)X h?2 M2r 2tiBM+iBQM `i2 Bb MQr β- ;Bp2M #v  ∞

p



eβt h(t)dt = 1 .

0

*H2`Hv- b 2tT2+i2/- β > βX h?2 2z2+i Q7 BbQHiBQM Q7 M BM72+i2/ BM/BpB/mH +M #2 bBKBH`Hv 2pHmi2/X AbQHiBQM K2Mb i?i i?2 72`iBHBiv `i2 Bb 0 7i2` bQK2 iBK2X AM i?2 bBKTH2 +b2 Q7  /2i2`KBMBbiB+ 72`iBHBiv `i2 h(t)- BbQHiBQM Bb KQ/2HH2/ #v  M2r 72`iBHBiv `i2 h(t)1t≤Z1 X >2`2 Z1 `2T`2b2Mib i?2 ivTB+H iBK2 i r?B+? QM2 Bb BbQHi2/ 7i2` BM72+iBQM Ui?i Bb- +QKTH2i2Hv BbQHi2/- #mi vQm Kv r2HH BK;BM2 KQ/2Hb rBi? H2bb bi`BM;2Mi +QM}M2K2MiVX

_i2 Q7 1tiBM+iBQM, i?2 am#2tTQM2MiBH *b2 q2 MQr +QMbB/2` i?2 +b2 Q7  K2M BM72+iBpBiv `i2 rBi? bHQr /2+vX JQ`2 T`2+Bb2Hvi?2 /Bbi`B#miBQM 7mM+iBQM G rBi? /2MbBiv E[h(t, Z1 )] g(t) =  ∞ E[h(t, Z1 )]dt 0 Bb bbmK2/ iQ #2 bm#2tTQM2MiBH-e i?i Bb- 7Q` HH n ∈ N- limt→∞ G(t) := 1 − G(t)- r2 ?p2 i?2 7QHHQrBM; `2bmHi,

URkXRjV 1−G∗n (t) 1−G(t)

= nX qBi?

h?2Q`2K RkXkXj lM/2` bbmKTiBQMb (12.5)- (12.7) M/ (12.9)- B7 KQ`2Qp2` G Bb bm#2tTQM2MiBH M/ i?2 7mM+iBQM ν Bb #QmM/2/ M/ bm+? i?i B = lim sup t→∞

ν(t) < ∞, G(t)

URkXR9V

i?2M 7Q` HH K bm+? i?i e a22 .2}MBiBQM 9XkXky M/ i?2 HBM2b 7QHHQrBM; Bi 7Q` i?2 KBM T`QT2`iB2b Q7 i?Bb ivT2 Q7 /Bbi`B@ #miBQMX

9dk

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a K>

1−

∞ 0

B E[h(t, Z1 )]dt

i?2`2 2tBbib  t0 = t0 (K) bm+? i?i 7Q` HH t ≥ t0  ∞ P (T > t) ≤ K G(s)ds. t

UP#b2`p2 i?i +QM/BiBQM (12.14) ;m`Mi22b i?i ν(t) pMBb?2b i H2bi b 7bi b G(t), i?2 2z2+i Q7 i?2 BMBiBH BM72+iBQM Bb H2bb T2`bBbi2Mi i?M i?2 BM72+iBQM `i2 Q7 M BM/BpB/mHXV S`QQ7X q2 }`bi Q#b2`p2 i?i i?2 #QmM/2/M2bb Q7 ν(t)- (12.7) M/ (12.14) BKTHv +QM/BiBQM URkXeVX AM/22/- #v URkXR9V Bi 7QHHQrb i?i 7Q` HH  > 0 i?2`2 2tBbib  t = t() bm+? i?i ν(t) ≤ (B + )G(t) 7Q` HH t ≥ t , URkXR8V M/ i?2`27Q`2





 ν(t)dt ≤ (B + )

t



G(t)dt.

URkXReV

0

∞ aBM+2 #v +QM/BiBQM URkXdV r2 ?p2 0 G(t)dt < ∞- i?2 BMi2;`#BHBiv Q7 ν(t) 7QHHQrb #v (12.16) M/ i?2 #QmM/2/M2bb Q7 ν(t)X LQr- `;mBM; b BM i?2 T`QQ7 Q7 h?2Q`2K ∞ RkXkXR- 7Q` HH t > 0- P (T > t) ≤ t λ(s)ds- M/ 

t

λ(t − s)dG(s) ,

λ(t) = ν(t) + r

URkXRdV

0



r?2`2



r :=

E[h(t, Z1 )]dt .

URkXR3V

0

h?2 mMB[m2 HQ+HHv #QmM/2/ bQHmiBQM Q7 URkXRdV Bb Ub22 am#b2+iBQM 9XkV    i ∗i (t). λ(t) = ν ∗ rG

URkXRNV

i≥0

lbBM; BM2[mHBiv URkXR8V r2 Q#iBM  0

t

 t ν(t − s) ∗i ν(t − s) ∗i dG (s) + dG (s) G(t) G(t) 0 t−t  t  t−t G(t − s) ∗i ν(t − s) ∗i ≤ (B + ) dG (s) + dG (s) G(t) G(t) 0 t−t  t−t G(t − s) ∗i G∗i (t − t) − G∗i (t) ≤ (B + ) dG (s) + V , G(t) G(t) 0

ν(t − s) ∗i dG (s) = G(t)



t−t

r?2`2 V Bb bm+? i?i ν(t) ≤ V QM R+ X "mi G∗i (t − t) − G∗i (t) = 0. t→∞ G(t) lim

RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL AM/22/

9dj

G∗i (t − t) − G∗i (t) G∗i (t − t) G(t − t) G∗i (t) = − , G(t) G(t − t) G(t) G(t) lim

t→∞

G∗i (t − t) G∗i (t) = lim =i t→∞ G(t − t) G(t)

UG2KK 9XkXkjV M/ UG2KK 9XkXkkV lim

t→∞

G(t − t) = 1. G(t)

6m`i?2`KQ`2

t 0

G(t − s) ∗i dG (s) = G(t)

t 0

1 − G(t − s) ∗i dG (s) G(t)

G∗i+1 (t) − G∗i (t) G∗i (t) − G∗i+1 (t) = G(t) G(t)

= M/ i?2`27Q`2





t

lim

t→∞

0

G(t − s) ∗i dG (s) = (i + 1) − i = 1 . G(t)

h?2`27Q`2-

t lim sup t→∞

0

ν(t − s)dG∗i (s) G(t)

≤ B.

URkXkyV

6BMHHv- #v URkXRNV M/ URkXkyV r2 ?p2 i?i- 7Q` HH K > B i?2`2 2tBbib  t0 = t0 (K) bm+? i?i t ≥ t0 BKTHB2b   t K λ(t) = ri ν(t − s)dG∗i (s) ≤ G(t) 1−r 0 i≥0 7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX



_i2 Q7 AMbiHHiBQM, i?2 GB;?i@iBH *b2 q2 MQr im`M iQ bim/v i?2 `i2 Q7 +QMp2`;2M+2 iQ 2[mBHB#`BmK Q7 i`MbB2Mi Ui?i Bb  MQM@biiBQM`vV TQBMi T`Q+2bb2b N - r?B+? `2 2KTiv 7Q` MQM@TQbBiBp2 iBK2b- rBi? HBM2` /vMKB+b QM R+ Q7 ivT2 URkXjV rBi? ν(t) = ν > 0X h?2 HB;?i@iBH +QM/BiBQM (12.8) 7Q` i?2 /Bbi`B#miBQM rBi? /2MbBiv URkXRjV Bb bbmK2/X h?2Q`2K RkXkX9 lM/2` bbmKTiBQM (12.8)- M/ B7 KQ`2Qp2` i?2 7mM+iBQM eβt H(t) Bb /B`2+iHv _B2KMM@BMi2;`#H2 QM R+ - r?2`2  ∞ ν ∞ H(t) = E[h(u, Z1 )]du, URkXkRV 1 − 0 E[h(s, Z1 )]ds t i?2M 7Q` HH K bm+? i?i

9d9

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a ∞

K>

eβt H(t)dt  ∞0 βt β 0 te E[h(t, Z1 )]dt

URkXkkV

i?2`2 2tBbib  t0 = t0 (K) bm+? i?i 7Q` HH t ≥ t0 P (T > t) ≤ Ke−βt . AM i?2 #Qp2 BM2[mHBiv URkXkkV i?2 `B;?i@?M/ bB/2 Bb /2}M2/ b x2`Q B7 i?2 /2MQK@ BMiQ` Bb BM}MBivX  bm{+B2Mi +QM/BiBQM 7Q` eβt H(t) #2BM; /B`2+iHv _B2KMM@BMi2;`#H2 QM R+ Bb  ∞ teβt E[h(t, Z1 )]dt < ∞. URkXkjV 0

h?Bb 7QHHQrb #v h?2Q`2K 9XkXRy #2+mb2 eβt H(t) Bb  +QMiBMmQmb 7mM+iBQM QM R+ r?B+? Bb H2bb i?M Q` 2[mH iQ  ∞ ν ∞ z(t) := eβu E[h(u, Z1 )]du . 1 − 0 E[h(s, Z1 )]ds t h?2`27Q`2- Bi Bb #QmM/2/ #v bbmKTiBQM (12.8)X JQ`2Qp2`- z(t) Bb /B`2+iHv _B2KMM@ BMi2;`#H2 bBM+2 Bi Bb MQM@BM+`2bBM; M/ G2#2b;m2 BMi2;`#H2 #v bbmKTiBQM URkXkjVX S`QQ7X "v BK#2//BM; M/ +QmTHBM;X G2i N #2  SQBbbQM T`Q+2bb QM R × L × R+  rBi? K2M K2bm`2 dt × Q(dz) × dsX *QMbi`m+i  MQM@biiBQM`v TQBMi T`Q+2bb N r?B+? Bb 2KTiv QM (−∞, 0]- M/  biiBQM`v QM2 N - #Qi? ?pBM; /vMKB+b (12.3) rBi? ν(t) = ν- #v 



NZ (dt × dz) = N (dt × dz × [0, λ (t)]) ((t, z) ∈ R+ × L) ,    λ (t) = ν + h(t − s, z)NZ (ds × dz) (t ∈ R+ ) ,

URkXk9V

(0,t)×L

NZ (dt × dz) = N (dt × dz × [0, λ(t)]), (t, z) ∈ R × L ,  h(t − s, z)NZ (ds × dz) (t ∈ R) . λ(t) = ν +

URkXk8V

(−∞,t)×L

G2i



NZ

Ft := FtN ∨ Ft

(t ∈ R) .

"v i?2 BK#2//BM; +QMbi`m+iBQM 

N ⊂ N,



λ (t) ≤ λ(t) (t > 0) ,





URkXkeV

{λ (t)}t>0 Bb M Ft @T`2/B+i#H2 biQ+?biB+ BMi2MbBiv Q7 N QM R+ M/ {λ(t)}t∈R Bb M  Ft @T`2/B+i#H2 biQ+?biB+ BMi2MbBiv Q7 N QM RX h?2 TQBMi T`Q+2bb N − N /KBib  QM R+ i?2 Ft @biQ+?biB+ BMi2MbBiv {λ(t) − λ (t)}t>0 X "v i?2 bKQQi?BM; 7Q`KmH 7Q` TQBMi T`Q+2bb2b rBi? biQ+?biB+ BMi2MbBiv ∞  E[(N − N )(t, ∞)] = δ(u)du, 7Q` HH t > 0 , t

RkXkX _h1a P6 1shAL*hAPL L. P6 ALahGGhAPL

9d8



r?2`2 δ(t) = E[λ(t) − λ (t)]X h?2`27Q`2-



P (T > t) ≤



URkXkdV

δ(u)du. t

AM/22/M/



P (T > t) = P ((N − N )(t, ∞) ≥ 1),

7Q` HH t > 0





P ((N − N )(t, ∞) ≥ 1) ≤ E[(N − N )(t, ∞)].

ai`B;?i7Q`r`/ +QKTmiiBQMb mbBM; URkXk9V M/ URkXk8V vB2H/     λ(t)−λ (t) = h(t−s, z)NZ (ds×dz)+ h(t−s, z)(NZ −NZ )(ds×dz). (−∞,0]×L

(0,t)×L

"v i?2 bKQQi?BM; 7Q`KmH ∞  t E[h(u, Z1 )]du + E[h(t − s, Z1 )]δ(s)ds δ(t) = λ t

0

M/ i?2`27Q`2- #v Q#b2`pBM; i?i λ=

1−

∞ 0

ν , E[h(t, Z1 )]dt

r2 Q#iBM i?2 /272+iBp2 `2M2rH 2[miBQM  t δ(t) = H(t) + δ(t − s)E[h(s, Z1 )]ds.

URkXk3V

URkXkNV

0

h?2M- #v i?2 HB;?i@iBH +QM/BiBQM URkX3V M/ i?2 `2bmHib Q7 am#b2+iBQM 9Xk UQ#b2`p2 i?i #Qi? i?2 7mM+iBQMb δ(t) M/ H(t) `2 #QmM/2/V  ∞ βt e H(t)dt lim eβt δ(t) =  ∞ 0 βt , t→∞ te E[h(t, Z1 )]dt 0 

7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX _i2 Q7 AMbiHHiBQM, i?2 am#2tTQM2MiBH *b2

q2 MQr +QMbB/2` i?2 +b2 r?2M i?2 /Bbi`B#miBQM 7mM+iBQM G rBi? /2MbBiv URkXRjV Bb bm#2tTQM2MiBHX h?2Q`2K RkXkX8 lM/2` bbmKTiBQMb (12.5) M/ (12.7)- B7 KQ`2Qp2` G Bb bm#2t@ TQM2MiBH- i?2M 7Q` HH K bm+? i?i ∞ ν 0 E[h(t, Z1 )]dt K> 2 ∞ 1 − 0 E[h(t, Z1 )]dt i?2`2 2tBbib  t0 = t0 (K) bm+? i?i 7Q` HH t ≥ t0  ∞ P (T > t) ≤ K G(u)du. t

URkXjyV

9de

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

S`QQ7X `;mBM; 2t+iHv b BM i?2 T`QQ7 Q7 h?2Q`2K RkXkX9 r2 ?p2  P (T > t) ≤



δ(u)du . t



G2i 7Q` HH t > 0- λ (t) := E[λ (t)] M/ λ := E[λ(t)]X "v i?2 bKQQi?BM; 7Q`KmH iFBM; i?2 K2M Q7 λ (t) M/ λ(t) Bi 7QHHQrb i?i 

t

λ (t) = ν + r

λ (t − s)dG(s) 0

M/ λ = ν + λr- r?B+? ;Bp2b ν = λ(1 − r) r?2`2 r Bb ;Bp2M #v URkXR3VX .BpB/BM; λ (t) #v λ- r2 ?p2 λ (t) = (1 − r) + r λ



t 0

λ (t − s) dG(s) . λ

URkXjRV

aBM+2 λ (t) Bb #QmM/2/ #v λ Ub22 URkXkeVV M/ r < 1 #v bbmKTiBQM- r2 Q#iBM #v Bi2`iBM; 1[X URkXjRV  λ (t) rn G∗n (t) . = (1 − r) λ n≥0 ai`B;?i7Q`r`/ +QKTmiiBQMb H2/ iQ 1−

 λ (t) rn G∗n (t) . = (1 − r) λ n≥0

URkXjkV

*QM/BiBQM r < 1 2Mbm`2b HbQ i?2 2tBbi2M+2 Q7  TQbBiBp2  bm+? i?i r(1 + ) < 1X JQ`2Qp2` UG2KK 9XkXk9V i?2`2 2tBbib  D > 0 bm+? i?i- 7Q` HH BMi2;2`b n ≥ 0rn

G∗n (t) ≤ D[r(1 + )]n . G(t)

aBM+2 n≥0 [r(1 + )]n < ∞- Bi 7QHHQrb #v i?2 /2}MBiBQM Q7 bm#2tTQM2MiBH /Bbi`B@ #miBQM M/ #v /QKBMi2/ +QMp2`;2M+2 i?i lim

t→∞

JmHiBTHvBM; URkXjkV #v

λ G(t)

 n≥0

rn

G∗n (t) r . = (1 − r)2 G(t)

M/ iFBM; i?2 HBKBi b t → ∞- r2 }MHHv ;2i lim

t→∞

δ(t) νr , = (1 − r)2 G(t)

7`QK r?B+? i?2 MMQmM+2/ `2bmHi 7QHHQrbX



RkXjX h>1 "_hG1hh aS1*h_lJ P6 h>1 >qE1a S_P*1aa

RkXj

9dd

h?2 "`iH2ii aT2+i`mK Q7 i?2 >rF2b S`Q+2bb

h?2 TQr2` bT2+i`H /2MbBiv Q7 i?2 >rF2b #`M+?BM; TQBMi T`Q+2bb BM i?2 mMBp`Bi2 M/ KmHiBp`Bi2 +b2b rBi? /2i2`KBMBbiB+ 72`iBHBiv `i2b M/ M M+2biQ` T`Q+2bb i?i Bb M ?TT rb +QKTmi2/ BM >rF2bǶ b2KBMH rQ`FXd h?Bb `2bmHi Bb MQr 2t@ i2M/2/ BM irQ /B`2+iBQMbX 6B`biHv- i?2 72`iBHBiv `i2 Bb `M/QK M/ b2+QM/Hv- i?2 M+2biQ` T`Q+2bb Bb  ;2M2`H b2+QM/@Q`/2` biiBQM`v TQBMi T`Q+2bb rBi? FMQrM TQr2` bT2+i`H K2bm`2X G2i N #2  >rF2b T`Q+2bb rBi?  `M/QK 72`iBHBiv `i2 h biBb7vBM;- BM //BiBQM  

E h (t, Z) dt < 1, URkXjjV

iQ

Rm



i?2 +QM/BiBQM

2  h (t, Z) dt

E Rm

< ∞.

URkXj9V

h?2 TQBMi T`Q+2bb Q7 M+2biQ`b N0 Bb bbmK2/ rB/2@b2Mb2 biiBQM`v rBi? TQr2` bT2+i`H K2bm`2 μ0 M/ bbQ+Bi2/ bT+2 Q7 i2bi 7mM+iBQMb BN0 X h?2 M2ti i?2Q`2K ;Bp2b i?2 TQr2` bT2+i`H K2bm`2 Q7 i?Bb TQBMi T`Q+2bb iQ;2i?2` rBi?  /2b+`BTiBQM Q7 i?2 bbQ+Bi2/ i2bi 7mM+iBQMbX h?2Q`2K RkXjXR U3 V lM/2` +QM/BiBQM URkXj9V- i?2 "`iH2ii bT2+i`H K2bm`2 Q7 i?2 >rF2b T`Q+2bb N Bb  

λ0 1 ρλ0 μN (dν) = & dν + Var h(ν, Z)dν ,  &2 μ0 (dν) + & & 1−ρ 1−ρ &1 − E h(ν, Z) & URkXj8V r?2`2 λ = λ0 /(1 − ρ)X JQ`2Qp2`- QM2 +M iF2 7Q` i?2 i2bi 7mM+iBQM bT+2 BN i?2 b2i Q7 7mM+iBQMb f ∈ L1C (Rm ) ∩ L2C (Rm ) bm+? i?i i?2 bQHmiBQM Q7  h(s − t, z)E [ϕ(s, Z)] ds = f (t) URkXjeV ϕ(t, z) − Rm

biBb}2b E [ϕ(·, Z)] ∈ BN0 ,

URkXjdV

r?2`2 BN0 Bb i?2 i2bi 7mM+iBQM bT+2 Q7 i?2 M+2biQ` T`Q+2bb N0 X 1tKTH2 RkXjXk, "`iH2ii bT2+i`mK Q7 i?2 Q`B;BMH >rF2b T`Q+2bbX AM i?2 T`iB+mH` +b2 r?2`2 h(t, Z) = h(t)- h(ν, Z) = h(ν)- M/ i?2`27Q`2

1 ρλ0 μN (dν) = dν . μ0 (dν) + 1−ρ |1 − h(ν)|2 d 3

(>rF2b- RNdR- RNdR#)X ("`ûKm/ M/ JbbQmHBû- kyyk)- ("`ûKm/- JbbQmHBû M/ _B/QH}- kyy8)X

9d3

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

A7 BM //BiBQM N0 Bb  SQBbbQM T`Q+2bb rBi? p2`;2 BMi2MbBiv λ0 - r2 Q#iBM i?2 Q`B;BMH 7Q`KmH Q7 >rF2b 7Q` i?2 bT2+i`H /2MbBiv, fN (ν) =

λ0 (1 − ρ)|1 − h(ν)|2

.

M /KBbbB#H2 bT+2 Q7 i2bi 7mM+iBQMb Bb BN = L1C (Rm ) ∩ L2C (Rm ) Ub22 M2ti 2tK@ TH2VX 1tKTH2 RkXjXj, >rF2b T`Q+2bb rBi? *QtBM M+2biQ` T`Q+2bbX q?2M N0 Bb  *Qt T`Q+2bb b BM 1tKTH2 NXkXRj UrBi?  biQ+?biB+ BMi2MbBiv T`Q+2bb {λ(t)}t∈Rm i?i Bb  rB/2@b2Mb2 biiBQM`v biQ+?biB+ T`Q+2bb rBi? K2M λ M/ TQr2` bT2+i`H K2bm`2 μλ V

1 λ0 dν . μN (dν) = &  &2 μλ (dν) + & & 1−ρ &1 − E h(ν, Z) & q2 Kv iF2 7Q` i2bi 7mM+iBQM bT+2 BN = L1C (Rm ) ∩ L2C (Rm )X AM/22/- B7 f ∈ L1C (Rm )∩L2C (Rm )- i?2 bQHmiBQM ϕ Q7 1[MX URkXjeV biBb}2b i?2 +QM/BiBQM E [ϕ(·, Z)] ∈ L1C (Rm ) ∩ L2C (Rm ) = BN0 U1tKTH2 NXkXRjVX h?2 Ti? iQr`/b i?2 T`QQ7 Q7 h?2Q`2K RkXjXR Bb T`2T`2/ #v i?2 M2ti irQ H2KKbX G2KK RkXjX9 6Q` ϕ ∈ L1C ( × Q) ∩ L2C ( × Q)     Var ϕ(t, z) M (dt × dz) = λ Rm

r?2`2 λ (t) :=

 Rm

K

Rm

E |ϕ(t, Z)|2 dt,

 ¯  (dt × dz) − λ (t) dt Q (dz) , M (dt × dz) := N    ¯0 (dt × dz  ) + ¯  (dt × dz  ) h (t − s, z  ) N h (t − s, z  ) N K

Ui?2 biQ+?biB+ BMi2MbBiv Q7

Rm

K

 n≥1 Nn V M/ λ = λ − λ0 X

    S`QQ7X aBKTHB}2/ MQiiBQM Q7 i?2 FBM/ ϕ(t, z) M (dt × dz) = ϕ dM rBHH #2 mb2/X q2 ?p2    ϕ dM = ϕ dM n , n≥1

¯n (dt × dz) − λn (t)dt Q(dz)X :Bp2M Fn−1 - N ¯n Bb  SQBbbQM r?2`2 M n (dt × dz) = N T`Q+2bb rBi? BMi2MbBiv K2bm`2 λn (t)Q(dz)dt- M/ i?2`27Q`2- #v i?2 +Qp`BM+2 7Q`KmH UjX9V&    & & o` ϕ dM n & Fn−1 = E ϕ2 (t, Z) λn (t)dt, Rm

RkXjX h>1 "_hG1hh aS1*h_lJ P6 h>1 >qE1a S_P*1aa 

M/ E

9dN

&

& ϕ dM n && Fn−1 = 0,

M/ +QMb2[m2MiHv- #v i?2 +QM/BiBQMH p`BM+2 7Q`KmH& &    

 

 & & o` ϕ dM n = E o` ϕ dM n && Fn−1 + o` E ϕ dM n && Fn−1  = λn E ϕ2 (t, Z) dt. Rm

HbQ 7Q` HH j, k ≥ 1&   

  

 & ϕ dM j+k =E ϕ dM j E ϕ dM j+k && Fj+k−1 = 0 . E ϕ dM j h?2`27Q`2- }MHHv o`

ϕ dM



 =



 o`



 ϕ dM n

=

n≥1

= λ

 n≥1





E ϕ2 (t, Z) dt

λn Rm

E ϕ2 (t, Z) dt . Rm

 G2KK RkXjX8 A7 i?2 72`iBHBiv `i2 h biBb}2b +QM/BiBQMb URkXjjV M/ URkXj9V- i?2M UBV i?2`2 2tBbib- 7Q` Mv ;Bp2M F ∈ L1C ( × Q) ∩ L2C ( × Q)-  mMB[m2 bQHmiBQM ϕ ∈ L1C ( × Q) ∩ L2C ( × Q) Q7  h(s − t, z)E [ϕ(s, Z)] ds = F (t, z), M/ ϕ(t, z) − Rm

L1C

UBBV 7Q` ;Bp2M f ∈ L1C (Rm ) ∩ L2C (Rm )- i?2`2 2tBbib  mMB[m2 bQHmiBQM ϕ ∈ ( × Q) ∩ L2C ( × Q) Q7 URkXjeVX

S`QQ7X X 6Q`  7mM+iBQM v : R×K → R- /2MQi2 E [v (t, Z)] #v v¯ (t) M/ v (−t, z) #v vˇ (t, z)X *QM/BiBQM v ∈ L1C ( × Q) ∩ L2C ( × Q) BKTHB2b i?i v¯ ∈ L1C (Rm ) ∩ L2C (Rm )X HbQ- i?2 6Qm`B2` i`Mb7Q`K ν → v(ν, z) Q7 i?2 7mM+iBQM  t → v(t, z) Bb mMB7Q`KHv ∗ #QmM/2/X 6BMHHv- #2+mb2 ρ < 1- ν → 1 − E h(ν, Z) Bb mMB7Q`KHv #QmM/2/ rv 7`QK 0X ˇ¯ ∗ gX aBM+2 F¯ ∈ L1 (Rm ) ∩ L2 (Rm )*QMbB/2` i?2 `2M2rH 2[miBQM g = F¯ + h C C

¯ˇ ∗n Bb i?2 mMB[m2 bQHmiBQM g ∈ M/ BM pB2r Q7 +QM/BiBQM URkXjjV- g := F¯ ∗ h n≥0

L1C (Rm ) ∩ L2C (Rm )XN h?2 6Qm`B2` i`Mb7Q`K Q7 g Bb

N h?2 +QMp2`;2M+2 Q7 i?2 b2`B2b BM L1C (Rm ) b r2HH b BM L2C (Rm ) Bb ;m`Mi22/ #v i?2 BM2[mHBiB2b 'a ∗ b'L1 ≤ 'a'L1 'b'L1 M/ 'a ∗ b'L2 ≤ 'a'L1 'b'L2 c mMB[m2M2bb 7QHHQrb 7`QK ˇ¯    i?2 2[mHBiv =g − = g = h ∗ (g − g )- r?2`2 g Bb MQi?2` +M/B/i2 bQHmiBQM- r?B+? BKTHB2b  =¯ = 'g − g 'L1 ≤ h L1 'g − g 'L1 X >2M+2 mM/2` +QM/BiBQM URkXjjV- M2+2bb`BHv 'g − g 'L1 = 0X

93y

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a   E F (ν, Z)  . g (ν) = 1 − E h (ν, Z)∗

URkXj3V

6`QK i?2 Q#b2`piBQMb K/2 BM i?2 }`bi HBM2b Q7 i?2 T`QQ7- r2 MQi2 i?i ν → g (ν) Bb mMB7Q`KHv #QmM/2/X .2}M2 MQr i?2 7mM+iBQM (t, z) → ϕ (t, z) #v  ϕ (t, z) := h (s − t, z) g (s) ds + F (t, z) . Rm

q2 ?p2 i?i ϕ ∈ L1C ( × Q)- bBM+2



 |ϕ (t, Z)| dt ≤ E |F (t, Z)| dt E Rm Rm 

 +E |h (t, Z)| dt Rm

Rm

URkXjNV

|g (t)| dt < ∞.

HbQ ϕ ∈ L2C ( × Q)X hQ T`Qp2 i?Bb- Bi bm{+2b iQ b?Qr i?i ϕ (ν, z) ∈ L2C ( × Q) #2+mb2 i?2M- #v i?2 SHM+?2`2HĜS`b2pH B/2MiBiv 



2 2 E |ϕ (t, Z)| dt = E |ϕ (ν, Z)| dν < ∞ . Rm

Rm

6Q` i?Bb Tm`TQb2- iF2 7Q` }t2/ z i?2 6Qm`B2` i`Mb7Q`K Q7 #Qi? bB/2b Q7 URkXjNV, ϕ (ν, z) = h (ν, z)∗ g (ν) + F (ν, z) , Q`- BM pB2r Q7 URkXj3V  h (ν, z)∗ E F (ν, Z)  .  ϕ (ν, z) = F (ν, z) + 1 − E h (ν, Z)∗ aBM+2 F (t, z) ∈ L2C ( × Q)- Bi 7QHHQrb #v i?2 SHM+?2`2HĜS`b2pH B/2MiBiv i?i F (ν, z) ∈ L2C ( × Q)X Ai `2KBMb iQ b?Qr i?i h (ν, z) g(ν) ∈ L2C ( × Q) . h?Bb 7QHHQrb 7`QK i?2 7+i i?i g Bb mMB7Q`KHv #QmM/2/- b r2 `2K`F2/ 2`HB2`M/ i?i & & &2  &2  & &2

& & & & & & 2iπνt E &h (ν, Z)& = E && h (t, Z) e dt&& ≤ E && h (t, Z) dt&& , Rm

Rm

 }MBi2 +QMbiMi U#v ?vTQi?2bBb URkXj9VVX "X h?Bb Bb +H2`Hv  T`iB+mH` +b2 Q7 X q2 MQi2 7Q` 7mim`2 `272`2M+2 i?i BM i?Bb +b2-

RkXjX h>1 "_hG1hh aS1*h_lJ P6 h>1 >qE1a S_P*1aa

93R



⎞ ∗ (ν, z) h  ⎠ . ϕ(ν, z) = f (ν) ⎝1 + 1 − E h∗ (ν, Z)

URkX9yV 

q2 MQr im`M iQ i?2 T`QQ7 Q7 h?2Q`2K RkXjXRX S`QQ7X 6Q` ϕ- i?2 mMB[m2 bQHmiBQM Q7 URkXjeV        ¯ (dt × dz) − λ (t)Q(dz)dt ϕ(t, z)M (dt × dz) = ϕ(t, z) N Rm K Rm K     ¯  (dt × dz) − = ϕ(t, z)N ϕ(t, z)λ (t)Q(dz)dt. Rm

HbQ



Rm

K

Rm

K

K

ϕ(t, z)λ (t)Q(dz)dt      ¯ (ds × dz  ) Q(dz)dt = ϕ(t, z) h(t − s, z  )N m Rm K R K ¯ (ds × dz  ) = h(t − s, z  )E [ϕ(t, Z)] dtN Rm ×Rm ×K     ˇ − ·, z  ) ∗ E [ϕ(·, Z)] (s) N ¯ (ds × dz  ). h(s = Rm

K

¯ =N ¯ + N ¯0 h?2`27Q`2- bBM+2 N    ϕ(t, z)M (dt × dz) Rm K       ˇ − ·, z) ∗ E [ϕ(·, Z)] (t) N ¯ (dt × dz) = ϕ(t, z) − h(t Rm  K ¯0 (dt × dz). − ϕ(t, z)N Rm

K

qBi? f ∈ BN UBM T`iB+mH` f ∈ L1C (Rm ) ∩ L2C (Rm )V- r2 ?p2       ¯0 (dt × dz) = ϕ(t, z)M (dt × dz) + ϕ(t, z)N Rm

K

Rm

K

f (t)N (dt). Rm

HbQ- #v G2KK RkXjX9      ϕ(t, z)M (dt × dz) = λ E |ϕ(t, Z)|2 dt o` m m R K R  =λ E |ϕ(ν, Z)|2 dν. Rm

LQr-

93k

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a 

E

   ¯0 (dt × dz) ϕ(t, z)M (dt × dz) ϕ(t, z)N Rm K Rm K &    

&  ¯0 (dt × dz) = 0. =E E ϕ(t, z)M (dt × dz)&& F0 ϕ(t, z)N 

Rm

Rm

K

K

h?2`27Q`2      ¯0 (dt × dz) = o` ϕ(t, z)M (dt × dz) + ϕ(t, z)N m Rm K  R K      ¯ o` ϕ(t, z)M (dt × dz) + o` ϕ(t, z)N0 (dt × dz) = Rm K Rm K     ¯0 (dt × dz) . λ E |ϕ(t, Z)|2 dt + o` ϕ(t, z)N Rm

Rm

K

PM i?2 Qi?2` ?M/- #v i?2 mMBp2`bH +Qp`BM+2 7Q`KmH UNX9RV   ¯ ϕ(t, z)N0 (dt × dz) o` Rm K   = |E [ϕ(ν, Z)]|2 μ0 (dν) + λ0 o` (ϕ(ν, Z)) dν. Rm

*QK#BMBM; i?2 #Qp2- r2 ?p2   o` f (t)N (dt) m R   =λ E |ϕ(ν, Z)|2 dν + Rm

Rm

 Rm

|E [ϕ(ν, Z)] |2 μ0 (dν)

+ λ0 Rm

o` (ϕ(ν, Z)) dν

= A + B + C. "v 6Q`KmH URkX9yV-

  1 + o` h(ν, Z)   dν, A = λ |f (ν)|2 Rm |1 − E h(ν, Z) |2  1   μ0 (dν), B= |f (ν)|2 Rm |1 − E h(ν, Z) |2    o` h(ν, Z)   dν. C= |f (ν)|2 λ0 Rm |1 − E h(ν, Z) |2 

_2+HHBM; i?i λ = ρλ0 /(1−ρ)- r2 Q#iBM }MHHv i?i 7Q` HH f ∈ L1C (Rm )∩L2C (Rm )  f (t)N (dt) o` Rm ⎛ ⎞      1   ⎠ μ0 (dν) + λρdν + λo` h(ν, Z) dν , = |f (ν)|2 ⎝ Rm |1 − E h(ν, Z) |2 M/ i?Bb HHQrb mb iQ B/2MiB7v μN b URkXj8VX



RkX9X 1s*h aJSGAL: P6 >qE1a S_P*1aa1a

RkX9

93j

1t+i aKTHBM; Q7 >rF2b S`Q+2bb2b

URy V aBM+2 i?2 >rF2b T`Q+2bb Bb  +Hmbi2` TQBMi T`Q+2bb- Bi Bb Mim`H iQ bF B7 i?2 2t+i bKTHBM; K2i?Q/ Q7 "`Bt M/ E2M/HH TTHB2bX AM i?Bb bT2+BH +b2- r2 b22F iQ Q#iBM QM  }MBi2 rBM/Qr W := [0, a] M 2t+i bKTH2 Q7  >rF2b T`Q+2bb N ≡ {Tn }n∈Z QM i?2 HBM2 rBi? `M/QK 72`iBHBiv `i2 h(t, Z)- r?2`2 Z Bb  `M/QK  ∞ 2H2K2Mi BM bQK2 K2bm`#H2 bT+2 (K, K)- M/ bm+? i?i ρ := E 0 h(t, Z) dt < 1X h?Bb T`Q+2bb Bb  +Hmbi2` TQBMi T`Q+2bb r?Qb2 ;2`K TQBMi T`Q+2bb N0 Bb  SQBbbQM T`Q+2bb rBi? BMi2MbBiv 7mM+iBQM μ  M/ bm+? i?i Zn Un ≥ 1V Bb  #`M+?BM; TQBMi T`Q+2bb Q7 `M/QK 72`iBHBiv `i2 h(t, Z) rBi?  bBM;H2 M+2biQ` HQ+i2/ i 0X AM T`BM+BTH2- i?2 "`BtĜE2M/HH T2`72+i bBKmHiBQM K2i?Q/ Q7 N QM i?2 }MBi2 rBM/Qr W = [0, a] TTHB2bX Ai +QMbBbib Q7 irQ bi2Tb, UBV :2M2`i2  SQBbbQM TQBMi T`Q+2bb QM (−∞, 0] rBi? BMi2MbBiv μ (t)P (L ≥ −t) r?2`2 L Bb i?2 ivTB+H H2M;i? U2tiBM+iBQM iBK2V Q7  #`M+?BM; TQBMi T`Q+2bb QM R+ rBi? `M/QK 72`iBHBiv `i2 h(t, Z) M/ rBi?  bBM;H2 M+2biQ` HQ+i2/ i 0X UBBV 6Q` 2+? Q7 i?2 TQBMib Q7 i?Bb SQBbbQM T`Q+2bb- ;2M2`i2 #`M+?BM; TQBMi T`Q+2bb2b Q7 `M/QK 72`iBHBiv `i2 h(t, Z) rBi?  bBM;H2 M+2biQ` HQ+i2/ i Tn mMiBH vQm Q#iBM QM2 rBi? TQBMib BM (0, ∞)X Uh?2 `iBQMH2 Bb i?i i?2 TQBMi Q7 N0 HQ+i2/ i −t ?b  T`Q##BHBiv P (L ≥ −t) Q7 ?pBM; TQBMib BM [0, ∞)XRR a22 ?Qr2p2` i?2 7Q`KH DmbiB}+iBQM BM am#b2+iBQM jXeXV 1t+i bKTHBM; `2[mB`2b i?i i?2 SQBbbQM T`Q+2bb QM (−∞, 0] Q7 BMi2MbBiv μ (t)P (L ≥ −t) #2 }MBi2X  Bb  ∞h?Bb BKTHB2b bQK2 `2bi`B+iBQMbX 6Q` BMbiM+2- B7 μ #QmM/2/- i?2 +QM/BiBQM 0 P (L > t) dt = E[L] < ∞ rBHH ;m`Mi22 i?iX AM T`iB+mH`- L Kmbi #2 }MBi2- i?i Bb +QMp2`;2M+2 Q7 i?2 +Hmbi2` iQ i?2 2KTiv TQBMi T`Q+2bb Ui?2 biiBQM`v bii2V Kmbi iF2 TH+2 BM }MBi2 iBK2X Bb  ∞ h?2 +QMp2`;2M+2 i?2M BM p`BiBQM- M/  bm{+B2Mi +QM/BiBQM 7Q` i?Bb Bb E 0 th(t, Z) dt < ∞X TTHB+iBQM Q7 i?2 "`BtĜE2M/HH K2i?Q/ `2[mB`2b i?2 +QMbi`m+iBQM Q7  SQBbbQM T`Q+2bb QM R+ Q7 BMi2MbBiv μ (t)P (L > t)XRk h?2 KBM T`Q#H2K Bb i?i bi2T UBV `2[mB`2b BM T`BM+BTH2 M 2t+i 2tT`2bbBQM 7Q` i?2 T`Q##BHBiv i?i L > t- r?2`2 L Bb i?2 H2M;i? Q7  #`M+?BM; TQBMi T`Q+2bb N rBi?  bBM;H2 M+2biQ` i 0 M/ rBi?  `M/QK 72`iBHBiv `i2 h(t, Z)X *H2`Hv P (L > t) = P (N ((t, ∞)) = 0) . M 2t+i 2tT`2bbBQM Q7 P (L > t) Bb MQi pBH#H2X >Qr2p2`- i?2 T`Q#H2K +M #2 +B`+mKp2Mi2/ BM i?2 7QHHQrBM; KMM2`X amTTQb2 i?i r2 FMQr 2tTHB+BiHv b2[m2M+2b Q7 MQM@M2;iBp2 7mM+iBQMb {ln }n≥1 M/ {un }n≥1 `2bT2+iBp2Hv MQM@/2+`2bBM; M/ MQM@BM+`2bBM;- M/ #Qi? +QMp2`;BM; TQBMirBb2 iQ P (L > t) BM bm+?  t  rv i?i ||un −ln ||∞ := supt≥0 |un (t)−ln (t)| dt → 0X amTTQb2 KQ`2Qp2` i?i 0 u0 (t) μ(t) dt < Ry

(JǠHH2` M/ _bKmbb2M- kyy9)X P7 +Qm`b2 W = [0, a] ⊆ [0, ∞)- #mi  HBiiH2 i?Qm;?i rBHH +QMpBM+2 i?2 `2/2` i?i r2 +M T`2i2M/ i?i i?2 ;QH Bb iQ bKTH2 N QM [0, ∞) rBi?Qmi //BiBQMH +Qbi bBM+2 i?2 TQBMib Q7 N0 7i2` a rBHH MQi #2 mb2/X Rk >2`2- iQ 7+BHBii2 MQiiBQM- iBK2 Bb `2p2`b2/- bQ i?i i?2 bm`pBpBM; TQBMib TT2` iQ #2 QM i?2 TQbBiBp2 ?H7@HBM2 `i?2` i?M QM i?2 M2;iBp2 ?H7@HBM2X RR

939

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

∞X h?Bb Bb i?2 +b2 BM T`iB+mH` B7 u0 (t) = 1−G(t) 7Q` bQK2 +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM QM R+ Q7 }MBi2 K2M M/ supt≥0 μ (t) < ∞X h?2 +QMbi`m+iBQM i?2M ;Q2b b 7QHHQrbX 6B`bi ;2M2`i2 i?2 TQBMib UBM }MBi2 MmK@ #2`V Q7  SQBbbQM T`Q+2bb Q7 BMi2MbBiv u0 (t) μ(t)X G2i t1 - Ę- tk #2 i?2b2 TQBMibX :2M@ 2`i2 M BB/ b2[m2M+2 V1 - Ę- Vk - Q7 `M/QK p`B#H2b mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X lM/2` i?2 +m`p2 y = u0 (t) μ(t)- i?2 TQBMib (t1 , V1 u0 (t1 ) μ(t1 ))- Ę- (tk , V1 u0 (tk ) μ(tk )) Q7 R2+ 7Q`K  SQBbbQM T`Q+2bb Q7 BMi2MbBiv 1X LQi2 i?i i?2 T`Q##BHBiv i?i Mv Q7 i?2b2 TQBMib HB2 QM i?2 +m`p2 y = μ (t)P (L > t) Bb MmHH- M/ i?2`27Q`2- Bi rBHH ?TT2M 7Q`  }MBi2 n i?i MQM2 Q7 i?2 TQBMib (t1 , V1 u0 (t1 ) μ(t1 ))- Ę- (tk , V1 u0 (tk ) μ(tk )) HB2b #2ir22M i?2 +m`p2b y = un (t) μ(t) M/ y = ln (t) μ(t)X 6Q` i?2 bBKmHiBQM- F22T QMHv i?2 TQBMib ti bm+? i?i Vi u0 (ti ) μ(ti ) ≤ n (ti ) μ(ti ) bBM+2 i?2b2 TQBMib `2 2t+iHv i?Qb2 HvBM; mM/2` i?2 +m`p2 y = P (L > t)X Ai `2KBMb iQ }M/ i?2 TT`QtBKiBM; 7mM+iBQMb un M/ ln X G2i Nk #2 N `2bi`B+i2/ iQ i?2 ;2M2`iBQMb 0- 1- Ę- kX AM T`iB+mH`- limk↑∞ Nk ((t, ∞)) = N ((t,  ∞))X *QK@   k ((t,∞))  −θ N 7Q` Mv θ > 0- iF2 i?2 HBKBi b k ↑ ∞ iQ Q#iBM E e−θN ((t,∞)) Tmi2 E e M/ i?2M Q#iBM i?2 `2bmHi 7`QK i?2 Q#b2`piBQM i?i    lim E e−θN ((t,∞)) = P (N ((t, ∞)) = 0) . θ↑+∞



  hQ +QKTmi2 E e−θNk ((t,∞)) - BMi`Q/m+2 i?2 `M/QK 2H2K2Mib Yk rBi? pHm2b BM (Mp (R+ ), Mp (R+ )) r?Qb2 /Bbi`B#miBQM Qk Bb i?i Q7 Nk X G2i MQr {Yk,n }n∈N #2 M BB/ b2[m2M+2 Q7 `M/QK 2H2K2Mib /Bbi`B#mi2/ b Yk X h?2 /Bbi`B#miBQM Q7 Nk ((t, ∞)) Bb i?2 bK2 b i?i Q7  1{τn ≤t} Yk−1,n (t − τn ) n≥0

r?2`2 i?2 τn Ƕb `2 i?2 TQBMib Q7 i?2 }`bi ;2M2`iBQM Q7 N - 7Q`KBM;  *Qt TQBMi T`Q+2bb Q7 `M/QK BMi2MbBiv h(t, Z)X h?2`27Q`2      E e−θNk ((t,∞)) = E e−θ n≥0 1{τn ≤t} Yk−1,n (t−τn )    = E e−θ n≥0 1{τn ≤t} f (Yk−1,n ,t−τn ) , r?2`2 7Q` μ ∈ Mp (R+ ) M/ u ∈ R+ - f (μ, u) := μ((u, ∞))- bQ i?i    E e−θ n≥0 1{τn ≤t} f (Yk−1,n ,t−τn )     t   −θf (μ,t−s) E exp − 1 h(s, Z) Qk−1 (dμ) ds e 

0

  t

Mp (R+ )

0

Mp (R+ )



E exp LQr-



e

−θμ((t−s,∞))





− 1 h(s, Z) Qk−1 (dμ) ds

    e−θμ((t−s,∞)) − 1 Qk−1 (dμ) = E e−θNk−1 ((t−s,∞))

 Mp (R+ )

.

()

RkX9X 1s*h aJSGAL: P6 >qE1a S_P*1aa1a

938

M/ i?2`27Q`2  t   

      E e−θNk ((t,∞)) = E exp E e−θNk−1 ((t−s,∞)) − 1 h(s, Z) ds . 0

hFBM; i?2 HBKBi b k ↑ ∞ vB2H/b  t   

     ((t,∞))  ((t−s,∞)) −θ N −θ N = E exp E e − 1 h(s, Z) ds E e 0

M/ iFBM; i?2 HBKBi b θ → +∞ ;Bp2b- rBi? f (t) = P (N ((t, ∞)) > 0)  

 t f (t) = E exp −ν(t, Z) + f (t − s)h(s, Z) ds , 0

t

r?2`2 ν(t, Z) := 0 h(s, Z) dsX h?2 `B;?i@?M/ bB/2 rBHH #2 /2MQi2/ #v Φ(f )X q2 `2 i?2`27Q`2 +QM+2`M2/ rBi? i?2 2[miBQM f = Φ(f )

(f ∈ A),

r?2`2 A := {f : (R+ , B(R+ )) → ([0, 1]), B([0, 1]))}X q2 b?Qr i?i i?2 bQHmiBQM F (t) := P (N ((t, ∞)) > 0 Bb i?2 mMB[m2 bQHmiBQM- M/ i?i i?2`2 2tBbi b2[m2M+2b {gn }n≥1 M/ {hn }n≥1 - `2bT2+iBp2Hv MQM@/2+`2bBM; M/ MQM@BM+`2bBM;- rBi? i?2 +QKKQM HBKBi F M/ bm+? i?i ||fn − gn ||∞ → 0 . h?2 7QHHQrBM; bm++2bbBQM Q7 `2K`Fb rBHH #2 mb27mHX UBV h?2 b2[m2M+2 Q7 7mM+iBQMb {fn }n≥1 /2}M2/ #v fn (t) = P (Nn ((t, ∞)) > 0) Bb MQM@/2+`2bBM;- ?b F 7Q` HBKBi- M/ r2 ?p2 b22M i?i Bi biBb}2b i?2 `2+m``2M+2 fn = Φ(fn−1 )

(n ≥ 1) ,

rBi? f0 (t) ≡ 1X UBBV G2i f ∈ A M/ H2i Φ(n) #2 /2}M2/ `2+m`bBp2Hv #v Φ(0) (f ) := f - Φ(n) (f ) := Φ (Φ(f ))- M/ H2i fn := Φ(n) (f )X q2 ?p2 i?i (n−1)

f ≤ g ⇒ fn ≤ g n , f ≤ Φ(f ) ⇒ {fn }n≥1 Bb MQM@/2+`2bBM;- M/ f ≥ Φ(f ) ⇒ {fn }n≥1 Bb MQM@BM+`2bBM;. h?2 b2+QM/ M/ i?B`/ bb2`iBQMb #Qp2 `2 BKK2/Bi2 +QMb2[m2M+2b Q7 i?2 }`bi QM2r?B+? Bb Bib2H7 M BKK2/Bi2 +QMb2[m2M+2 Q7 i?2 /2}MBiBQM Q7 ΦX UBBBV Φ Bb  +QMi`+iBQM QM A rBi? `2bT2+i iQ i?2 bmT MQ`KX JQ`2 T`2+Bb2Hv f, g ∈ A ⇒ ||Φ(f ) − Φ(g)||∞ ≤ ρ||f − g||∞ , r?2`2 ρ :=

 R

E [h(t, Z)] dt U < 1VX

93e

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

S`QQ7X "v i?2 K2M@pHm2 i?2Q`2K- 7Q` Mv `2H MmK#2`b a M/ b- ea −eb = (b−a)ec 7Q` bQK2 c ∈ [a, b]X AM T`iB+mH` &

&  t & & ||Φ(f ) − Φ(g)||∞ = sup &&E e−ν(t,Z) ec (f (t − s) − g(t − s))h(s, Z) ds && , t≥0

0

t r?2`2 i?2 [mMiBiv c = c(t, Z, f, g) HB2b #2ir22M 0 f (t − s)h(s, Z) ds M/ t g(t − s)h(s, Z) dsX AM T`iB+mH`- bBM+2 f, g ≤ 1- ec ≤ eν(t,Z) M/ i?2`27Q`2 0 &  t

& & & ||Φ(f ) − Φ(g)||∞ ≤ sup &&E (f (t − s) − g(t − s))h(s, Z) ds && t≥0

 ∞ 0 ||f − g||∞ h(s, Z) ds = ρ||f − g||∞ . ≤E 0

 UBpV F Bb i?2 mMB[m2 }t2/ TQBMi Q7 ΦX S`QQ7X h?2 bT+2 Q7 7mM+iBQMb A Bb +QKTH2i2 rBi? `2bT2+i iQ i?2 bmT MQ`K M/ Φ Bb  +QMi`+iBQM QT2`iQ` QM i?Bb bT+2X h?2`27Q`2- #v i?2 }t2/ TQBMi i?2Q`2K- Bi ?b  mMB[m2 }t2/ TQBMi- r?B+? r2 FMQr iQ #2 F X  UpV qBi? fn := Φ(n) (f )-

||fn − F ||∞ → 0 .

S`QQ7X ||fn − F ||∞ = ||Φ(fn−1 ) − Φ(F )||∞ ≤ ρ||fn−1 − F ||∞ ≤ ρn ||f − F ||∞ .  UpBV ||fn − F ||∞ ≤

ρn ||Φ(f ) 1−ρ

− f ||∞ X

S`QQ7X aBKBH`Hv iQ UpV- r2 ?p2 i?i ||fn − fn−1 )||∞ ≤ ρn−1 ||f1 − f )||∞ X HbQ #v UpV- ||F − f ||∞ = limm↑∞ ||fm − f ||∞ X h?2`27Q`2 ||F − f ||∞ ≤ lim (||f1 − f ||∞ + ||f2 − f1 ||∞ + · · · + ||fm − fm−1 ||∞ ) m↑∞

≤ lim ||f1 − f ||∞ (1 + ρ + · · · + ρm−1 ) ≤ m↑∞

||f1 − f ||∞ . 1−ρ 

UpBBV A7 f ≤ Φ(f ) Q` f ≥ Φ(f )- i?2M fn → F 7`QK #2HQr Q` 7`QK #Qp2 `2bT2+iBp2HvX h?Bb 7QHHQrb 7`QK UBBVX h?2 7mM+iBQMb un M/ n `2 MQr /2}M2/X aBM+2 0 ≤ Φ(0)- fnu := Φ(n) (0) ↓ F X hF2 n (t) = 1 − fnu (t) bQ i?i n ↑ 1 − F X JǠHH2` M/ _bKmbb2M 2t?B#Bi  +mKmHiBp2 /Bbi`B#miBQM 7mM+iBQM G rBi? }MBi2 K2M bm+? i?i G ≤ Φ(G)X h?2M fn := Φ(n) (G) ↑ F X hF2 un (t) = 1 − fn (t) bQ i?i un ↓ 1 − F X h?Bb Bb +`m+BH

RkX9X 1s*h aJSGAL: P6 >qE1a S_P*1aa1a

93d

bBM+2 i?2 H;Q`Bi?K bi`ib #v +QMbi`m+iBM; UbKTHBM;V  SQBbbQM T`Q+2bb QM R+ Q7 BMi2MbBiv u0 (t) μ(t) = (1 − G(t)) μ(t) Q7 }MBi2 KbbXRj

h?2 *b2 Q7 MQM@SQBbbQMBM :2`K S`Q+2bb2b  bHB;?i KQ/B}+iBQM Q7 i?2 JǠHH2`Ĝ_bKmbb2M i?BMMBM; T`Q+2/m`2 HHQrb mb iQ i`2i i?2 +b2 Q7  ;2`K T`Q+2bb i?i Bb  `2M2rH T`Q+2bbXR9 G2i N0 #2 M mM/2Hv2/ `2M2rH b2[m2M+2 QM (0, +∞), 7Q` n ≥ 1- X0,n = S1 + · · · + Sn r?2`2 i?2 b2[m2M+2 Q7 MQM@M2;iBp2 `M/QK p`B#H2b {Sn }n≥1 Bb BB/rBi?  +QKKQM /Bbi`B#miBQM /KBiiBM;  /2MbBiv f - rBi?  +Q``2bTQM/BM; 7BHm`2 `i2 r(t) := 1− f f(t)(s) ds mMB7Q`KHv #QmM/2/ #v M < ∞X h?2 b2[m2M+2 Q7 TQBMib Q7 0 i?Bb `2M2rH T`Q+2bb Kv #2 i?Qm;?i Q7 b #2BM; Q#iBM2/ #v i?BMMBM;  SQBbbQM T`Q+2bb N 0 Q7 `i2 M - rBi?  b2[m2M+2 Q7 2p2Mi@iBK2b {X 0,n }n≥1 X h?i Bb {X0,n , n ≥ 1} ⊂ {X 0,n , n ≥ 1}X q2 ?p2 iQ i?BM N0 rBi? i?2 i?BMMBM; T`Q##BHBiv p(x)X q2 TTHv i?Bb i?BMMBM; iQ i?2 /QKBMiBM; TQBMi T`Q+2bb N 0 - M/ r2 MQr FMQr ?Qr iQ /Q i?Bb #v i?2 TT`QtBKiBM; b2[m2M+2 K2i?Q/ /2b+`B#2/ #Qp2X h?2 MmK#2` Q7 TQBMib Q7 i?2 i?BMM2/ /QKBMi2/ SQBbbQM T`Q+2bb Bb }MBi2X h?2 TQBMib Q7 i?2 i?BMM2/ N0 `2 i?Qb2 TQBMib Q7 N0 i?i ?p2 bm`pBp2/ M/ `2 Q7 i?2 ivT2 X0,n X LQr- bi`i rBi? i?2 bKTH2 Q7  SQBbbQM T`Q+2bb Q7 BMi2MbBiv M P (L > t) Dmbi Q#iBM2/X G2i t1 - Ę- tk #2 Bib TQBMibX G2i mb HQQF i i?2 }`bi TQBMi- t1 M/ /`r  `M/QK MmK#2` V1 mMB7Q`KHv /Bbi`B#mi2/ QM [0, 1]X amTTQb2 r2 FM2r i?2 pHm2 P (L > t1 )X h?2M r2 rQmH/ +?2+F B7 V1 M P (L > t1 ) Bb #2HQr Q` #Qp2 r(t1 )M P (L > t1 )- M/ /2+B/2 i?i t1 Bb Q` Bb MQi i?2 }`bi TQBMi Q7 i?2 i?BMM2/ `2M2rH T`Q+2bbX hQ /2+B/2 i?Bb- r2 +?2+F i?2 `2HiBp2 TQbBiBQMb Q7 V1 M un (t) M/ V1 M ln (t) rBi? `2bT2+i iQ i?2 i?`2b?QH/ r(t1 ) rBi? n bm{+B2MiHv H`;2 iQ ?p2 #Qi? TQBMib 2Bi?2` #Qp2 Q` #2HQr i?2 i?`2b?QH/X A7 #2HQr- iF2 t1 iQ #2 i?2 }`bi 2p2Mi iBK2 Q7 i?2 i?BMM2/ `2M2rH T`Q+2bbX A7 MQi- `2TH+2 t1 #v t2 M/ `2T2i i?2 T`Q+2@ /m`2 b HQM; b vQm ?p2  M2;iBp2 Mbr2`X Ai Kv ?TT2M i?i vQm 2M/ mT 7i2` k i`BHb rBi? MQ TQBMiX amTTQb2 MQr i?i i?2 }`bi TQBMi i?i Bb `2iBM2/ b  TQBMi Q7 i?2 i?BMM2/ `2M2rH T`Q+2bb KQM; i?2 HBbi t1 - Ę- tk Bb t - < kX q2 b?HH 2tKBM2 t+1 b r2 /B/ 7Q` t1 - i?Bb iBK2 rBi? i?2 i?`2b?QH/ r(t+1 − t )- M/ bQ QMX h?2 +b2 Q7  ;2`K TQBMi T`Q+2bb- i?i Bb- r?2M i?2 iBK2 tBb Bb `2p2`b2/ /2Hv2/ TQBMi T`Q+2bb Bb bBKBH`- KmiiBb KmiM/BbX JQ`2 ;2M2`HHv- i?2 +b2 r?2`2 i?2 ;2`K T`Q+2bb Bb- r?2M i?2 iBK2 tBb Bb `2p2`b2/-  TQBMi T`Q+2bb rBi? biQ+?biB+ BMi2MbBiv rBi? `2bT2+i iQ Bib BMi2`MH ?BbiQ`v λ(t) ≤ M `2+2Bp2b  bBKBH` i`2iK2Mi- mbBM; i?2 `2;2M2`iBp2 7Q`K Q7 i?2 biQ+?biB+ BMi2MbBivX lM/2` //BiBQMH +QM/BiBQMb i?i rBHH MQi #2 /Bb+mbb2/ ?2`2- i?2 #QmM/2/M2bb bbmKTiBQM +M HbQ #2 bmTT`2bb2/X h?2 +?QB+2 fn := Φ(n) (1) rQmH/ H2/ iQ i?2 +Q``2+i HBKBi fn ↑ F - #mi BM i?Bb +b2 i?2 SQBbbQM T`Q+2bb Q7 BMi2MbBiv u0 (t) μ(t) = μ (t) +QmH/ ?p2 M BM}MBi2 MmK#2` Q7 TQBMib- 7Q` BMbiM+2 B7 μ (t) Bb  TQbBiBp2 +QMbiMiX q2 `272` iQ (JǠHH2` M/ _bKmbb2M- kyy9) 7Q` i?2 /2iBHb +QM+2`MBM; i?2 Q#i2MiBQM Q7 bm+? BMBiBH pHm2 Q7 i?2 mTT2` #QmM/BM; T`Q+2bbX R9 ("`ûKm/- kyRe)X Rj

933

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

RkX8 "`M+?BM; SQBMi S`Q+2bb2b qBi?Qmi M+2biQ` h?Bb b2+iBQM 72im`2b +`BiB+H >rF2b TQBMi T`Q+2bb2b U?TTV rBi? }MBi2 p2`;2 BM@ i2MbBiB2bX *`BiB+HBiv K2Mb i?i i?2 72`iBHBiv `i2 BMi2;`i2b iQ 1 Ui?Bb +Q``2bTQM/b iQ i?2 mbmH +`BiB+H #`M+?BM; T`Q+2bbV M/- BM i?2 +QMi2ti Q7 >rF2b TQBMi T`Q@ +2bb2b rBi?  }MBi2 p2`;2 BMi2MbBiv- Bi Bb 2[mBpH2Mi iQ i?2 #b2M+2 Q7 M+2biQ`bX h?2 }`bi [m2biBQM Bb- /Q bm+? T`Q+2bb2b 2tBbi\ h?2 p2`;2 BMi2MbBiv λ Q7  HBM2` biiBQM`v >rF2b TQBMi T`Q+2bb QM i?2 HBM2 biBb}2b  ∞ λ=ν+λ h(t)dt , URkX9RV 0

M/ i?2`27Q`2- r?2M ν > 0- i?2 bi#BHBiv +QM/BiBQM ρ < 1 Bb M2+2bb`v M/ bm{+B2Mi 7Q` N iQ ?p2  }MBi2 p2`;2 BMi2MbBivX  >rF2b #`M+?BM; TQBMi T`Q+2bb rBi?Qmi M+2biQ`b Bb  biiBQM`v TQBMi T`Q@ +2bb N rBi? i?2 biQ+?biB+ BMi2MbBiv  h(t − s)N (ds) . URkX9kV λ(t) = (−∞,t)

 A7 bm+?  TQBMi T`Q+2bb 2tBbib- M2+2bb`BHv λ = λ h- M/ i?2`27Q`2- i?2 `2[mB`2K2Mi Q7  }MBi2 M/ MQM@MmHH p2`;2 BMi2MbBiv BKTHB2b i?i  ∞ h(t)dt = 1 , URkX9jV 0

r?B+? +Q``2bTQM/b iQ i?2 +`BiB+H +b2 Q7 #`M+?BM; T`Q+2bb i?2Q`vX *`BiB+H /vMKB+b Ui?i Bb- rBi?  biQ+?biB+ BMi2MbBiv Q7 ivT2 URkX9kVV rBi?  }MBi2 MQM@MmHH p2`;2 BMi2MbBiv Kv ?p2 MQ bQHmiBQM- b i?2 7QHHQrBM; `2bmHi b?QrbX h?2Q`2K RkX8XR UR8 V G2i N #2  biiBQM`v 2`;Q/B+ >rF2b TQBMi T`Q+2bb rBi?Qmi M+2biQ`b Ui?i Bb- bm+? i?i ν = 0V- M/ bbmK2 i?i i?2 72`iBHBiv `i2 7mM+iBQM h biBb}2b i?2 b?Q`i@`M;2 +QM/BiBQM  ∞ th(t)dt < ∞ . URkX99V 0

h?2M i?2 p2`;2 BMi2MbBiv λ := EN ([0, 1) Bb 2Bi?2` 2[mH iQ y Q` +∞- i?i Bb- i?2`2 /Q MQi 2tBbi MQM@i`BpBH +`BiB+H >rF2b T`Q+2bb2b rBi? b?Q`i@`M;2 72`iBHBiv `i2X S`QQ7X q`Bi2 + P (N (R+ ) = 0) = E [P  (N (R,) = 0 | F0 )] - ∞ = E exp − 0 dt (−∞,0) N (ds)h(t − s)  , -  ∞ ≥ exp −E 0 dt (−∞,0) N (ds)h(t − s)   ∞ = exp −λ 0 th(t)dt , R8

("`ûKm/ M/ JbbQmHBû- kyyR)X

RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_

93N

r?2`2 i?2 b2+QM/ 2[mHBiv Bb  /B`2+i TTHB+iBQM Q7 i?2 `2bmHi Q7 1tKTH2 8XRXR9- i?2 BM2[mHBiv 7QHHQrb 7`QK C2Mb2MǶb BM2[mHBiv- M/ i?2 Hbi bi2T Bb  bBKTH2 TTHB+iBQM Q7 6m#BMBǶb i?2Q`2KX bbmKBM; i?2 p2`;2 BMi2MbBiv λ Bb }MBi2- mM/2` i?2 b?Q`i `M;2 ?vTQi?2bBb URkX99V- i?Bb +H+mHiBQM vB2H/b P (N (R+ ) = 0) > 0. LQi2 i?i 2`;Q/B+Biv Q7 N BKTHB2b i?i i?2 2p2Mi {N (R+ ) = 0} ?b T`Q##BHBiv 2Bi?2` y Q` Rc Bi i?mb 7QHHQrb i?i- rBi? T`Q##BHBiv R- N (R+ ) = 0 M/- +QMb2[m2MiHvλ = 0X  _2K`F RkX8Xk 6B`bi Q#b2`p2 i?i B7 r2 ?p2 irQ BM/2T2M/2Mi >rF2b T`Q+2bb2b N1 M/ N2 rBi?Qmi M+2biQ`b +Q``2bTQM/BM; iQ i?2 bK2 72`iBHBiv `i2- i?2 TQBMi T`Q+2bb N = N1 + N2 Bb HbQ  >rF2b T`Q+2bb rBi?Qmi M+2biQ`bX HbQ- B7  >rF2b T`Q+2bb rBi?Qmi M+2biQ` Q7 p2`;2 BMi2MbBiv λ Bb BM/2T2M/2MiHv i?BMM2/ rBi? T`Q#@ #BHBiv p- i?2 `2bmHiBM; TQBMi T`Q+2bb Bb Q7  bBKBH` Mim`2- rBi? p2`;2 BMi2MbBiv pλX AM T`iB+mH`- Mv >rF2b T`Q+2bb rBi?Qmi M+2biQ`b Bb M BM}MBi2Hv /BpBbB#H2 TQBMi T`Q+2bbX _2+HH i?i i?2 TQr2` bT2+i`H K2bm`2 μN Q7 i?2 TQBMi T`Q+2bb N /KBib  /2MbBiv fN ν fN (ω) := . URkX98V  2π(1 − h)|1 − h(ω)|2 Uh?2 6Qm`B2` i`Mb7Q`Kb `2 7`QK MQr QM BM i2`Kb Q7 i?2 TmHbiBQM ω = 2πν 7Q` MQiiBQMH +QMp2MB2M+2XV h?2 bT2+i`H /2MbBiv i2M/b iQ  }MBi2 +QMbiMi fN (0) = ν/(2π(1 − h)3 ) b ω i2M/b iQ x2`QX h?Bb Kv #2 pB2r2/ b M BM/B+iBQM Q7  b?Q`i `M;2 /2T2M/2M+2 T`QT2`iv, 7Q` BMbiM+2- B7 QM2 BMp2biB;i2b i?2 #2?pBQm` Q7 p`N (0, T ] b T → ∞- mbBM; i?2 2tT`2bbBQM (eiωT −1)/(iω) Q7 i?2 6Qm`B2` i`Mb7Q`K Q7 1[0,T ] - Bi Bb 2bBHv b22M i?i  p`N (0, T ] ∼ T fN (0)

R

|eiu − 1|2 du = 2πT fN (0) u2

Ui?2 BMi2;`H 2[mHb 2π- #v i?2 SHM+?2`2HĜS`b2pH B/2MiBivV r?B+? Bb [mHBiiBp2Hv B/2MiB+H iQ i?2 #2?pBQm` Q7 i?2 SQBbbQM T`Q+2bb- M/ i?2 Hii2` +2`iBMHv /2b2`p2b iQ #2 +HH2/ b?Q`i@`M;2 /2T2M/2Mi- #2+mb2 Bi ?b BM/2T2M/2Mi BM+`2K2MibX h?mb >rF2b T`Q+2bb2b 7Q` r?B+? i?2 bi#BHBiv +QM/BiBQM h < 1 ?QH/b `2 b?Q`i@`M;2 /2T2M/2MiX >Qr2p2`- i ν = 0fN (0) :=

λ  , 2π|1 − h|2

M/ i?2`27Q`2- b QM2 TT`Q+?2b i?2 +`BiB+H +b2- fN (0) #2+QK2b BM+`2bBM;Hv H`;2X q2 MQr T`Qp2 i?2 2tBbi2M+2 Q7 +`BiB+H >rF2b T`Q+2bb2b rBi?  }MBi2 M/ MQM@ MmHH p2`;2 BMi2MbBivX h?2 bT2+B}+ +QM/BiBQMb i?i r2 BKTQb2 QM i?2 72`iBHBiv `i2- #2bB/2b URkX9jV- `2

9Ny

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a sup t1+α h(t) ≤ R ,

URkX9eV

lim t1+α h(t) = r ,

URkX9dV

t≥0

t→∞

7Q` }MBi2 +QMbiMib r, R > 0 M/ r?2`2 α ∈ (0, 12 )X h?2 7QHHQrBM; `2bmHi rBHH #2 T`Qp2/ i i?2 2M/ Q7 i?Bb bm#b2+iBQMX h?2Q`2K RkX8Xj h?2`2 2tBbib  MQMi`BpBH biiBQM`v >rF2b TQBMi T`Q+2bb rBi?@ Qmi M+2biQ`b rBi? }MBi2 BMi2MbBiv- T`QpB/2/ i?2 72`iBHBiv `i2 BMi2;`i2b iQ 1 M/ biBb}2b i?2 +QM/BiBQMb URkX9eV M/ URkX9dV- r?2`2 α ∈ (0, 12 )X h?2 T`QQ7 Q7 2tBbi2M+2 Bb MQM@+QMbi`m+iBp2 M/ mb2b r2F +QMp2`;2M+2 `;m@ K2MibX 6Q` i?Bb- r2 b?HH T`Q+22/ b 7QHHQrb, 7Q` ε ∈ (0, 1)- +QMbB/2` i?2 biiBQM`v >rF2b T`Q+2bb rBi? T`K2i2`b hε = (1 − ε)h- M/ ν ε = ελ- 7Q` bQK2 }t2/ +QM@ biMi λ > 0X Ai Bb r2HH /2}M2/- bBM+2 i?2 bi#BHBiv +QM/BiBQM ?QH/b- M/ ?b p2`;2 BMi2MbBiv λ 7Q` HH ε > 0X b h?2Q`2K RkX8XR b?Qrb- B7 r2 dzb2M/ i?2 M+2biQ`b iQ BM}MBivǴ- i?2B` 72`iBHBiv `i2 Kmbi 2t?B#Bi bQK2 HQM;@iBH T`QT2`iv B7 QM2 rMib  MQMi`BpBH HBKBi T`Q+2bbX h?2 b2i Q7 +QM/BiBQMb +?Qb2M Bb QM2 Q7 i?2KX G2KK RkX8X9 (i) q?2M URkX9eV M/ URkX9dV `2 biBb}2/ eiu − 1 −α lim (h(ω) − 1)ω = r du, 0 < α < 1 . α+1 ω→0 R+ u

URkX93V

(ii) A7 BM //BiBQM α < 12 - i?2M i?2 p`BM+2b V ε = p`N ε [0, T ] `2 #QmM/2/ b ε → 0- M/ +QMp2`;2 b ε → 0 iQ i?2 }MBi2 HBKBi  |eiωT − 1|2 λ V0 = dω . URkX9NV 2π R ω 2 |1 − h(ω)|2 S`QQ7X (i) 7i2`  +?M;2 Q7 p`B#H2b- i?2 H27i@?M/ bB/2 Q7 URkX93V `2/b  ω −1−α (eiu − 1)h(u/ω)du . R+

h?2 KQ/mHmb Q7 i?2 BMi2;`M/ Bb- BM pB2r Q7 URkX9eV- H2bb i?M R|eiu −1|u−α−1 - r?B+? Bb BMi2;`#H2 r?2M α < 1X h?2`27Q`2 URkX93V 7QHHQrb 7`QK URkX9dV #v /QKBMi2/ +QMp2`;2M+2X (ii) h?2 bT2+i`H 7Q`KmH TTHB2/ iQ N ε rBi? i?2 7mM+iBQM f = 1[0,T ] vB2H/b  |eiωT − 1|2 λ Vε = dω. 2π R ω 2 |1 − (1 − ε)h(ω)|2 h?2 BMi2;`M/ Bb KtBKBx2/ 7Q` 1 − ε = _2(h(ω))/|h(ω)|2 X h?Bb 2Mbm`2b i?i i?2 BMi2;`M/ Bb #QmM/2/ #Qp2 #v |eiωT − 1|2 |h(ω)|2 × , 2 ω |AKh(ω)|2

RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_

9NR

r?2`2 AKz /2MQi2b i?2 BK;BM`v T`i Q7 i?2 +QKTH2t MmK#2` zX AM i?2 M2B;?#Qm`@ ?QQ/ Q7 y- bv BM bQK2 /QKBM |ω| ≤ ωc - i?Bb mTT2` #QmM/ Bb BMi2;`#H2- bBM+2 BM pB2r Q7 URkX93V i?2 b2+QM/ 7+iQ` Bb Q7 Q`/2` |ω|−2α - M/ r2 ?p2 bbmK2/ α < 12 X AM i?2 +QKTH2K2Mi`v /QKBM |ω| > ωc - MQiB+2 }`bi i?i i?2 7mM+iBQM ω → |h(ω)| Bb +QMiBMmQmb- ;Q2b iQ x2`Q i BM}MBiv #v i?2 _B2KMMĜG2#2b;m2 H2KK- M/ Bb i?2`2@ 7Q`2 #QmM/2/ #Qp2 #v bQK2 +QMbiMi θ(ωc ) > 0X Ai Bb 2bBHv b22M i?i M2+2bb`BHvθ(ωc ) < 1X PM2 i?mb ?b |ω| > ωc ⇒ |1 − (1 − ε)h(ω)| ≥ 1 − (1 − ε)θ(ωc ) ≥ 1 − θ(ωc ) > 0 . h?2 BMi2;`M/ BM i?2 /2}MBiBQM Q7 V ε Bb i?mb #QmM/2/ #Qp2 #v bQK2 BMi2;`#H2 7mM+iBQM r?B+? /Q2b MQi /2T2M/ QM εX h?2 +QM+HmbBQMb Q7 i?2 H2KK i?2M 7QHHQr #v /QKBMi2/ +QMp2`;2M+2X  _2K`F RkX8X8 LQi2 i?i- BM T`iB+mH`- i i?2 HBKBi- i?2 TQr2` bT2+i`H /2MbBiv Bb +QMbiMi × |ω −2α | BM i?2 M2B;?#Qm`?QQ/ Q7 i?2 x2`Q 7`2[m2M+v- M/ i?2 HBKBi Q7 o`N [0, T ] = O(T 1+2α ) b T → ∞X AM i?Bb b2Mb2- i?2 HBKBi T`Q+2bb 2t?B#Bib HQM;@ `M;2 /2T2M/2M+2X >Qr2p2`- BM Q`/2` iQ KF2 i?Bb `;mK2Mi `B;Q`Qmb- r2 ?p2 iQ T`Qp2 i?i i?2 HBKBi Q7 i?2 p`BM+2 Bb BM/22/ i?2 p`BM+2 Q7 i?2 HBKBiX h?Bb rBHH #2 /QM2 Hi2` Uh?2Q`2K RkX8XRRVX S`QQ7X UQ7 h?2Q`2K RkX8XjV AM pB2r Q7 G2KK RkX8X9- i?2 7KBHv Q7 `M/QK p`B@ #H2b N ε [0, T ] Bb iB;?iX h?Bb 2Mbm`2b i?i i?2 }MBi2@/BK2MbBQMH /Bbi`B#miBQMb Q7 i?2 T`Q+2bb2b N ε HbQ 7Q`K  iB;?i 7KBHvX b  +QMb2[m2M+2- i?2 Hrb Q7 i?2 N ε `2 iB;?iXRe *QMbB/2` i?2M Mv HBKBi /Bbi`B#miBQM b ε → 0X q2 }`bi 2bi#HBb? i?i i?Bb /Bbi`B#miBQM Bb BM/22/ i?i Q7  >rF2b T`Q+2bb rBi? T`K2i2`b ν = 0 M/ hX hQ b22 i?Bb- Bi Bb 2MQm;? iQ T`Qp2 i?i





b

E [1A N ((a, b])] = E 1A

λ(s) ds a

7Q` HH a < b- M/ HH 2p2Mib A BM bQK2 b2KB`BM; ;2M2`iBM; FaN X hF2 A Q7 i?2 7Q`K {N (C1 ) = k1 , . . . , N (Cn ) = kn } 7Q` BMi2;2`b k1 , . . . , kn M/ K2bm`#H2 bm#@ b2ib C1 , . . . , Cn ∈ (−∞, a]X AM Q`/2` iQ b?Qr i?i i?Bb 7Q`KmH ?QH/b i`m2 r?2M N Bb i?2 +Q``2bTQM/BM; HBKBiBM; T`Q+2bb- QM2 b?QmH/ TTHv i?2 7Q`KmH iQ N ε M/ +?2+F i?i QM2 +M ;Q iQ i?2 HBKBi ε → 0 BM i?2 7Q`KmHX "mi N (a, b]1A Bb 2[mH iQ g(N (a, b], N (C1 ), . . . , N (Cn )) 7Q` bQK2 +QMiBMmQmb 7mM+iBQM g UiFBM; /pMi;2 Q7 i?2 7+i i?i i?2b2 `M/QK p`B#H2b `2 BMi2;2`@pHm2/VX h?2`27Q`2g(N ε (a, b], N ε (C1 ), . . . , N ε (Cn )) +QMp2`;2b r2FHv iQ g(N (a, b], N (C1 ), . . . , N (Cn ))X hQ b?Qr i?i i?2 HBKBiBM; T`Q+2/m`2 Bb pHB/ 7Q` i?2 H27i@?M/ bB/2- Bi bm{+2b iQ b?Qr i?i i?2 p`B#H2b g(N ε (a, b], N ε (C1 ), . . . , N ε (Cn )) `2 mMB7Q`KHv BMi2;`#H2XRd h?Bb Bb i`m2 bBM+2 i?2v `2 #QmM/2/ #v i?2 `M/QK p`B#H2b N ε (a, b] r?Qb2 b2+QM/ KQK2Mib `2 #QmM/2/- b b?QrM BM G2KK RkX8X9X  Re Rd

(.H2v M/ o2`2@CQM2b- RN33)- h?2Q`2K NXRXoA- TX kd9X ("BHHBM;bH2v)- h?2Q`2K 8X9X

9Nk

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a G2i





b

ϕ(N ) :=

h(t − s)N (ds)1A .

dt a

(−∞,t)

hQ T`Qp2 i?i i?2 HBKBiBM; T`Q+2/m`2 Bb pHB/ 7Q` i?2 `B;?i@?M/ bB/2- QM2 ?b iQ b?Qr i?i i?2 2tT2+iiBQM Q7 ϕ(N ε ) i2M/b iQ i?i Q7 ϕ(N )X LQr ϕ(N ) Bb  +QMiBMmQmb 7mM+iBQM Q7 N 7Q` i?2 TT`QT`Bi2 iQTQHQ;v QM i?2 +MQMB+H bT+2 Q7 HQ+HHv }MBi2 TQBMi T`Q+2bb2bc b22 a2+iBQM RX9X h?2`27Q`2 ϕ(N ε ) +QMp2`;2b r2FHv iQ ϕ(N )- bBM+2 N ε +QMp2`;2b r2FHv iQ N X Ai `2KBMb iQ b?Qr i?i i?2 p`B#H2b ϕ(N ε ) `2 mMB7Q`KHv BMi2;`#H2X .2}MBM;  b  dt h(t − s)N ε (ds)1A , Y ε := a

(−∞,t)

Bi Bb 2MQm;? iQ b?Qr i?i i?2 Y ε ?p2  #QmM/2/ b2+QM/ KQK2MiX TTHvBM; i?2 /2}MBM; 7Q`KmH UNXR9V Q7 i?2 "`iH2ii bT2+i`H K2bm`2- r2 Q#iBM  E(Y ε )2 = [λ(b − a)]2 + |f (ω)|2 fN ε (ω)dω, R

r?2`2



b−s

f (s) :=

h(u)du. a−s

h?2 b2+QM/ KQK2Mi Q7 Y ε Bb mMB7Q`KHv #QmM/2/ r?2M i?2 7mM+iBQM |f (ω)|2 Bb BMi2@ ;`#H2 i BM}MBivX lbBM; i?2 7+i i?i f Bb /Bz2`2MiB#H2 rBi? BMi2;`#H2 /2`BpiBp2Bi 7QHHQrb 7`QK i?2 _B2KMMĜG2#2b;m2 H2KK i?i |f (ω)| = O(ω −1 ) b ω → ∞?2M+2 i?2 mMB7Q`K BMi2;`#BHBiv Q7 i?2 Y ε - rBi? i?2 bK2 T`QQ7 b i?i Q7 T`i UiiV Q7 G2KK RkX8X9X *QmTHBM; S`QT2`iB2b q2 MQr //`2bb irQ BMiBKi2Hv +QMM2+i2/ Bbbm2bX h?2 }`bi QM2 Bb i?2 #2?pBQm` Q7 i`MbB2Mi +`BiB+H T`Q+2bb2b- M/ BM T`iB+mH`- i?2B` +QMp2`;2M+2 BM bQK2 b2Mb2 iQ biiBQM`BivX h?2 b2+QM/ QM2 Bb i?2 [m2biBQM Q7 mMB[m2M2bbX LQi2 rBi? `2bT2+i iQ i?2 Hii2` Bbbm2 i?i i?2 bmK Q7 irQ BM/2T2M/2Mi +`BiB+H >rF2b T`Q+2bb2b rBi? i?2 bK2 72`iBHBiv `i2 h Bb  +`BiB+H >rF2b T`Q+2bb2b rBi? i?2 bK2 72`iBHBiv `i2 hX h?mb mMB[m2M2bb +M #2 2tT2+i2/ iQ ?QH/ QMHv B7 QM2 //b  +QMbi`BMi- bv- iQ ?p2  }t2/ p2`;2 BMi2MbBivX h?2 Bbbm2 Q7 +QMp2`;2M+2 rBHH #2 i`2i2/ }`bi- pB +QmTHBM;X *QMbB/2` irQ /Bb@ iBM+i TQBMi T`Q+2bb2b N1 - N2 M/ bbmK2 i?i 7Q` i = 1, 2- Ni /KBib QM R+ i?2 biQ+?biB+ BMi2MbBiv  λi (t) = νi (t) + h(t − s)Ni (ds) , URkX8yV (0,t)

r?2`2 νi Bb bQK2 MQM@M2;iBp2 F0 @K2bm`#H2 T`Q+2bbX 6Q` BMbiM+2- B7 i?2b2 T`Q@ +2bb2b `2 /2}M2/ QM R− b r2HH- i?2 T`iB+mH` 7Q`K  νi (t) = h(t − s)Ni (ds) URkX8RV (−∞,0]

RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_

9Nj

+Q``2bTQM/b iQ i?2 +HbbB+H >rF2b T`Q+2bb /vMKB+bX "v :`B;2HBQMBbǶb BK#2//BM; i?2Q`2K- Bi Kv #2 bbmK2/ i?i i?2`2 2tBbib M ?TT N QM R2 - BM/2T2M/2Mi Q7 F0 bm+? i?i 7Q` i = 1, 2- QM R+ Ni (dt) = N (dt × [0, λi (t)]), r?2`2 λi (t) = νi (t) +

 (0,t)

URkX8kV

h(t − s)Ni (ds)X

LQi2 i?i i?Bb +QMbi`m+iBQM Kv vB2H/ BKT`QT2` Ui?i Bb- MQi HQ+HHv }MBi2V TQBMi T`Q+2bb2bX >Qr2p2` B7 i?2 biQ+?biB+ BMi2MbBiB2b `2 HQ+HHv BMi2;`#H2- i?2 TQBMi T`Q+2bb2b `2 T`QT2`X hQ ;m`Mi22 i?Bb- Bi bm{+2b i?i ν(t) U7Q` +QMp2MB2M+2 r2 KQK2Mi`BHv /`QT i?2 BM/B+2bV #2 HQ+HHv BMi2;`#H2X AM/22/ λ(t) = ν(t) + h(t − s)N (ds). (0,t)

hFBM; 2tT2+iiBQMb M/ Bi2`iBM; i?Bb 2[miBQM- QM2 Q#iBMb i?2 7QHHQrBM; #QmM/,  ν ∗ h∗n (t) . URkX8jV E[λ(t) | F0 ] = ν(t) + n>0

AMi2;`i2/ #2ir22M y M/ T - i?2 `B;?i@?M/ bB/2 2[mHb T ν(t)dt 0 , T 1 − 0 h(t)dt r?B+? Bb }MBi2 mM/2` Qm` bbmKTiBQMb QM i?2 7mM+iBQM hX >2M+2- E[λ(t) | F0 ] Bb XbX HQ+HHv BMi2;`#H2X q2 MQr ;Bp2 +QM/BiBQMb mM/2` r?B+? i?2 irQ b?B7i2/ T`Q+2bb2b St N1 M/ St N2 #2+QK2 +HQb2 BM bQK2 b2Mb2 b t → ∞X G2KK RkX8Xe bbmK2 i?i #Qi? ν1 M/ ν2 `2 HQ+HHv BMi2;`#H2- M/ i?i lim

t→∞

|ν1 (t) − ν2 (t)| ∞ = 1 XbX h(s)ds t

URkX89V

6Q` HH a < blim P [N1 = N2 QM (t + a, t + b] | F0 ] = 0 XbX

t→∞

URkX88V

S`QQ7X h?2 TQBMi T`Q+2bb Δ /2}M2/ #v Δ(dt) = |N1 (dt) − N2 (dt)| /KBib QM R+ i?2 biQ+?biB+ BMi2MbBiv  δ(t) = |ν1 (t) − ν2 (t) + h(t − s)(N1 (ds) − N2 (ds))| . (0,t)

AM T`iB+mH`-

 δ(t) ≤ |ν1 (t) − ν2 (t)| +

h(t − s)Δ(ds) . (0,t)

9N9

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

hFBM; 2tT2+iiBQMb M/ Bi2`iBM; i?Bb 2[miBQM vB2H/b i?2 #QmM/,  |ν1 − ν2 | ∗ h∗n (t) . E[δ(t) | F0 ] ≤ |ν1 − ν2 |(t) +

URkX8eV

n>0

∞ 6Bt ε > 0- M/ b2H2+i T > 0 bQ i?i 7Q` u ≥ T - |ν1 − ν2 |(u) ≤ ε u h(s)ds Ubm+? T 2tBbib XbX 7Q` HH ε > 0 #v bbmKTiBQM i URkX89VVX h?2M 7Q` u ≥ T - URkX8eV BKTHB2b  ∞  u  T E[δ(u) | F0 ] ≤ ε ds h∗n (u − s) h(v)dv + h∗n (u − s)|ν1 − ν2 |(s)ds. n≥0

0

s

n>0

0

b h BMi2;`i2b iQ 1- i?2 ;2M2`B+ i2`K BM i?2 }`bi bmKKiBQM `2/b  u  u−v  u h∗n (v)dv − h∗n (v)dv h(s)ds . 0

0

0

 Q7 p`B#H2 `;mK2Mi b?Qrb i?i i?2 b2+QM/ i2`K i?2`2 2[mHb  u +?M;2 ∗(n+1) h (v)dvX h?Bb 2M#H2b mb iQ +M+2H i2`Kb BM i?Bb }`bi bmKKiBQM- r?B+? 0 i?2M `2/m+2b 2t+iHv iQ εX AMi2;`iBM; i?Bb BM2[mHBiv #2ir22M t + a M/ t + b vB2H/b    t+b  T   t+b ∗n E[δ(u) | F0 ]du ≤ (b − a)ε + |ν1 − ν2 |(s) h (u − s) du ds . t+a

0

n>0

t+a

.2MQiBM; #v U i?2 `2M2rH K2bm`2 bbQ+Bi2/ rBi? i?2 T`Q##BHBiv /2MbBiv h- i?2 b2+QM/ i2`K BM i?Bb `B;?i@?M/ bB/2 Bb #QmM/2/ #Qp2 #v  T |ν1 − ν2 |(s)U ([t + a − s, t + b − s]) ds. 0

 Ai 7QHHQrb #v /QKBMi2/ +QMp2`;2M+2 i?i (t+a,t+b) E[δ(u) | F0 ]du ;Q2b iQ y b  t → ∞X AM/22/- U ([t + a − s, t + b − s] i2M/b iQ 0 = 1/ th(t)dt #v i?2 `2M2rH i?2Q`2K- M/ Bb #QmM/2/ #v i?2 }MBi2 [mMiBiv U ([0, b − a]) b  bBKTH2 `;mK2Mi b?QrbX h?2 +QM+HmbBQM Q7 i?2 H2KK 7QHHQrb #v MQiB+BM; i?i  E[|N1 − N2 | (t + a, t + b] | F0 ] = E[δ(u) | F0 ]du , (t+a,t+b)

r?B+? ;Q2b iQ x2`Q b t → ∞- #v i?2 T`2pBQmb MHvbBbX h?2`27Q`2 lim E [|N1 − N2 | (t + a, t + b] | F0 ] = 0 XbX

t→∞



7`QK r?B+? URkX88V 7QHHQrbX

G2KK RkX8Xd *QMbB/2` TQBMi T`Q+2bb2b Ni (i = 1, 2) QM R- /KBiiBM; QM R+ i?2 `2bT2+iBp2 biQ+?biB+ BMi2MbBiB2b  h(t − s)Ni (ds) (i = 1, 2) . λi (t) = (−∞,t)

bbmK2 BM //BiBQM i?i 7Q` bQK2 λ > 0-

RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_

9N8

Ni [−t, 0] = λ a.s. (i = 1, 2) . URkX8dV t  h?2M- 7Q` i = 1, 2- νi (t) := (−∞,0] h(t − s)Ni (ds) Bb HQ+HHv BMi2;`#H2 M/  ∞ h(s)ds ∼ α−1 λrt−α b t → ∞ , νi (t) ∼ λ lim

t→∞

t

rBi? r b BM URkX9dVX AM T`iB+mH`- i?2 bbmKTiBQMb Q7 G2KK k `2 HH biBb}2/X S`QQ7X 6Q` }t2/ t > 0- QM2 ?b  t  νi (s)ds = 0



t−u

Ni (du) (−∞,0]

h(s)ds . −u

aTHBi i?2 BMi2;`H Q7 i?2 `B;?i@?M/ bB/2 ++Q`/BM; iQ r?2i?2` u ≥ −1 Q` u < −1 Ui?2 +?QB+2 −1  ∞Bb `#Bi``vVX h?2 }`bi BMi2;`H bQ Q#iBM2/ Bb i?2M #QmM/2/ #Qp2 #v Ni [−1, 0] 0 h(s)ds- r?B+? Bb }MBi2X AM pB2r Q7 URkX9eV- i?2 b2+QM/ BMi2;`H Bb bKHH2` i?M  Ni (du)Rt(−u)−1−α . _2TH+2 (−u)−1−α #v i?Bb 2tT`2bbBQM 2[mHb

∞

(−∞,−1)

(1 + α)z −2−α dz M/ TTHv 6m#BMBǶb i?2Q`2K iQ b22 i?i −u 



Rt(1 + α)

z −2−α Ni [−z, −1)dz .

1

AM pB2r Q7 URkX8dV- i?2 BMi2;`M/ Bb 2[mBpH2Mi iQ λz −1−α b z → ∞- bQ i?i i?2 BMi2;`H Bb }MBi2X h?mb νi Bb HQ+HHv BMi2;`#H2X q`Bi2   νi (t) = h(t − s)Ni (ds) + h(t − s)Ni (ds) . (−∞,−1]

(−1,0]

h?2 b2+QM/ i2`K Q7 i?2 `B;?i@?M/ bB/2 Bb- 7Q` H`;2 t- Q7 Q`/2` t−1−α X 6Bt ε > 0M/ +QMbB/2` T bm+? i?i 7Q` t ≥ T U+7X URkX9dVV- r−1 t1+α h(t) ∈ (1 − ε, 1 + ε)X 6Q` t ≥ T - i?2 }`bi i2`K Bb 2[mH iQ  ∞  r(t − s)−1−α Ni (ds) = r(1 + α) u−2−α duNi [t − u, −1] , C t+1

(−∞,−1]

7Q` bQK2 +QMbiMi C BM (1 − ε, 1 + ε)X "2+mb2 Q7 i?2 bbmKTiBQM URkX8dV- i?Bb Hbi i2`K Bb 2[mBpH2Mi iQ  ∞ u−2−α (u − t)du , Cr(α + 1)λ t+1 −1

−α

Bib2H7 2[mBpH2Mi iQ Cα rλt X _2+HHBM; i?i εBb `#Bi``v- i?Bb Hii2` i2`K Bb ∞ 2[mBpH2Mi iQ νi (t) b t → ∞- M/ ?2M+2 Q7 Q`/2` t h(s)dsX  h?2 #Qp2 H2KKb HHQr mb iQ 2bi#HBb? i?2 7QHHQrBM; `2bmHiX

9Ne

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

h?2Q`2K RkX8X3 amTTQb2 i?i i?2 72`iBHBiv `i2 h biBb}2b URkX9jV- URkX9eV M/ URkX9dVX h?2 7QHHQrBM; T`QT2`iB2b i?2M ?QH/, UV 6Q` HH λ > 0- i?2`2 2tBbib i KQbi QM2 /Bbi`B#miBQM 7Q`  biiBQM`v >rF2b T`Q+2bb rBi? p2`;2 BMi2MbBiv λ M/ bm+? i?i limt→∞ N [−t, 0]/t = λ XbXc H2i πλ (·) /2MQi2 i?Bb /Bbi`B#miBQM r?2M Bi 2tBbibX U#V πλ Bb 2`;Q/B+ M/ KBtBM;X U+V Mv biiBQM`v >rF2b T`Q+2bb bm+? i?i Λ := limt→∞ N [−t, 0]/t Bb XbX }MBi2 /KBib i?2 2`;Q/B+ /2+QKTQbBiBQM  P (N ∈ ·) = πλ (·)P (Λ ∈ dλ) . U/V Mv TXTX N QM R+ rBi?  biQ+?biB+ BMi2MbBiv  λ(t) = νi (t) + h(t − s)N (ds) , (0,t)

∞

r?2`2 νi (t) ∼ Λ t h(s)ds b t → ∞ 7Q` M XbX }MBi2 `XpX Λ bm+? i?i i?2 /Bbi`B#miBQM π(Λ) 2tBbib XbX biBb}2b i?2 r2F +QMp2`;2M+2 T`QT2`iv  w P (St N ∈ ·) → πλ (·)P (Λ ∈ dλ) .

S`QQ7X UV G2i N1 M/ N2 #2 irQ biiBQM`v >rF2b T`Q+2bb2b bm+? i?i lim Ni [−t, 0]/t = λ .

t→∞

G2i N1 M/ N2 #2 b BM i?2 +QmTHBM; H2KK UG2KK kV  lim P N1 = N2 QM (t + a, t + b] | F0 = 1 XbX t→∞

6`QK i?Bb Bi 7QHHQrb i?i i?2 }MBi2@/BK2MbBQMH /Bbi`B#miBQMb Q7 N1 M/ N2 `2 i?2 bK2- ?2M+2 i?2B` /Bbi`B#miBQMb +QBM+B/2, i?2`2 Bb i KQbi QM2 /Bbi`B#miBQM 7Q`  biiBQM`v >rF2b T`Q+2bb N bm+? i?i limt→∞ Ni [−t, 0]/t = λ XbX- M/ r2 /2MQi2 Bi #v πλ r?2M Bi 2tBbibX U#V πλ Bb KBtBM; UM/ ?2M+2 2`;Q/B+V, *QMbB/2` bQK2 `#Bi``v }MBi2 BMi2`pH [a, b]- M/ H2i A- B #2 irQ K2bm`#H2 bm#b2ib Q7 i?2 bT+2 N Q7 TQBMi T`Q+2bb2bBMpQHpBM; QMHv i?2 `2bi`B+iBQM Q7 i?2 TQBMi T`Q+2bb iQ i?2 BMi2`pH [a, b]X JBtBM; ?QH/b B7 7Q` HH bm+? a < b- A M/ Blim P (N ∈ A, St N ∈ B) = P (N ∈ A)P (N ∈ B) .

t→∞

*QMbB/2` irQ BM/2T2M/2Mi +QTB2b Ni Q7 N - M/ TTHv i?2 +QMbi`m+iBQM Q7 G2KK RkX8Xe iQ i?2K- bi`iBM; MQi 7`QK iBK2 y #mi 7`QK iBK2 bX Ai i?2M ?QH/b i?i

RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_

9Nd

P (N ∈ A, St N ∈ B)

, = P (N1 ∈ A, St N2 ∈ B) + P (N1 ∈ A, St N1 ∈ B) − P (N1 ∈ A, St N2 ∈ B) .

LQi2 i?i N1 +QBM+B/2b rBi? N1 QM (−∞, b]- r?BH2 N2 Bb +QMbi`m+i2/ 7`QK N2 M/ ¯ QMHv- M/ Bb i?mb BM/2T2M/2Mi Q7 i?2 2p2Mi {N1 ∈ A}X h?2 }`bi i2`K BM i?2 N T`2pBQmb 2[miBQM Bb i?2M 2t+iHv 2[mH iQ P (N ∈ A)P (N ∈ B)X h?2 i2`K BM +m`Hv #`+F2ib ;Q2b iQ y b t → ∞- #2+mb2 Bib KQ/mHmb Bb bKHH2` i?M i?2 p`BiBQM /BbiM+2 #2ir22M i?2 /Bbi`B#miBQMb Q7 N1 M/ N2 r?2M `2bi`B+i2/ iQ [t + a, t + b]M/ #v G2KK RkX8Xe i?2 Hii2` ;Q2b iQ x2`Q b t → ∞X U+V 1`;Q/B+ /2+QKTQbBiBQM, "v bbmKTiBQM- limt→∞ N [−t, 0]/t = Λ- r?B+? Bb M XbX }MBi2 `M/QK p`B#H2 U2tBbi2M+2 Q7 i?2 XbX HBKBi Λ Bb BKTHB2/ #v "B`F?QzǶb 2`;Q/B+ i?2Q`2Kc Bib }MBi2M2bb M22/b iQ #2 bbmK2/ i?Qm;?VX Ai Bb 2bv iQ b?Qr i?i i?2 /Bbi`B#miBQM Q7 N +QM/BiBQMHHv QM Λ = l Bb biBHH i?2 /Bbi`B#miBQM Q7  biiBQM`v ?TT- mM/2` r?B+? limt→∞ N [−t, 0]/t = l XbX Ai i?2M 7QHHQrb 7`QK UV i?i i?2 /Bbi`B#miBQM Q7 N +QM/BiBQMHHv QM Λ = l Kmbi +QBM+B/2 rBi? πl X aii2K2Mi U/V 7QHHQrb 7`QK i?2 +QmTHBM; H2KK BM M Q#pBQmb rvX  a2+QM/@Q`/2` S`QT2`iB2b h?2 2tBbi2M+2 Q7 biiBQM`v +`BiB+H >rF2b T`Q+2bb2b Nλ rBi? p2`;2 BMi2MbBiv λ 7Q` Mv λ > 0 ?b #22M 2bi#HBb?2/- #mi i i?Bb TQBMi i?2B` b2+QM/@Q`/2` bi`m+im`2 Bb v2i iQ #2 MHvb2/X b K2MiBQM2/ Dmbi 7i2` i?2 T`QQ7 Q7 G2KK RkX8X9- r2 M22/ iQ b?Qr i?i i?2 p`BM+2 Q7 i?2 HBKBi Bb 2[mH iQ i?2 HBKBi Q7 i?2 p`BM+2X K P#b2`p2 i?i 7Q` rBi? +QKT+i  HH g ∈ C0 Ui?2 bT+2 Q7 +QMiBMmQmb 7mM+iBQMb  bmTTQ`iV- pB2rBM; g(x)Nλ (dx) b i?2 r2F HBKBi Q7 i?2 g(x)N ε (dx) r?2M ε 7QHHQrb  b2[m2M+2 /2+`2bBM; iQ y HQM; r?B+? i?2 N ε +QMp2`;2 r2FHv- 6iQmǶb H2KK BKTHB2b i?i   |g(ω)|2 λ p` g(x)Nλ (dx) ≤ dω . URkX83V 2π |1 − h(ω)|2

h?2 M2ti H2KK ;Bp2b M BM2[mHBiv BM i?2 +QMp2`b2 /B`2+iBQMX G2KK RkX8XN amTTQb2 i?i i?2 72`iBHBiv `i2 h biBb}2b URkX9jV- URkX9eV- M/ URkX9dVX 6Q` HH g ∈ L1C (R) ∩ L2C (R)- M/ 2p2`v biiBQM`v >rF2b T`Q+2bb N bm+? i?i Λ := limt→∞ N [−t, 0]/t XbX }MBi2

 p`

g(x)N (dx) ≥ p`(Λ)

2 g(x)dx

+

E(Λ) 2π



|g(ω)|2 |1 − h(ω)|2

dω.

URkX8NV

S`QQ7X Ai bm{+2b iQ 2bi#HBb? URkX8NV BM i?2 +b2 r?2`2 Λ `2/m+2b iQ  +QMbiMibBM+2 i?2 ;2M2`H 7Q`KmH 7QHHQrb 7`QK i?i +b2 #v mbBM; i?2 +QM/BiBQMH p`BM+2  7Q`KmHX q2 i?mb bbmK2 Λ = λ XbX 7Q` bQK2 +QMbiMi λ > 0X A7 g(x)N (dx) ?b  BM}MBi2 p`BM+2- i?2 H2KK ?QH/b i`BpBHHvX >2M+2 bbmK2 i?i o` g(x)N (dx)

9N3

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

Bb }MBi2X 6Q`  biiBQM`v >rF2b T`Q+2bb rBi?  }MBi2 p2`;2 BMi2MbBiv λ- M/ f ∈ L1C (R) ∩ L2C (R)  ˜ ∗ f (x)]N (dx) = λ |f (ω)|2 dω, URkXeyV p` [f (x) − h 2π  ˜ = h(−t)- M/ h ˜ ∗ f (x) = h(x ˜ − t)f (t)dtX TTHvBM; 7Q`KmH URkXeyV iQ r?2`2 h(t) i?2 7mM+iBQM ˜ + ··· + g ∗ h ˜ ∗n f := g + g ∗ h vB2H/b

&2 & n  & & λ ∗(n+1) 2& k& ˜ p` [g(x) − h |g(ω)| & ∗ g(x)]N (dx) = h(ω) & dω . & & 2π k=0  ˜ ∗n (x)N (dx) → 0 b n → ∞X 1tTM/BM; i?2 H27i@?M/ b Bb b?QrM #2HQr- o` g ∗ h bB/2 Q7 i?2 T`2pBQmb 2tT`2bbBQM- M/ H2iiBM; n → ∞- QM2 i?mb b22b i?i URkX8NV ?QH/b rBi?  2[mHBiv BMbi2/ Q7 BM2[mHBiv- mM/2` i?2 bbmKTiBQM i?i Λ = λ XbXM/ p` g(x)N (dx) Bb }MBi2X  

 ˜ ∗n (x)N (dx) +QMp2`;2b iQ x2`Q b n → ∞X *QMbB/2` q2 MQr b?Qr i?i o` g ∗h    ˜ ∗n ∗ g (x + t)N (dx)X LQi2 i?i QM2 T`Q+2bb Bb Q#iBM2/ h i?2 T`Q+2bb X n (t) := 7`QK i?2 T`2pBQmb QM2 #v +QMpQHmiBQMH }Hi2`BM;,  n ds. X n+1 (t) = h(s)Xt+s

URkXeRV

G2i C 0 (u) /2MQi2 *Qp(X 0 (t), X 0 (t))X LQi2 i?i C 0 (u) Bb #QmM/2/ #Qp2 #v p`X 0 (0)- r?B+? Bb }MBi2 #v bbmKTiBQMX G2i mb MQr b?Qr i?i i?2 T`Q+2bb X 1 /KBib  bT2+i`H K2bm`2 μX "Q+?M2`Ƕb i?2Q`2K 2Mbm`2b Bib 2tBbi2M+2 T`QpB/2/ i?2 T`Q+2bb Bb rB/2@b2Mb2 biiBQM`vX Ai Bb  biiBQM`v b2+QM/@Q`/2` T`Q+2bb- M/ Bi `2KBMb iQ T`Qp2 +QMiBMmBiv BM i?2 [m/`iB+ K2MX q`Bi2   p` X 1 (y) − X 1 (0)    0 0 = *Qp (h(s − y) − h(s))X (s) ds, (h(t − y) − h(t))X (t) dt   = C 0 (t − s)(h(s − y) − h(s))(h(t − y) − h(t)) ds dt . 2  h?2 Hbi i2`K Bb bKHH2` BM #bQHmi2 pHm2 i?M C 0 (0) |h(t − y) − h(t)| dt r?B+? i2M/b iQ x2`Q b y → 0 bBM+2 h Bb BMi2;`#H2X AM pB2r Q7 URkXeRV- X n HbQ /KBib  bT2+i`H K2bm`2- ;Bp2M #v μ(dω)|h(ω)|2n−2 X Aib BMi2;`H Bb 2[mH iQ i?2  ˜ ∗n (x)N (dx)- r?B+? r2 rMi iQ T`Qp2 ;Q2b iQ x2`QX aBM+2 h Bb p`BM+2 Q7 g ∗ h +QMiBMmQmb M/ |h(ω)| < 1 7Q` HH ω = 0- i?Bb +QMp2`;2M+2 ?QH/b B7 r2 +M T`Qp2 i?i μ({0}) = 0X h?2 7QHHQrBM; H2KK- r?Qb2 T`QQ7 Bb ;Bp2M Hi2`- rBHH #2 mb27mHX G2KK RkX8XRy G2i {Xn }n>0 #2  b2[m2M+2 Q7 +2Mi2`2/ `M/QK p`B#H2b- +QM@ p2`;BM; iQ y BM T`Q##BHBiv b n → ∞X bbmK2 7m`i?2` i?i 7Q` HH n > 0- Xn ≤cx Y r?2`2 Y Bb  `M/QK p`B#H2 rBi? }MBi2 p`BM+2- M/ ≤cx /2MQi2b i?2 +QMp2t biQ+?biB+ Q`/2`BM;X h?2M limn→∞ p`(Xn ) = 0X

RkX8X "_L*>AL: SPALh S_P*1aa1a qAh>Plh L*1ahP_

9NN

n G2i Xn := (1/n) 0 X 1 (t)dt−E(X01 )X "v "B`F?QzǶb 2`;Q/B+ i?2Q`2K- Xn +QMp2`;2b XbX- M/ i?mb  7Q`iBQ`B BM T`Q##BHBiv- iQ y b n → ∞X HbQ- iFBM; Y := X 1 (0) − E(X 1 (0))- Bi ?QH/b i?i Xn ≤cx Y - M/ Y ?b }MBi2 p`BM+2X G2KK RkX8XRy i?mb TTHB2b- M/ ?2M+2 limn→∞ p`(Xn ) = 0X "v i?2 bT2+i`H 7Q`KmH- QM2 ?b &  & iωn & e − 1 &2 & μ(dω) . & p`(Xn ) = & iωn & 6`QK i?Bb Bi 7QHHQrb i?i limn→∞ p`(Xn ) = μ({0}) M/ i?2`27Q`2 M2+2bb`BHv μ({0}) = 0X S`QQ7X amTTQb2 i?i p`(Xn ) /Q2b MQi +QMp2`;2 iQ yc #v +QMbB/2`BM;  bm#b2[m2M+2bbmK2 rBi?Qmi HQbb Q7 ;2M2`HBiv i?i p`(Xn ) Bb #QmM/2/ 7`QK #2HQr #v σ 2 > 0X lbBM; +QMp2`;2M+2 BM T`Q##BHBiv Q7 Xn iQ y- 7Q` HH δ > 0- i?2`2 2tBbib M n(δ) bm+? i?i   P Xn2 > 13 σ 2 ≤ δ, n ≥ n(δ) . ∞ q`Bi2 p`(Xn ) b 0 P (Xn2 > t)dt- M/ bTHBi i?Bb BMi2;`H Qp2` i?`22 BMi2`pHb [0, σ 2 /3)- [ 13 σ 2 , 13 σ 2 (1 + δ −1 ))- [ 13 σ 2 (1 + δ −1 ), +∞)X "QmM/BM; P (Xn2 > t) 7`QK #Qp2 #v R QM i?2 }`bi BMi2`pH- M/ #v P (Xn2 > σ 2 /3) QM i?2 b2+QM/- QM2 Q#iBMb,  ∞ P (Xn2 > t)dt ≥ σ 2 − 13 σ 2 − 13 σ 2 = 13 σ 2 , n ≥ n(δ) . (σ 2 /3)(1+δ −1 )

h?Bb BM im`M Bb 2[mBpH2Mi iQ +  E Xn2 − 13 σ 2 (1 + δ −1 ) ≥ 13 σ 2 ,

n ≥ n(δ) .

>Qr2p2`- #v bbmKTiBQM- i?2 H27i@?M/ bB/2 Bb #QmM/2/ #Qp2 #v 1 E(Y 2 − σ 2 (1 + δ −1 ))+ , 3 bBM+2 i?2 7mM+iBQM x → (x2 − t)+ Bb +QMp2t 7Q` HH tX aBM+2 δ +M #2 iF2M `#Bi``BHv bKHH- i?Bb BKTHB2b i?i limt→∞ E(Y 2 − t)+ ≥ 13 σ 2 > 0- r?B+? Bb MQi +QKTiB#H2 rBi? i?2 bbmKTiBQM i?i o`(Y ) Bb }MBi2- ?2M+2  +QMi`/B+iBQMX  h?2 7QHHQrBM; `2bmHiR3 Bb Q#iBM2/ b  +Q`QHH`v iQ G2KK RkX8XNX Ai +QKTH2@ K2Mib h?2Q`2K RkX8X3- r?B+? H27i QT2M i?2 [m2biBQM Q7 2tBbi2M+2 Q7 i?2 /Bbi`B#miBQMb πλ X h?2Q`2K RkX8XRR lM/2` i?2 bbmKTiBQMb Q7 G2KK RkX8XN- i?2`2 2tBbib M 2`@ ;Q/B+ /Bbi`B#miBQM πλ 7Q` 2p2`v λ > 0X HbQ- 7Q`KmH URkX8NV ?QH/b rBi? 2[mHBivBX2X 7Q` HH g ∈ L1C (R) ∩ L2C (R)- M/ 2p2`v biiBQM`v >rF2b T`Q+2bb N bm+? i?i Λ := limt→∞ N [−t, 0]/t XbX }MBi2

 p` R3

g(x)N (dx) = p`(Λ)

("`ûKm/ M/ JbbQmHBû- kyyR)X

2 g(x)dx

+

E(Λ) 2π



|g(ω)|2 |1 − h(ω)|2

dω .

URkXekV

8yy

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

S`QQ7X *QK#BMBM; i?2 HQr2` #QmM/ URkX8NV Q7 G2KK RkX8XN rBi? i?2 mTT2` #QmM/ URkX83V r2 ?/ 7Q` Nλ 2M#H2b mb iQ /2/m+2- 7Q` g ∈ C0K - i?i p`(Λ) = 0- r?2`2 Λ := limt→∞ Nλ ([−t, 0]/tX h?mb #v h?2Q`2K RkX8X3 UV- Nλ Kmbi #2 /Bbi`B#mi2/ b πλ - r?B+? i?mb 2tBbib UMQi2 i?i bBM+2 Mv /?2`2Mi /Bbi`B#miBQM iQ i?2 b2[m2M+2 {N ε } ?b iQ #2 /Bbi`B#mi2/ b πλ - {N ε } Bb MQi QMHv iB;?i #mi HbQ r2FHv +QMp2`@ ;2MiVX h?Bb +QK#BMiBQM Q7 HQr2` M/ mTT2` #QmM/b HbQ b?Qrb i?i  p`

g(x)Nλ (dx) =

λ 2π



|g(ω)|2 |1 − h(ω)|2

dω .

0 )c i?2 ;2M2`H +b2 h?mb URkXekV ?QH/b r?2M N Bb /Bbi`B#mi2/ b πλ (M/ 7Q` g ∈ CK 0 7QHHQrb 7`QK i?2 +QM/BiBQMH p`BM+2 7Q`KmH UM/ #v /2MbBiv Q7 CK VX 

RkXe E2`biM SQBMi S`Q+2bb2b h?2 bBKTH2bi 2tKTH2 Bb  TQBMi T`Q+2bb N QM R rBi? i?2 FtN @BMi2MbBiv   λ(t) := ϕ h(t − s)N (ds) ,

URkXejV

(−∞,t)

r?2`2 ϕ : R → R Bb  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM M/ h : R → R Bb  K2bm`#H2 7mM+iBQM +HH2/ U#mbBp2Hv #mi +QMp2MB2MiHvV i?2 72`iBHBiv `i2- bm+? i?i  t < 0 ⇒ h(t) = 0 M/

R+

|h(t)|dt < ∞ .

am+?  TQBMi T`Q+2bb Bb bQK2iBK2b +HH2/  MQM@HBM2` >rF2b T`Q+2bbXRN  aiQ+?biB+ .Bz2`2MiBH avbi2K oB BK#2//BM;-  MQM@HBM2` >rF2b T`Q+2bb +M #2 pB2r2/ b i?2 bQHmiBQM Q7  biQ+?biB+ /Bz2`2MiBH 2[miBQM /`Bp2M #v  SQBbbQM K2bm`2X AM 7+i- #v i?2 BMp2`b2 BK#2//BM; i?2Q`2K Uh?2Q`2K 8XdXeV- bm+?  TQBMi T`Q+2bb +M #2 `2T`2b2Mi2/ b  N (dt) = 1(0,ϕ(X(t))] (z)N (dt × dz) ,  R+ h(t − s)N (ds) , URkXe9V X(t) = (−∞,t)

r?2`2 N Bb  ?QKQ;2M2Qmb SQBbbQM T`Q+2bb QM R × R+ rBi? BMi2MbBiv RX h?Bb ivT2 Q7 `2T`2b2MiiBQM rBHH THv  +`m+BH `QH2 BM i?2 7Q`i?+QKBM; MHvbBbX JQ`2 ;2M2`H KQ/2Hb Q7  bBKBH` Mim`2 rBHH 2p2MimHHv #2 +QMbB/2`2/- bi`i@ BM; rBi? 1tKTH2 RkXeXk- rBi? Ft @BMi2MbBiB2b 7Q` ?BbiQ`B2b H`;2` i?M i?2 BMi2`MH ?BbiQ`vX RN q?2M ϕ Bb #QmM/2/- bm+? T`Q+2bb2b `2 BM 7+i bT2+BH +b2b Q7 i?Qb2 +QMbB/2`2/ BM i?2 b2KBMH `iB+H2 (E2`biM- RNe9)X

RkXeX E1_ahL SPALh S_P*1aa1a

8yR

1tKTH2 RkXeXR,  M2m`QM 7B`BM; KQ/2HX >2`2   t  λ(t) = λ0 exp − h(t − s) N (ds) , −∞

r?2`2 h : R+ → R+ U TQbBiBp2 7mM+iBQM QM (0, +∞)- MmHH QM (−∞, 0]V r?Qb2 bmT@ TQ`i Bb BM+Hm/2/ BM [0, a] 7Q` bQK2 TQbBiBp2 }MBi2 aX h?Bb }ib i?2 ;2M2`H 7`K2rQ`F Q7 i?2 T`2pBQmb 2tKTH2X >2`2 i?2 `2;2M2`iBQM TQBMib `2 i?2 TQBMib t Q7 M ?TT N0 Q7 BMi2MbBiv λ0 bm+? i?i N0 ([t − a, t)) = 0X h?Bb KQ/2H Bb Q7 BMi2`2bi BM i?2 M2m`Qb+B2M+2b #2+mb2 Q7 Bib TiBim/2 iQ KQ/2H `27`+iQ`v T2`BQ/b BM M2m`QMH bTBF2 +iBpBivX 6Q` BMbiM+2- B7 h(t) = α1 1[0,1) (t) + α2 1[1,2) (t) + α3 1[2,3) (t) + α4 1[3,4) (t) r2 ?p2 λ(t) =λ0 exp{−α1 N ((t, t − 1]) + α2 N ((t − 1, t − 2]) + α3 N ((t − 2, t − 3]) + α4 N ((t − 3, t − 4])} , r?2`2 iBK2 Bb ?2`2 K2bm`2/ BM bm{+B2MiHv bKHH mMBib +QKTiB#H2 rBi? M2m`Q#BQ@ HQ;B+H TTHB+iBQMbX hF2 7Q` BMbiM+2 λ0 = 10 , α1 = 100 , α2 = 2 , α3 = 0.5 , α4 = 0.1. A7 i iBK2 t- i?2`2 Q++m``2/  bTBF2 BM i?2 T`2pBQmb QM2 iBK2 mMBi BMi2`pH- i?2 M2m`QM Bb BM?B#Bi2/ BM i?i Bib }`BM; BMi2MbBiv Bb i?2M H2bb i?i 10e−100 - r?B+? Bb M2;HB;B#Hv bKHHX aTBF2b Q++m``BM; BM i?2 iBK2 BMi2`pH (t − 4, t − 1] BM i?2 #b2M+2 Q7  bTBF2 BM i?2 BMi2`pH (t − 1, t] HbQ ?p2 M BM?B#BiQ`v 2z2+i- #mi MQi bQ `/B+HX h?2 bi`B+i BM?B#BiQ`v T2`BQ/ i?2`27Q`2 ?b H2M;i? 1- r?2`2b i?2 `2HiBp2 BM?B#BiQ`v T2`BQ/ ?b H2M;i? 3X h?2 #bB+ KQ/2H URkXejV rBHH #2 2M`B+?2/X 6Q` BMbiM+2- QM2 Kv rBb? iQ +QMbB/2` KmHiBp`Bi2 MQM@HBM2` >rF2b T`Q+2bb2bX 1tKTH2 RkXeXk,  M2m`H M2irQ`F KQ/2HX >2`2 N := (Ni ; 1 ≤ i ≤ M ) Bb  KmHiBp`Bi2 TQBMi T`Q+2bbX h?2 BMi2MbBiv Q7 Ni Bb Q7 i?2 7Q`K λi (t) =

M 

  cji (t) exp −

j=1





t −∞

hii (t − s) Ni (ds) 

t

cji (t) = αji (t)ϕji −∞

,

hji (t − s) Nj (ds)

.

h?2 hii 7mM+iBQMb `2 MQM@M2;iBp2 BMi2;`#H2 M/ Q7 #QmM/2/ bmTTQ`i- i?2 hij 7mM+iBQMb `2 7Q` i = j BMi2;`#H2 M/ Q7 #QmM/2/ bmTTQ`i Q7 `#Bi``v bB;MX h?2 7mM+iBQMb ϕij `2 #QmM/2/ MQM@M2;iBp2 MQM@/2+`2bBM; 7mM+iBQMbX h?2B` +?QB+2 M/ i?2 +?QB+2 Q7 i?2 bB;Mb Q7 i?2 hij ›b /2i2`KBM2 i?2 bi`2M;i? M/ 2t+BiiQ`vfBM?B#BiQ`v biimb Q7 i?2 bvMTb2 i → jX h?2 biQ+?biB+ T`Q+2bb2b {αij }t≥0 - 1 ≤ i, j ≤ M - `2

8yk

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

MQM@M2;iBp2 M/ #QmM/2/- iQ r?B+? +M #2 bbB;M2/ p`BQmb `QH2b- BM T`iB+mH` i?2 BM~m2M+2 Q7 2ti2`MH biBKmHBX PM2 Kv HbQ rBb? iQ +QMbB/2` MQM@HBM2` >rF2b T`Q+2bb2b rBi?  `M/QK 72`iBHBiv `i2X 1tKTH2 RkXeXj,  /2Hv bvbi2KX i 2p2Mi@iBK2 Tn Q7  TQBMi T`Q+2bb N QM R  +mbiQK2` b?Qrb mT i  b2`pB+2 7+BHBiv `2[mB`BM;  íb2`pB+2 iBK2í σn X h?2 b2[m2M+2 {σn }n∈Z Bb BB/ M/ BM/2T2M/2Mi Q7 N X  +mbiQK2` ``BpBM; i iBK2 Tn /2T`ib 7`QK i?2 bvbi2K i iBK2 Tn + σn X h?2`27Q`2 i?2 MmK#2` Q7 +mbiQK2`b T`2b2Mi BM i?2 bvbi2K i iBK2 t Bb  1{Tn t X(t) = n∈Z

Q`- 2[mBpH2MiHvX(t) =



1{Tn 2`2 & & & 1 , . . . , N M ) − ϕi (N  1 , . . . , N  M )&& &ϕi (N & &M   M  0  & & 0  & j (ds × dz) −  j (ds × dz)&& hji (−s, z)N hji (−s, z)N =& & & −∞ K −∞ K j=1



M  0  j=1

j=1



−∞

& & &  & hji (−s, z) &N j − N j & (ds × dz) , K

M/ i?2`27Q`2 E2`biMǶb +QM/BiBQM Bb biBb}2/X *QM/BiBQM ρ(A) < 1 ;m`Mi22b i?2 2tBbi2M+2 Q7  biiBQM`v KmHiB@ivT2 >rF2b T`Q+2bb rBi? }MBi2 p2`;2 BMi2MbBiB2bX

1tKTH2 RkXeXe, h?2 hvTB+H 1tKTH2X h?2 T`2pBQmb 2tKTH2b M/ BM 7+i i?2 KDQ`Biv Q7 2tKTH2b +QM+2`M TQBMi T`Q+2bb2b N rBi?  biQ+?biB+ BMi2MbBiv Q7 i?2 7Q`K UmMBp`Bi2 +b2- 7Q` bBKTHB+BivV 



 (ds × dz h(t − s, z) N

λ(t) = ϕ (−∞,t)

 ,

K

r?2`2 ϕ Bb 1@GBTb+?BixX E2`biMǶb +QM/BiBQM Bb i?2M biBb}2/ bBM+2 ky

(E2`biM- RNe9)X

8y9

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

&  & &ϕ &

  &  & h(t − s, z) μ (ds × dz) − ϕ h(t − s, z) μ  (ds × dz) && (−∞,t) K (−∞,t) K & &    & &  & h(t − s, z) μ (ds × dz) − h(t − s, z) μ  (ds × dz)&& ≤& (−∞,t) K (−∞,t) K & &  & &  h(t − s, z) | μ−μ  | (ds × dz)&& = && (−∞,t) K   |h(t − s, z)| | μ−μ  | (ds × dz) . ≤ 

(−∞,t)

K

 HBiiH2 i?Qm;?i rBHH +QMpBM+2 i?2 `2/2` i?i ϕ B7 α@GBTb+?Bix 7Q` `#Bi``v TQbBiBp2 α- E2`biMǶb +QM/BiBQM rQmH/ biBHH #2 biBb}2/ UDmbi  Kii2` Q7 MQiiBQMVX h?2 7QHHQrBM; [m2biBQMb `Bb2 Mim`HHv, M ) rBi? i?2 #Qp2 1 , . . . , N X .Q2b i?2`2 2tBbi  DQBMiHv biiBQM`v 7KBHv (N /vMKB+b\ "X Ab Bi mMB[m2\ *X Ab Bi bvKTiQiB+HHv bi#H2\ BM r?i b2Mb2\ mM/2` r?i BMBiBH +QM/BiBQMb\ bvKTiQiB+ bi#BHBiv Bb /2}M2/ BM i?2 7QHHQrBM; rvX 6B`bi r2 /2}M2  i`MbB2Mi bQHmiBQMX M ) rBi? i?2 /v@ 1 , . . . , N .2}MBiBQM RkXeXd  i`MbB2Mi bQHmiBQM Bb Mv 7KBHv (N MKB+b UDV QM (0, ∞)X M BMBiBH +QM/BiBQM Bb Mv T`QT2`iv (P) `2HiBp2 iQ i?2 1 , . . . , N M ) iQ (−∞, 0]X `2bi`B+iBQM Q7 (N 1 , . . . , N M QM (−∞, 0] M/ r2 i?2M h?2`27Q`2 r2 `2 ;Bp2M i?2 TQBMi T`Q+2bb2b N   +QMbi`m+i N1 , . . . , NM QM (0, +∞] rBi? i?2 biQ+?biB+ Ft @F2`M2Hb /2b+`B#2/ #Qp2X h?2 /vMKB+b UDV `2 bB/ iQ #2 bvKTiQiB+HHv bi#H2 mM/2` i?2 BMBiBH +QM@ /BiBQM (P) B7 i?2`2 2tBbib  mMB[m2 biiBQM`v bQHmiBQM M/ B7 i?2 +Q``2bTQM/BM;  )+ Ui?2 `2bi`B+iBQM iQ (0, +∞) Q7 St N  V +QM@ i`MbB2Mi bQHmiBQM N  Bb bm+? i?i (St N p2`;2b b t → ∞ iQ i?2 biiBQM`v /Bbi`B#miBQM QM (0, +∞)X "v +QMp2`;2M+2 r2 K2M +QMp2`;2M+2 BM /Bbi`B#miBQM Q` +QMp2`;2M+2 BM p`BiBQM UiQ #2 K/2 T`2+Bb2 7Q` 2+? `2bmHiVX "QmM/2/ 62`iBHBiv _i2 rBi? "QmM/2/ amTTQ`i h?2 [m2biBQMb - " M/ * }M/ 2bv M/ TQbBiBp2 Mbr2`b BM i?2 bT2+BH +b2 r?2`2 h : R → R ?b  bmTTQ`i {t ∈ R; h(t) = 0} BM+Hm/2/ BM [0, a] 7Q` bQK2 a ∈ R+ - M/ r?2M KQ`2Qp2` ϕ Bb #QmM/2/ Ubv- #v Λ < ∞VX AM 7+i- HQ+H }MBi2M2bb Bb ;m`Mi22/ ˜ bBM+2 7`QK URkXe9V- i?2 TQBMib Q7 N `2 M2+2bb`BHv TQBMib Q7 i?2 TQBMi T`Q+2bb N ˜ QM R /2}M2/ #v N (C) = N (C × [0, Λ])- r?B+? Bb BM/22/  HQ+HHv #QmM/2/ bBKTH2 T`Q+2bbX h?2 2tBbi2M+2 Q7  biiBQM`v TQBMi T`Q+2bb N rBi? i?2 /vMKB+b URkXe9V +M ˜ a #2 i?2 TQBMi T`Q+2bb 7Q`K2/ #v i?2 TQBMib T˜n Q7 N ˜ #2 b?QrM b 7QHHQrbX G2i N

RkXeX E1_ahL SPALh S_P*1aa1a

8y8

˜ a X h?2b2 `2 bm+? i?i T˜n − T˜n−1 > a- M/ +HH {T˜na }n∈Z i?2 b2[m2M+2 Q7 TQBMib Q7 N a ˜a ˜ dz`2;2M2`iBQM TQBMibǴ- #2+mb2 7Q` HH t ∈ [Tn , Tn+1 )   λ(t) = ϕ h(t − s)N (ds) (T˜na ,t)

/Q2b MQi /2T2M/ QM N #27Q`2 T˜na X h?mb r2 ?p2 M 2tTHB+Bi 7Q`K 7Q` λ(t) 7Q` HH t i?i /Q2b MQi `2[mB`2 FMQrH2/;2 Q7 i?2 r?QH2 Tbi Q7 N X lMB[m2M2bb Q7 i?2 biiBQM`v bQHmiBQM Bb +H2` 7`QK i?Bb `;mK2MiX AM/22/- H2i N  #2  biiBQM`v bQHmiBQM- M/ +QMbi`m+i 7`QK Bi N b BM i?2 BMp2`bBQM i?2Q`2K- M/ MQr +QMbi`m+i N b #Qp2mbBM; N X h?2 BMi2MbBiv Q7 N  Bb HbQ Q7 i?2 7Q`K   h(t − s)N  (ds) λ (t) = ϕ (T˜nα ,t)

a a 7Q` t ∈ [T˜na , T˜n+1 )X Ai Bb i?2M +H2` i?i N = N  QM HH (T˜na , T˜n+1 ]- n ∈ Z- i?i Bb N ≡NX

:Bp2M Mv BMBiBH +QM/BiBQM U TQBMi T`Q+2bb QM (−∞, 0]V-  i`MbB2Mi bQHmiBQM N  rBHH +QMp2`;2 BM p`BiBQM iQ i?2 biiBQM`v QM2X AM/22/- H2i N  #2 i?2 i`MbB2Mi bQHmiBQM bQ +?Qb2M- M/ +QMbi`m+i 7`QK N  i?2 SQBbbQM T`Q+2bb N rBi? i?2 p2`;2 BMi2MbBiv R QM R+ × R+ - b BM i?2 BMp2`bBQM i?2Q`2KX 6`QK N QM R × R+ - +QMbi`m+i i?2 biiBQM`v bQHmiBQM N X h?2M N M/ N  +QmTH2 i iBK2 T˜1a X G2i mb #2 KQ`2 bT2+B}+ #Qmi i?2 bBKmHiBQMX h?2 ;QH Bb iQ bBKmHi2 QM i?2 TQbBiBp2 ?H7@HBM2  biiBQM`v TQBMi T`Q+2bb rBi? i?2 `2[mB`2/ /vMKB+bX 6Q` i?Bb QM2 }`bi +QMbi`m+ib QM i?2 M2;iBp2 ?H7@HBM2  SQBbbQM T`Q+2bb rBi? BMi2MbBiv Λ, t(0) < 0- t(−i) < 0- XXX- mMiBH i?2 }`bi TQBMi t(−L) < 0 bm+? i?i t(−L)−t(−L−1) > aX ai`i i?2 +QMbi`m+iBQM 7`QK t(−L) B;MQ`BM; i?2 TQBMib #27Q`2 bBM+2 i?2v /Q MQi z2+i i?2 b?Q`i `M;2 biQ+?biB+ BMi2MbBivX PM2 i?2`27Q`2 ?b  b2[m2M+2 t(−L)XXX- t(0)- iQ r?B+? QM2 //b i?2 TQBMib t(1) > 0- t(2) > 0 Q7  SQBbbQM BMi2MbBiv 1 QM i?2 TQbBiBp2 ?H7@HBM2X PM2 Kmbi MQr b2H2+i i?2 TQBMib Q7 i?2 #Qp2 b2[m2M+2 i?i rBHH #2 `2iBM2/ b TQBMib Q7 i?2 TQBMi T`Q+2bb Q7 BMi2`2biX amTTQb2 i?i i?Bb b2H2+iBQM ?b #22M T2`7Q`K2/ 7Q` i?2 TQBMib t(−L)- XXX- t(n)X PM2 i?2M FMQrb i?2 TQBMi T`Q+2bb rBi? i?2 /2bB`2/ /vMKB+b #27Q`2 USh1_ RkX >qE1a SPALh S_P*1aa1a

r?2`2 a > 0X Uh?Bb [mMiBiv /2T2M/b QMHv QM i?2 `2bi`B+iBQM N− Q7 N iQ (−∞, 0]XV 6Q` 7mim`2 `272`2M+2- r2 ;Bp2 H#2Hb iQ i?`22 T`iB+mH` BMBiBH +QM/BiBQMb, − ) < ∞ M/ lim εa (t, N− ) = 0 , sup εa (t, N

(Pd,1 )

  − ) < ∞ M/ lim E [εa (t, N− )] = 0 , sup E εa (t, N

(Pd,2 )

t↑∞

t≥0

t↑∞

t≥0

M/

M   i,j=1

∞ 0



j (ds × dz) hji (t − s, z)N

 dt < ∞ .

(Pv )

(−∞,0]×K

h?2 KBM `2bmHi Q7 i?Bb b2+iBQM Bb, h?2Q`2K RkXeXN UkR V A7 ρ(A) < 1- i?2`2 2tBbib  mMB[m2 biiBQM`v KmHiBp`Bi2 1 , . . . , N M ) rBi? i?2 /vMKB+b UDV biBb7vBM; E2`biMǶb K`F2/ TQBMi T`Q+2bb (N +QM/BiBQM URkXe8V M/ rBi? }MBi2 p2`;2 BMi2MbBiB2b (λ1 , · · · , λM )X JQ`2Qp2`- mM/2` 2Bi?2` QM2 Q7 +QM/BiBQMb UPd,1 V Q` UPd,2 V- i?2 /vMKB+b `2 bvKTiQiB+HHv bi#H2 BM /Bbi`B#miBQMX A7 KQ`2Qp2` 7Q` HH i, j- (1 ≤ i, j ≤ M )  ∞  t hji (t, z) Qj (dz) dt , URkXedV 0

K

mM/2` i?2 BMBiBH +QM/BiBQM UPv V- i?2 /vMKB+b `2 bvKTiQiB+HHv bi#H2 BM p`B@ iBQMX  #2 i?2 TQBMi T`Q+2bb QM R×K 1tKTH2 RkXeXRy, h?2 mMBp`Bi2 +b2X G2i N rBi? biQ+?biB+ Ft @F2`M2H     (ds × dz) Q(dz) , h(t − s, z)N λ(t, dz) = ϕ (−∞,t)

K

r?2`2 ϕ Bb α@GBTb+?Bix M/ ∞h  : R × K → R Bb  7mM+iBQM bm+? i?i h(t, z) > 0 BKTHB2b i?i t ≥ 0- M/ 0 K |h(t, z)| dt Q(dz) < ∞X h?2 Ki`Bt A `2/m+2b iQ ∞ i?2 b+H` α 0 K |h(t, z)| dt Q(dz)X h?2`27Q`2 B7 i?Bb [mMiBiv Bb bi`B+iHv bKHH2` i?M 1- i?2`2 2tBbib  mMB[m2 biiBQM`v /Bbi`B#miBQM rBi? i?2 ;Bp2M /vMKB+bX h?2 BMBiBH +QM/BiBQMb `2 bQK2iBK2b ;Bp2M BM  /Bz2`2Mi rv- MQi BMpQHpBM; i?2 T`2?BbiQ`v Q7 i?2 TQBMi T`Q+2bbX q2 Kv HQQF i i?2 7QHHQrBM; dzi`MbB2MiǴ /vMKB+b      (ds × dz) Q(dz) . λ(t, dz) = ϕ ν(t) + h(t − s, z)N (0,t)

K

 QM (−∞, 0] h?Bb +M #2 `2/m+2/ iQ i?2 biM/`/ +b2 /2b+`B#2/ #Qp2- #v ;BpBM; N b 7QHHQrbX h?2`2 Bb QMHv QM2 TQBMi i T0 ≡ 0 M/ h(t, Z0 ) = ν(t) UBM 7+i- i?2`2  − ) = t |ν(s)| ds M/ i?2`27Q`2 +QM/BiBQMb Bb MQ K`F i T0 VX AM i?Bb +b2- εa (t, N t−a UPd,1 V M/ UPd,2 V `2 i?2 bK2 M/ `2/ kR

(JbbQmHBû- RNN8 M/ RNN3)- ("`ûKm/ M/ JbbQmHBû- RNNe)X

RkXeX E1_ahL SPALh S_P*1aa1a 



t

t

|ν(s)| ds < ∞ M/ lim

sup t≥0

8yd

t↑∞

t−a

r?2`2b +QM/BiBQM UPv V Bb





|ν(s)| ds = 0 , t−a

|ν(s)| ds < ∞ ,

0

M/ URkXedV Bb







 t|h(t, z)| Q(dz)

0

dt < ∞ .

K

q2 MQr T`Q+22/ iQ i?2 T`QQ7 Q7 h?2Q`2K RkXeXNX S`QQ7X ai2T R, 1tBbi2M+2 Q7  biiBQM`v bQHmiBQMX G2i N i U1 ≤ i ≤ M V #2 BM@ /2T2M/2Mi SQBbbQM T`Q+2bb2b QM R × K × R- Q7 `2bT2+iBp2 BMi2MbBiv K2bm`2b dt×Qi (dz)×du U1 ≤ i ≤ M VX G2i {Ft }t∈R #2 i?2 BMi2`MH ?BbiQ`v Q7 (N 1 , . . . , N M )X Ai Kv #2 bbmK2/ i?i i?2b2 TQBMi T`Q+2bb2b `2 /2}M2/ QM  T`Q##BHBiv bT+2 (Ω, F, P ) 2M/Qr2/ rBi?  K2bm`#H2 ~Qr {θt }t∈R - i?i (P, θt ) Bb 2`;Q/B+ M/ N i U1 ≤ i ≤ M V Bb θt @+QKTiB#H2X 6Q` i?Bb- QM2 Kv +?QQb2 iQ rQ`F QM bQK2 +MQMB+H bT+2 Q7 TQBMi T`Q+2bb2bX .m2 iQ i?2 BM/2T2M/2M+2 T`QT2`iB2b Q7 SQBbbQM T`Q+2bb2b(P, θt ) rBHH MQi Dmbi #2 2`;Q/B+- #mi HbQ KBtBM;X  (n) U1 ≤ i ≤ M V QM R × K× M/ *QMbi`m+i `2+m`bBp2Hv TQBMi T`Q+2bb2b N i (n) MQM@M2;iBp2 biQ+?biB+ T`Q+2bb2b {λi (t)}t∈R U1 ≤ i ≤ M V- n ≥ 0V #v (n)  (n) (dt × dz) = N (n) N i (dt × dz × [0, λi (t)]) i

M/ (n+1)

λi

  1(n) , . . . , St N  (n) , (t) = ϕi St N M

URkXe3V

URkXeNV

(0)

bi`iBM; rBi? λi (t) ≡ 0 U1 ≤ i ≤ M VX "v E2`biMǶb +QM/BiBQM &  & & (n+1) & (n+1) (0)& − λi (0) E &λi

  M & &  &  (n)  (n−1) && (dt × dz) . E hji (−t, z) &N − N ≤ i i i=1

(−∞,0)

K

& & &  (n)  (n−1) && +QmMib i?2 TQBMib (t, z, u) Q7 N i bm+? i?i u 7HHb h?2 TQBMi T`Q+2bb &N −N i i (n)

(n−1)

#2ir22M λi (t) M/ λi (t)X "v `2+m``2M+2- Bi Bb 2bBHv b?QrM i?i i?2 T`Q+2bb2b (n) {λi (t)}t∈R U1 ≤ i ≤ M V `2 Ft @T`2/B+i#H2X "v h?2Q`2K 8XdXj- i?2 F2`M2H & & & & (n) (n−1) (t)& Qi (dz) &λi (t) − λi & & &  (n)  (n−1) &&X AM T`iB+mH`- #v i?2 bKQQi?@ Bb  biQ+?biB+ Ft @BMi2MbBiv F2`M2H Q7 &N − N i i BM; i?2Q`2K Uh?2Q`2K 8XRXkyV-

8y3

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a & &  & (n+1) & (n) E &λi (0)& − λi (0) 

 M & &  & (n) & (n−1) E hji (−t, z) &λi (t) − λi (t)& dt × Qi (dz) . ≤ (−∞,0)

i=1

K

h?Bb `2+m`bBp2 +QMbi`m+iBQM Bb θt @+QKTiB#H2 M/ (P, θt ) Bb 2`;Q/B+ UKBtBM;V- i?2 (n) biQ+?biB+ T`Q+2bb2b {λi (t)}t∈R U1 ≤ i ≤ M V- n ≥ 0V `2 DQBMiHv biiBQM`v- M/ i?2`27Q`2 & &  & (n+1) & (n) E &λi (0)& − λi (0)  M &  &  & (n) & (n−1) E &λi (0) − λi (0)& hji (−t, z) dt × Qi (dz) , ≤ (−∞,0)

i=1

i?i Bb-

K

& & & & & (n+1) & & (n) & (n) (n−1) E &λi (0) − λi (0)& ≤ AE &λi (0) − λi (0)& .

6Q` HH k ≥ 0- i?2 MQ`K Q7 An Bb 2[mBpH2Mi iQ nk ρ(A)n b n ↑ ∞X aBM+2 #v ?vTQi?2bBb ρ(A) < 1- Bi 7QHHQrb i?i &  && (n) & (n−1) E &λi (0) − λi (0)& < ∞ (1 ≤ i ≤ M ) n≥0

Q`- 2[mBpH2MiHv #v biiBQM`Biv&  && (n) & (n−1) E &λi (t) − λi (t)& < ∞

(t ∈ R) .

n≥0 (n)

h?Bb BKTHB2b i?i 7Q` HH t ∈ R- λi (t) +QMp2`;2b HKQbi bm`2Hv M/ BM L1 b n → ∞X JQ`2Qp2`- 7Q` HH #QmM/2/ C ∈ B(R) &    && (n) (n−1) &   P − N N (dt × dz) =  0 & i & i C

n≥0



K



E

n≥0

= (C)

  &

& &  (n)  (n−1) && (dt × dz) &Ni − N i C



K

& & & (n) & (n−1) E &λi (0) − λi (0)& < ∞ .

n≥0

"v i?2 "Q`2HĜ*Mi2HHB H2KK   & & &  (n)  (n−1) && (dt × dz) = 0 BXQX = 0 . P &Ni − N i C

K

 (n) (C × K) `2KBMb +QMbiMi 7i2`  AM T`iB+mH`- 7Q` HH #QmM/2/ C ∈ B(R)- N i i }MBi2 `M/QK iBK2X h?Bb K2Mb i?i i?2`2 2tBbib  HQ+HHv }MBi2 TQBMi T`Q+2bb N (n) (  C × K) 7Q` HH #QmM/2/ C ∈ B(R)X  (C × K) = N bm+? i?i limn↑∞ N i i i +QmMib i?2 TQBMib (t, z) bm+? i?i |N | ({t, z} × [0, λi (t)]) > 0X q2 b?Qr i?i N i AM 7+i- #v 6iQmǶb H2KK-

RkXeX E1_ahL SPALh S_P*1aa1a E

8yN

  & &

& & &Ni (dt × dz) − N i (dt × dz × [0, λi (t)])& C K   & &

& & (n) ≤ lim E &N i (dt × dz × [0, λi (t)]) − N i (dt × dz × [0, λi (t)])& n↑∞ C K & & & (n) & = (C) lim E &λi (0) − λi (0)& = 0 . n↑∞

i /KBib i?2 biQ+?biB+ Ft @BMi2MbBiv F2`M2H λi (t)Qi (dz)X Ai `2KBMb iQ h?2`27Q`2- N   1 , . . . , St N M , p2`B7v i?i λi (t) = ϕi St N  & & & 1 , . . . , St N M && E &λi (t) − ϕi St N & & & & (n) ≤ E &λi (t) − λi (t)&

 & &   &  (n−1) && (ds × dz) + E hji (t − s, z) &N j − Nj j

& & & & (n) = E &λi (t) − λi (t)&   + E

(−∞,t)







0

j

L

& & & & (n−1) hji (s, z) ds Qi (dz) E &λi (0) − λi (0)& . L

 & & & 1 , . . . , St N M && = 0- i?mb +QM+Hm/BM; i?2 G2iiBM; n → ∞ vB2H/b E &λi (t) − ϕi St N T`QQ7X  Bb θt @+QKTiB#H2- M/ i?2`27Q`2 2`;Q/B+X LQi2 i?i i?2 HBKBiBM; T`Q+2bb N ai2T kX q2 MQr b?Qr +QMp2`;2M+2 BM /Bbi`B#miBQM iQ i?2 #Qp2 biiBQM`v bQHmiBQM   }`bi rBi? BMBiBH +QM/BiBQM UPd,1 VX h?Bb rBHH ;m`Mi22 Q7  i`MbB2Mi bQHmiBQM N mMB[m2M2bb Q7  biiBQM`v /Bbi`B#miBQM rBi? }MBi2 p2`;2 BMi2MbBiB2b biBb7vBM;   Bb bm+? biiBQM`v bQHmiBQM rBi? }MBi2 BMi2MbBiB2b +QM/BiBQM UPd,1 V #2+mb2 B7 N   λ1 , . . . , λM     t  εa (t, N− ) = λj hji (s − u, z) du Qi (dz) ds i,j

=



λj



 i,j

t



λj a

(−∞,0]



K



hji (u, z) du Qi (dz) ds t−a

i,j



t−a



(s,∞)



K

hji (u, z) du Qi (dz) , (s,∞)

K

  biBb}2b +QM/BiBQM UPd,1 VX 7`QK r?B+? r2 b22 i?i N   QM (−∞, 0] biBb7vBM; +QM/BiBQM UPd,1 VX h?2M G2i #2 ;Bp2M  TQBMi T`Q+2bb N   QM (0, ∞) rBi? i?2 /2bB`2/ /vMKB+b b 7QHHQrbX G2i N i U1 ≤ i ≤ M V +QMbi`m+i N #2 BM/2T2M/2Mi SQBbbQM T`Q+2bb2b QM (0, ∞)×K rBi? `2bT2+iBp2 BMi2MbBiv K2bm`2b   QM (−∞, 0]X *QMbi`m+i

× Qi U1 ≤ i ≤ M V M/ BM/2T2M/2Mi Q7 i?2 #Qp2 ;Bp2M N  (n)  `2+m`bBp2Hv i?2 TQBMi T`Q+2bb2b N U1 ≤ i ≤ M V QM R × K #v BKTQbBM; i?i i?2v i   +QBM+B/2 rBi? Ni U1 ≤ i ≤ M V QM (−∞, 0] × K- M/ i?i QM (0, +∞)-

8Ry

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a (n)  (n) (dt × dz) = N (n) (t)]) N i (dt × dz × [0, λi i

M/

(n+1)

λi

  1(n) , . . . , St N  (n) , (t) = ϕi St N M

URkXdyV

(0)

bi`iBM; rBi? λi (t) ≡ 0 U1 ≤ i ≤ M VX AM pB2r Q7 E2`biMǶb +QM/BiBQM  0

T

&  &  & (n+1) & (n) E &λi (t) − λi (t)& | F0N dt

   ∞  hji (t, z) dt Qj (dz) ≤ j

0

T

& &   & (n) & (n−1) E &λi (t) − λi (t)& | F0N dt .

0

K

URkXdRV E2`biMǶb +QM/BiBQM HbQ ;m`Mi22b i?i   (1) hji (t − s, z)tNj (ds × dz) , λi (t) ≤ ϕi (∅) + (−∞,0]

j

K (1)

M BM2[mHBiv r?B+?- BM pB2r Q7 i?2 BMBiBH +QM/BiBQM UPd,1 V- BKTHB2b i?i λi (t) Bb (n) HQ+HHv BMi2;`#H2X AM2[mHBiB2b URkXdRV i?2M b?Qr i?i λi (t) Bb HQ+HHv BMi2;`#H2 7Q` HH n ≥ 1 U1 ≤ i ≤ M VX *QM/BiBQM ρ(A) < 1 M/ BM2[mHBiB2b URkXdRV ;m`Mi22 (n)  (n) /KBi `2bT2+iBp2 HBKBib λ (t) M/ N   bm+? i?i 7Q` i?i i?2 λi (t)Ƕb M/ i?2 N i i i HH C ∈ B((0, ∞)) M/ HH L ∈ K    (C × L) = N N i (dt × dz × [0, λi (t)]) . i C

L

  )- M/ qBi? i?2 bK2 `;mK2Mib b BM ai2T R- QM2 b?Qrb i?i λi (t) = ϕi (St (N   ?b i?2 `2[mB`2/ /vMKB+bX h?Bb BKTHB2b i?i i?2`27Q`2 i?2 HBKBiBM; T`Q+2bb N    T     N dt < ∞ M/ i?2`27Q`2 E λi (t) | FtN Bb HQ+HHv BMi2;`#H2 Ui?Bb E λi (t) | Ft 0 rBHH #2 M22/2/ Hi2` QMVX  #2 i?2 biiBQM`v bQHmiBQM +QMbi`m+i2/ 7`QK i?2 SQBbbQM T`Q+2bb N G2i MQr N b BM ai2T RX 6Q` HH i U1 ≤ i ≤ M V H2i    fi (t) := E |λi (t) − λi (t)| | FtN 1(0,∞) (t) ,  )X hFBM; E2`biMǶb +QM/BiBQM BMiQ ++QmMir?2`2 λi (t) = ϕi (St (N     (ds × dz) hji (t − s, z)N fi (t) ≤ j j

+

(−∞,0)

 j

+

K







λj t

 t j

0

hji (s, z) ds Qj (dz)

URkXdkV

hji (t − s, z)fj (s) ds Qj (dz) ,

URkXdjV

K

K

RkXeX E1_ahL SPALh S_P*1aa1a

8RR

r?2`2 λj Bb i?2 p2`;2 BMi2MbBiv Q7 Nj X G2i a > 0 #2 }t2/ M/ H2i Fi (t) := t f (s) dsX AMi2;`iBM; i?2 T`2pBQmb BM2[mHBiv #2ir22M t − a M/ t- r2 Q#iBM t−a i Fi (t) ≤ εi (t) +

 t j

0

hji (t − s, z)Fj (s) ds Qj (dz) ,

URkXd9V

K

r?2`2 εi (t) :=

 j

t



t−a

 (−∞,0)

  (ds × dz) hji (u − s, z)N j K

+

 j



 du



λj a

hji (s, z) ds Qj (dz) . t−a

h?2  fi Ƕb `2 HQ+HHv BMi2;`#H2- #2+mb2 r2 ?p2  b22M  72r  HBM2b #Qp2 i?i    N N E λi (t) | Ft Bb HQ+HHv BMi2;`#H2- M/ bQ Bb E λi (t) | Ft #2+mb2 E [λi (t)] = λi < ∞X h?2`27Q`2 i?2 7mM+iBQMb Fi M/ εi U1 ≤ i ≤ M V `2 #QmM/2/ QM }MBi2 BMi2`pHbX h?2`27Q`2- #v Bi2`iBM; BM2[mHBiv URkXd9V M/ TbbBM; iQ i?2 HBKBi Fi (t) ≤

 n≥0

j

t (n)

εj (t − s)gij (s) ds ,

URkXd8V

0

(0)

r?2`2 gij (t) = 1{i=j} 1{t=0} M/ (n+1) gij (t)

=

 t k

0

hki (t − s, z) ds Qk (dz) . K

h?2 `B;?i@?M/ bB/2 Q7 URkXd8V 72im`2b i?2 KmHiBp`Bi2 `2M2rH K2bm`2 bbQ+Bi2/  rBi? i?2 /Bbi`B#miBQM 7mM+iBQMb K hji (t, z) Qj (dz)X aBM+2 ρ(A) < 1- i?2 `B;?i@?M/ bB/2 Q7 URkXd8V Bb }MBi2X aBM+2 εj (t) Bb #QmM/2/ M/ i2M/b iQ 0 b t → +∞- #v /QKBMi2/ +QMp2`;2M+2 TTHB2/ iQ URkXd8V- limt↑∞ Fi (t) = 0X P#b2`p2 i?i 

 & &  & & N  Fi (t) = E &Ni − Ni & (ds × dz) | F0 t−a,t]

K

  QM [t − a, t] × K | F N  ) . i = N ≥ 1 − P (N i 0   +QMp2`;2 BM /Bbi`B#m@ h?Bb BKTHB2b i?i i?2 }MBi2@/BK2MbBQMH /Bbi`B#miBQMb Q7 St N   +QMp2`;2b BM /Bbi`B#miBQM iBQM iQ i?Qb2 Q7 N - r?B+? Bb 2MQm;? iQ ;m`Mi22 i?i St N iQ N Uh?2Q`2K RX9XdVX h?2 bK2 +QM+HmbBQM ?QH/b B7 i?2 i`MbB2Mi bQHmiBQM ?b i?2 BMBiBH +QM/BiBQM UPd,2 V BMbi2/ Q7 UPd,1 VX Ai bm{+2b iQ TTHv i?2 `;mK2Mib #Qp2 rBi? fi (t) M/ Fi (t) `2TH+2/ #v i?2B` 2tT2+iiBQMbX ai2T jX q2 MQr b?Qr +QMp2`;2M+2 BM p`BiBQM iQ i?2 biiBQM`v bQHmiBQM Ur?Qb2   rBi? BMBiBH +QM/BiBQMb 2tBbi2M+2 rb T`Qp2/ BM ai2T RV Q7  i`MbB2Mi bQHmiBQM N UPv V M/ mM/2` +QM/BiBQM URkXedVX Ai bm{+2b iQ b?Qr i?i i?2 i`MbB2Mi bQHmiBQM M/

8Rk

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

i?2 biiBQM`v bQHmiBQM +QmTH2 BM }MBi2 iBK2X 6Q` i?Bb- BMi2;`i2 BM2[mHBiv URkXdjV 7`QK 0 iQ T 7Q` }MBi2 TQbBiBp2 T iQ Q#iBM  T  ∞  T fi (t) dt ≤ εi + fi (t) dt × hji (t, z) dt Qj (dz) , 0

0

j

r?2`2 εi =

 j

+





0

j

 (−∞,0]





0



K

  (ds × dz) hji (t − s, z) N j

 dt

K

thji (t, z) dt Qj (dz) 0

K

T ++Q`/BM; iQ UPv V M/ URkXedV- 0 fi (t) dt M/ εi `2 }MBi2 7Q` HH T > 0X Ai2`iBM; i?2 Hbi BM2[mHBiv vB2H/b- BM p2+iQ` 7Q`K T  f (t) dt ≤ An ε . 0

n≥0

∞ h?2`27Q`2- bBM+2 i?2 #QmM/ Bb BM/2T2M/2Mi Q7 T - 0 fi (t) dt < ∞X "mi

  ∞  & &  & & N  fi (t) dt = E &Ni − Ni & (dt × dz) | F0 (0,+∞]

0

K

i M/ N i Bb }MBi2X aB/ Qi?@ M/ i?2`27Q`2 i?2 MmK#2` Q7 TQBMib MQi +QKKQM iQ N      +QMp2`;2b iQ N  2`rBb2- Ni M/ Ni +QmTH2 BM }MBi2 iBK2- r?B+? BKTHB2b i?i St N BM p`BiBQMX  _i2 Q7 1tiBM+iBQM *QMbB/2`  MQM@HBM2` >rF2b TQBMi T`Q+2bb N r?B+? Bb 2KTiv QM (−∞, 0]- rBi? /vMKB+b QM R+   λ(t) = ν(t) + ϕ h(t − s, z)NZ (ds × dz) , URkXdeV (0,t)×K

r?2`2 ν: R+ →[0, ∞)- ν = 0- Bb  HQ+HHv BMi2;`#H2 7mM+iBQM- ϕ: R+ → [0, ∞)ϕ(0) = 0- Bb  GBTb+?Bix 7mM+iBQM rBi? GBTb+?Bix +QMbiMi R- M/ h : R × L → R Bb  K2bm`#H2 7mM+iBQM UMQM@M2+2bb`BHv MQM@M2;iBp2V r?B+? biBb}2b URkX9VX bbmKBM;  ∞

E[|h(t, Z1 )|]dt < 1 , 0

i?2 mMB[m2 biiBQM`v T`Q+2bb N (0) +Q``2bTQM/BM; iQ i?2 /vMKB+b   (0) h(t − s, z)NZ (ds × dz) ϕ (0,t)×K

Bb i?2 2KTiv T`Q+2bbX A7 KQ`2Qp2` r2 bmTTQb2 URkXeV M/

URkXddV

RkXeX E1_ahL SPALh S_P*1aa1a 



tE[|h(t, Z1 )|]dt < ∞

8Rj URkXd3V

0

2+? T`Q+2bb N rBi? /vMKB+b URkXdeV Bb bm+? i?i St N i2M/b BM p`BiBQM iQ i?2 2KTiv T`Q+2bbX >2`2 ;BM- +QMp2`;2M+2 BM p`BiBQM iF2b TH+2 pB +QmTHBM;- M/ +QM/BiBQMb URkXddV M/ URkXeV `2 bm{+B2Mi 7Q` i?2 BMi2;`#BHBiv Q7 E[λ(t)]X AM i?2 2tTQM2MiBH +b2- i?i Bb r?2M bQK2 β > 0 2tBbib bm+? i?i  ∞ eβt E[|h(t, Z1 )|]dt = 1 , URkXdNV 0

r2 ?p2 i?2 7QHHQrBM; `2bmHi, h?2Q`2K RkXeXRR lM/2` bbmKTiBQMb (12.79) M/ URkXNV- M/ B7 i?2 7mM+iBQM eβt ν(t) Bb /B`2+iHv _B2KMM@BMi2;`#H2 QM R+ - i?2M 7Q` HH K bm+? i?i  ∞ βt e ν(t)dt URkX3yV K >  ∞ 0 βt β 0 te E[|h(t, Z1 )|]dt i?2`2 2tBbib  t0 = t0 (K) bm+? i?i 7Q` HH t ≥ t0 P (T > t) ≤ Ke−βt . AM i?2 #Qp2 BM2[mHBiv (12.80) i?2 `B;?i@?M/ bB/2 Bb /2}M2/ b 0 B7 i?2 /2MQKBMiQ` Bb BM}MBivX S`QQ7X h?2 T`QQ7 Bb bBKBH` iQ i?i Q7 h?2Q`2K RkXkXRX h?2 mMB[m2 /Bz2`2M+2 HB2b BM i?i- BM i?2 MQM@HBM2` +b2- r2 ?p2- #v i?2 GBTb+?Bix T`QT2`iv Q7 ϕ- i?2 /272+iBp2 `2M2rH BM2[mHBiv  t λ(t) ≤ ν(t) + λ(t − s)E[|h(s, Z1 )|]ds , URkX3RV 0

r?B+? vB2H/b- #v Bi2`iBM; M/ iFBM; BMiQ ++QmMi i?i λ(t) Bb HQ+HHv #QmM/2/ QM R+ λ(t) ≤ ξ(t) , r?2`2 ξ(t) biBb}2b i?2 /272+iBp2 `2M2rH 2[miBQM  t ξ(t − s)E[|h(s, Z1 )|]ds. ξ(t) = ν(t) + 0

h?2 `2bmHi 7QHHQrb #v TTHvBM; i?2 `2bmHib Q7 am#b2+iBQM 9XkX



AM i?2 bm#2tTQM2MiBH +b2, h?2Q`2K RkXeXRk lM/2` bbmKTiBQMb URkXddV- URkXd3V M/ URkXNV- B7 KQ`2Qp2` i?2 /Bbi`B#miBQM 7mM+iBQM G rBi? /2MbBiv E[|h(t, Z1 )|] g(t) =  ∞ E[|h(t, Z1 )|]dt 0

8R9

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

Bb bm#2tTQM2MiBH M/ i?2 7mM+iBQM ν Bb #QmM/2/ M/ bm+? i?i B = lim sup t→∞

ν(t) < ∞, G(t)

i?2M 7Q` HH K bm+? i?i K>

1−

∞ 0

B E[|h(t, Z1 )|]dt

i?2`2 2tBbib  t0 = t0 (K) bm+? i?i 7Q` HH t ≥ t0  ∞ P (T > t) ≤ K G(s)ds . t

h?2 T`QQ7 Bb MHQ;Qmb iQ i?i Q7 h?2Q`2K RkXkXjX Ai bm{+2b iQ MQi2 i?i- BM i?2 T`2b2Mi +b2- BM2[mHBiv URkX3RV biBHH ?QH/bX "QmM/2/ GBTb+?Bix .vMKB+b h?2Q`2K RkXeXRj Ukk V G2i ϕ #2 α@GBTb+?Bix 7Q` bQK2 α > 0 M/ #QmM/2/ #v Λ > 0- M/ H2i h #2 BMi2;`#H2 QM R+ M/ bm+? i?i  ∞ t|h(t)| dt > ∞ . URkX3kV 0

h?2M i?2`2 2tBbib  mMB[m2 biiBQM`v /Bbi`B#miBQM Q7 N rBi? /vMKB+b  t  λ(t) = ϕ h(t − s) N (ds) .

URkX3jV

0

JQ`2Qp2`- i?2 /vMKB+b URkX3jV `2 bi#H2 BM p`BiBQM rBi? `2bT2+i iQ i?2 BMBiBH +QM/BiBQM  t+s  lim |h(s − u)| N (du) ds = 0 a.s. URkX39V t↑∞

t

(−∞,0]

S`QQ7X G2i N #2 M ?TT QM R × R Q7 BMi2MbBiv 1X G2i {Ft }t∈R #2 i?2 BMi2`MH ?BbiQ`v Q7 N - i?i BbFt = σ(N (C) ; C ⊆ (−∞, t] × R) . Ai Kv #2 bbmK2/ i?i N Bb /2}M2/ QM  T`Q##BHBiv bT+2 (Ω, F, P ) 2M/Qr2/ rBi?  K2bm`#H2 ~Qr {θt }t∈R - i?i (P, θt ) Bb 2`;Q/B+ M/ N Bb θt @+QKTiB#H2X 6Q` i?Bb- QM2 Kv +?QQb2 iQ rQ`F QM bQK2 +MQMB+H bT+2 Q7 TQBMi T`Q+2bb2bX .m2 iQ i?2 BM/2T2M/2M+2 T`QT2`iB2b Q7 SQBbbQM T`Q+2bb2b- (P, θt ) rBHH MQi Dmbi #2 2`;Q/B+#mi HbQ KBtBM;X ai2T R, 1tBbi2M+2 Q7  biiBQM`v bQHmiBQMX *QMbi`m+i `2+m`bBp2Hv i?2 FtN @T`2/B+i#H2 T`Q+2bb2b {λn (t)}t∈R M/ i?2 TQBMi T`Q+2bb2b Nn ++Q`/BM; iQ kk

(E2`biM- RNe9)X

RkXeX E1_ahL SPALh S_P*1aa1a

8R8

 λn (t) (C ∈ B(R)) , ds × 0, Λ C   λn+1 (t) = ϕ h(t − s) Nn (ds) 



Nn (C) =

N

URkX38V URkX3eV

(−∞,t)

M/ λ0 (t) ≡ 0X amTTQb2 Ui?Bb rBHH #2 T`Qp2/ Hi2`V i?i- 7Q` HH #QmM/2/ C ∈ B(R)i?2 T`Q+2bb2b Nn XbX `2KBM 2p2MimHHv +QMbiMi QM C b n BM+`2b2b- M/ H2i N /2MQi2 i?2 HBKBiBM; T`Q+2bbX h?Bb T`Q+2bb- r?B+? Bb θt @+QKTiB#H2- Bb i?2`27Q`2 biiBQM`v UM/ KBtBM;V- M/ rBHH ?p2 i?2 2tT2+i2/ /vMKB+b B7 Bi +QmMib i?2  TQBMib Q7 N #2HQr i?2 +m`p2 y(t) = λ(t) = ϕ (−∞,t) h(t − s) N (ds) X G2i C #2 bQK2 #QmM/2/ b2i BM RX "v 6iQmǶb H2KK  &

&

& & &N (ds) − N (ds × 0, λ(s) )& E & & Λ C

&

 & & λn (s) λ(s) && & ≤ lim E &N (ds × 0, Λ ) − N (ds × 0, Λ )& n↑∞ 

 C |λn (0) − λ(0)| . URkX3dV ds lim E n↑∞ Λ C "v i?2 GBTb+?Bix T`QT2`iv Q7 ϕ  |λn (0) − λ(0)| ≤ α

|h(−s)| |Nn − N | (ds) . (−∞,0)

h?2 `B;?i@?M/ bB/2 i2M/b iQ0 b n → ∞- M/ i?2`27Q`2- #v /QKBMi2/ +QMp2`;2M+2 Ui?2 /QKBMiBM; p`B#H2 Bb (−∞,0) |h(−s)| N (ds × [0, Λ]V- i?2 }`bi i2`K BM URkX3dV Bb MmHH- bQ i?i N ?b i?2 `2[mB`2/ BMi2MbBivX h?2 b2[m2M+2 {Nn }n≥1 +QMp2`;2b B7  /2}M2/ #v i?2 TQBMi T`Q+2bb N  ({t}) := lim sup Nn ({t}) − lim inf Nn ({t}) (t ∈ R) N n↑∞

n↑∞

 Bb Bb 2KTivX P#b2`p2 i?i i?2 FtN @BMi2MbBiv Q7 N  := lim sup λn (t) − lim inf λn (t) λ(t) n↑∞

n↑∞

(t ∈ R)

 +QmMib i?2 TQBMib Q7 N #2ir22M i?2 T`2/B+i#H2 +m`p2b t → lim supn↑∞ λn (t) bBM+2 N  b M/ t → lim inf n↑∞ λn (t)X q`BiBM; λ(t) lim sup (λi (t) − λj (t)) ,

n↑∞ i,j≥n

Bi 7QHHQrb 7`QK i?2 GBTb+?Bix bbmKTiBQM i?i     ≤ α lim sup λ(t) |h(t − s)| Ni (ds) − inf n

αA − αB ,

i≥n

(−∞,t)

j≥n

 |h(t − s)| Nj (ds) (−∞,t)

8Re

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

r?2`2- rBi? `#Bi``v TQbBiBp2 a   A = lim sup |h(t − s)| Ni (ds) − inf n

M/

i≥n

  B = lim sup n

j≥n

(−∞,t−a)

i≥n

 |h(t − s)| Nj (ds) (−∞,t−a)



 |h(t − s)| Ni (ds) − inf

j≥n

[t−a,t)

|h(t − s)| Nj (ds)

.

[t−a,t)

aBM+2 i?2`2 `2 QMHv  }MBi2 MmK#2` Q7 TQBMib Q7 N BMpQHp2/ BM B- r2 ?p2   (ds) . B≤ |h(t − s)| N [t−a,t)

b 7Q` A- r2 ?p2 i?2 BKK2/Bi2 #QmM/  |h(t − s)| N (ds × [0, Λ]) . A≤ (−∞,t−a)

h?2`27Q`2- 7Q` `#Bi``v a > 0   ≤α |h(t − s)| N (ds × [0, Λ]) − λ(t) (−∞,t−a)

 (ds) |h(t − s)| N

 .

[t−a,t)

G2iiBM; a → ∞ ≤α λ(t)



 (ds) . |h(t − s)| N

URkX33V

(−∞,t)

 rBHH #2 HKQbi bm`2Hv 2KTiv B7 P (N  (0, +∞) = 0) = 1X aBM+2 h?2 TQBMi T`Q+2bb N   {N (t, +∞) = 0} ⊆ {N (0, +∞) = 0}- #v 2`;Q/B+Biv Q7 (P, θt )- i?Bb rBHH ?QH/ b  (0, +∞) = 0) > 0X 6Q` i?Bb Bi bm{+2b i?i rBi? TQbBiBp2 T`Q##BHBiv bQQM b P (N  (0, +∞) = 0 | F N ) > 0X h?2 F N BMi2MbBiv Q7 N  - r?B+? Bb Q7 i?2 7Q`K v(t, N  − )P (N t  0  | F N M/ i?2`27Q`2 biBb}2b KDQ`iBQM URkX33VX Ai 7QHHQrb i?i 2[mHb E λ(t) t

 −) ≤ v(t, N

 |h(t − s)| N (ds × [0, Λ]) . (−∞,0)

h?2`27Q`2

   (0, +∞) | F N ) = 0) ≥ exp − P (N 0







0

|h(t − s)| N (ds × [0, Λ]) ds

.

(−∞,0)

h?2 `;mK2Mi Q7 i?2 2tTQM2MiBH Bb }MBi2 BM pB2r Q7 bbmKTiBQM URkX3kV- M/ i?2`2@  (0, +∞) | F N ) = 0) > 0- r?B+? +QM+Hm/2b i?2 T`QQ7 Q7 2tBbi2M+2 Q7  7Q`2 P (N 0 biiBQM`v bQHmiBQMX G2i N  #2  TQBMi T`Q+2bb rBi? BMBiBH +QM/BiBQM URkX39VX lb2 h?2Q`2K 8XdX3 iQ +QMbi`m+i M ?TT N bm+? i?i N  +QmMib- QM R+ - i?2 TQBMib Q7 N #2HQr i?2 +m`p2   h(t − s) N  (ds) . t → λ (t) = ϕ (−∞,t]

RkXdX 1s1_*Aa1a

8Rd

*QMbi`m+i i?2 biiBQM`v TQBMi T`Q+2bb N 7`QK N b BM i?2 }`bi T`i Q7 i?2 T`QQ7X  G2i Ft := FtN ∧ FtN Ut ∈ RVX AM T`iB+mH`- {Ft }t∈R Bb  ?BbiQ`v Q7 #Qi? N M/ N  X 6Q` HH s ∈ R M/ HH t > 0- /2}M2 fs (t) =:= P (|N − N  |(s, s + t] = 0 | Fs ) M/ `2+HH i?i i?2 TQBMi T`Q+2bb |N − N  | /KBib i?2 Ft @BMi2MbBiv |λ(t) − λ (t)| QM R+ X "v i?2 GBTb+?Bix bbmKTiBQM QM ϕ     |h(t − s)| N (ds) + |h(t − s)| N (ds) |λ(t) − λ (t)| ≤ α (−∞,0] (−∞,0]  |h(t − s)||N − N  |(ds) . +α (0,t)

`;mBM; 2t+iHv b BM 1tKTH2 8XRXR9 vB2H/b i?2 KBMQ`iBQM   ∞  gs (u) du , fs (∞) ≥ exp − s

r?2`2

 (−∞,s]

G2iiBM;

|h(u − v)| N  (dv)

.

(−∞,0]







 |h(u − v)| N (dv) du

Z(s) = exp s

M/



 |h(u − v)| N (dv) +

gs (u) = α

(−∞,s]

  ε(s) = Z(s) − exp −





gs (u) du

,

s

i?2 Hbi BM2[mHBiv `2/b P (N = N  QM (s, +∞) | Fs ) ≥ Z(s) − ε(s) . h?2 +QM+HmbBQM U+QmTHBM; Q7 N M/ N  V i?2M 7QHHQrb 7`QK h?2Q`2K RX9XR8X AM/22/ Z Bb 2`;Q/B+ bBM+2 N Bbc Bi biBb}2b URX9XR8V #2+mb2 i?2 `;mK2Mi Q7 i?2 2tTQM2MiBH /2}MBM; Z(s) Bb }MBi2- #v bbmKTiBQM URkX3kVc M/ }MHHv- BM pB2r Q7 bbmKTiBQM ∞ URkX39V- s (−∞,0] |h(u − v)| N  (dv) du → 0 b → ∞X 

RkXd

1t2`+Bb2b

1t2`+Bb2 RkXdXRX 1tiBM+iBQM *QMbB/2` i?2 >rF2b #`M+?BM; T`Q+2bb QM [0, ∞) rBi? QM2 M+2biQ`  ∞ i i?2 Q`B;BM M/ rBi? 72`iBHBiv `i2 7mM+iBQM h(t) = ce−αt rBi? c < α UbQ i?i 0 h(t) dt < 1VX TTHv h?2Q`2K RkXeXRR iQ }M/ β M/ M mTT2` #QmM/ 7Q` K bm+? i?i P (T > t) ≤ Ke−βt X

8R3

*>Sh1_ RkX >qE1a SPALh S_P*1aa1a

1t2`+Bb2 RkXdXkX o++BMiBQM M/ +QM7BM2K2Mi aK2 [m2biBQM b BM 1t2`+Bb2 RkXdXR BM i?2 +b2 Q7 p++BMiBQM Q` +QM}M2K2Mi Ub22 1tKTH2 RkXkXkc 7Q` +QM}M2K2Mi- iF2 7Q` Z1  `M/QK p`B#H2 mMB7Q`KHv /Bbi`B#mi2/ BM [0, T ] 7Q` bQK2 T > 0VX 1t2`+Bb2 RkXdXjX AKT`Qp2/ `i2 Q7 2tiBM+iBQM lb2 i?2 #QmM/b ;Bp2M BM 1tKTH2 8XRXR8 iQ ;Bp2 mTT2` M/ HQr2` #QmM/b Q7 i?2 iBH /Bbi`B#miBQM Q7 i?2 iBK2 iQ 2tiBM+iBQM- i?mb BKT`QpBM; M/ +QKTH2K2MiBM; h?2Q`2K RkXkXRX 1t2`+Bb2 RkXdX9X AM7Q`KiBQM i`Mb72` #v M2m`QMb *QMbB/2` i?2 KQ/2H Q7 1tKTH2 RkXeXR- bHB;?iHv KQ/B}2/   t  λ(t) = Λ exp − h(t − s) N (ds) , −∞

r?2`2 h : R+ → R+ U TQbBiBp2 7mM+iBQM QM (0, +∞- MmHH QM (−∞, 0]V r?Qb2 bmTTQ`i Bb BM+Hm/2/ BM [0, a] 7Q` bQK2 TQbBiBp2 }MBi2 a- M/ r?2`2 Bb  TQbBiBp2 `M/QK p`B#H2 rBi? +/7 F X h?Bb `M/QK p`B#H2 Kv #2 +QMbB/2`2/ i?2 BMi2MbBiv Q7 i?2 bB;MH TTHB2/ iQ i?2 M2m`QMX :Bp2 M 2tT`2bbBQM 7Q` UV E Λ | FtN - M/ U#V i?2 KmimH BM7Q`KiBQM #2ir22M Λ M/ N X

TT2M/Bt XR

J2bm`#BHBiv M/ J2bm`2

.vMFBMǶb h?2Q`2K h?2 i2+?MB+H `2bmHi FMQrM b .vMFBMǶb i?2Q`2K Qr2b Bib BKTQ`iM+2 iQ i?2 7+i i?i σ@}2H/b `2 Q7i2M /2}M2/ BM i2`Kb Q7 i?2B` ;2M2`iQ`b- M/ i?2`27Q`2- QM2 M22/b bQK2 iQQHb iQ /2i2`KBM2 r?2M  ;Bp2M T`QT2`iv- T`Qp2/ 7Q` bQK2 +Hbb Q7 b2ib- 2ti2M/b iQ HH i?2 b2ib BM i?2 σ@}2H/ ;2M2`i2/ #v i?Bb T`iB+mH` +HbbX .2}MBiBQM XRXR G2i X #2 M `#Bi``v b2iX h?2 +QHH2+iBQM S ⊆ P(X) Bb +HH2/  π@bvbi2K UQ7 b2ibV B7 Bi Bb +HQb2/ mM/2` }MBi2 BMi2`b2+iBQMbX .2}MBiBQM XRXk G2i X #2 M `#Bi``v b2iX  MQM2KTiv +QHH2+iBQM Q7 b2ib S ∈ P(X) Bb +HH2/  /@bvbi2K UQ7 b2ibV B7 UV X, ∅ ∈ SX U#V S Bb bi#H2 mM/2` bi`B+i /Bz2`2M+2 Ui?i Bb- B7 A, B ∈ S M/ A ⊆ B- i?2M B − A ∈ SVX U+V S Bb bi#H2 mM/2` b2[m2MiBH BM+`2bBM; HBKBib Ui?i Bb- i?2 HBKBi Q7  MQM@ /2+`2bBM; b2[m2M+2 Q7 b2ib BM S Bb BM SVX AM i?2 HBi2`im`2-  /@bvbi2K Bb bQK2iBK2b +HH2/  .vMFBM bvbi2KX h?2Q`2K XRXj G2i X #2 M `#Bi``v b2iX A7 i?2 +QHH2+iBQM S ∈ P(X) Bb  π@ bvbi2K M/  /@bvbi2K- Bi Bb  σ@}2H/X h?2 bKHH2bi /@bvbi2K +QMiBMBM;  MQM@2KTiv +QHH2+iBQM Q7 bm#b2ib C ⊆ P(X) Bb /2MQi2/ #v d(C)X q2 MQr bii2 .vMFBMǶb i?2Q`2KX h?2Q`2K XRX9 G2i S #2  π@bvbi2K /2}M2/ QM XX h?2M d(S) = σ(S)X h?2 M2ti `2bmHi Bb +HH2/ i?2 7mM+iBQMH .vMFBMǶb i?2Q`2KX Ai Bb mb27mH BM T`QpBM; i?i  ;Bp2M +Hbb Q7 7mM+iBQMb +QMiBMb HH i?2 7mM+iBQMb K2bm`#H2 rBi? `2bT2+i iQ  +2`iBM σ@}2H/X

© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9

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X k−1

S`QQ7X .2MQi2 #v T i?2 7QHHQrBM; #BD2+iBp2 HBM2` i`Mb7Q`KiBQM 7`QK X k QMiQ Bib2H7, (x1 , . . . , xk ) → (x1 , x1 + x2 , . . . , x1 + xk ) . aBM+2



 f (x)μ(dx) = X

X

f (T x)(μ ◦ T )(dx) ,

r2 ?p2 iQ T`Qp2 i?i i?2 K2bm`2 ν := μ ◦ T Bb Q7 i?2 7Q`K ν(dx1 × · · · × dxk ) = dx1 × μ(dx2 × · · · × dxk ) 7Q` bQK2 HQ+HHv }MBi2 K2bm`2 μ QM X k−1 X 6Q` HH MQM@M2;iBp2 K2bm`#H2 7mM+@ iBQMb ϕ : X k → R   ϕ(x1 , . . . , xk )ν(dx1 × · · · × dxk ) = ϕ(x)(μ ◦ T )(dx) = ϕ(T −1 x)μ(dx) Xk Xk Xk  = ϕ(x1 , x2 − x1 , . . . , xk − x1 )μ(dx1 × · · · × dxk ) k X = ϕ(x1 + h, x2 − x1 , . . . , xk − x1 )μ(dx1 × · · · × dxk ), Xk

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Xk

i?i Bb- ν Bb BMp`BMi rBi? `2bT2+i iQ i?2 b?B7i +iBM; QM i?2 }`bi +QQ`/BMi2X AM T`iB+mH`- 7Q` HH C1 ∈ X - HH B ∈ X ⊗(k−1) M/ HH h ∈ Xν((C1 + h) × B) = ν(C1 × B) . h?mb 7Q` }t2/ B ∈ X k−1 - i?2 K2bm`2 ν(· × B) Bb BMp`BMi mM/2` i?2 b?B7i- M/ Bb i?2`27Q`2  KmHiBTH2 Q7 i?2 G2#2b;m2 K2bm`2, ν(C1 × B) = m (C1 ) × μ(B) , r?2`2 i?2 +QMbiMi U+QMbiMi rBi? `2bT2+i iQ C1 V μ(B) +H2`Hv /2}M2b  HQ+HHv }MBi2 K2bm`2 μ QM X k X 

*QMp2`;2M+2 BM o`BiBQM q2 b?HH [mQi2 rBi?Qmi T`QQ7 i?2 KBM #bB+ ;2M2`H `2bmHib Q7 i?2 i?2Q`v Q7 +QMp2`;2M+2 BM p`BiBQMXkj .2}MBiBQM XRXR8 G2i P1 M/ P2 #2 irQ T`Q##BHBiv K2bm`2b QM i?2 bK2 K2@ bm`#H2 bT+2 (X, X )X h?2 [mMiBiv dV (P1 , P2 ) := sup |P1 (A) − P2 (A)| A∈X

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h?2Q`2K XRXRe q2 ?p2 i?i dV (P1 , P2 ) =

1 2

 |f1 − f2 | dQ .

URkXNyV

X

Ai 7QHHQrb 7`QK i?2 2tT`2bbBQM URkXNyV i?i dV Bb BM/22/  /BbiM+2 7mM+iBQMX kj

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dV (P1 , P2 ) = dV (P1 , P2 ) .

h?2Q`2K XRXR3 G2i h : Ω → Ω #2  K2bm`#H2 TTHB+iBQM Q7 Ω QMiQ Bib2H7 M/ H2i P1 M/ P2 #2 i?2 BK;2b #v h Q7- `2bT2+iBp2Hv- i?2 T`Q##BHBiB2b P1 M/ P2 QM (Ω, F)X h?2M dV (P1 , P2 ) ≤ dV (P1 , P2 ) , rBi? 2[mHBiv B7 KQ`2Qp2` h /KBib  K2bm`#H2 BMp2`b2 h−1 X .2}MBiBQM XRXRN G2i X M/ Y #2 irQ `M/QK 2H2K2Mib rBi? pHm2b BM i?2 K2bm`#H2 bT+2 (U, U)X h?2 p`BiBQM /BbiM+2 dV (X, Y ) #2ir22M X M/ Y Bb- #v /2}MBiBQM- i?2 p`BiBQM /BbiM+2 #2ir22M i?2B` /Bbi`B#miBQMb, dV (X, Y ) := sup (P (X ∈ A) − P (Y ∈ A)) . A∈U

h?2Q`2K XRXky 6Q` Mv TB` (X, Y ) ∈ D(α, β)- r2 ?p2 i?2 7mM/K2MiH +Qm@ THBM; BM2[mHBiv dV (α, β) ≤ P (X = Y ) , URkXNRV r?2`2 2[mHBiv Bb iiBM2/ #v bQK2 TB` (X, Y ) ∈ D(α, β)- r?B+? Bb i?2M bB/ iQ `2HBx2 KtBKH +QBM+B/2M+2X

.2}MBiBQM XRXkR X h?2 b2[m2M+2 {Pn }n≥1 Q7 T`Q##BHBiv K2bm`2b QM (X, X ) Bb bB/ iQ +QMp2`;2 BM p`BiBQM iQ i?2 T`Q##BHBiv P QM (X, X ) B7 lim dV (Pn , P ) = 0 .

n↑∞ V ar.

h?Bb Bb /2MQi2/ Pn → P X "X G2i {Xn }n≥1 #2 `M/QK 2H2K2Mib rBi? pHm2b BM i?2 K2bm`#H2 bT+2 (U, U)- rBi? `2bT2+iBp2 /Bbi`B#miBQMb α M/ {αn }n≥1 X h?2 b2[m2M+2 {Xn }n≥1 Bb bB/ iQ +QMp2`;2 V ar. BM p`BiBQM iQ X B7 limn↑∞ dV (Xn , X) = 0X h?Bb Bb /2MQi2/ #v Xn → XX k9

A#B/2K- TX R3y- KmiiBb KmiM/BbX

XRX J1al_"AGAhu L. J1al_1

8k8

G2i Q #2  T`Q##BHBiv K2bm`2 bm+? i?i 7Q` HH n ≥ 1- Pn  Q- 7Q` BMbiM+2 Q=

 1 Pn . 2n n≥1

.2MQi2 #v fn U`2bTX- f V i?2 _/QMĜLBFQ/ɷK /2`BpiBp2 Q7 Pn U`2bTX- P V rBi? `2bT2+i L1

V ar.

iQ QX "v h?2Q`2K XRXRe- Pn → P B7 M/ QMHv B7 fn → f - r?2`2 L1 = L1C (Q)X LQi2 HbQ i?i B7 ϕ : X → C Bb  #QmM/2/ 7mM+iBQM- i?2M   ϕ dPn → ϕ dP , X

X

b 7QHHQrb 7`QK i?2 7+i i?i    ϕ dPn − ϕ dP = ϕ × (fn − f ) dQ X

X

X

M/ /QKBMi2/ +QMp2`;2M+2X h?2Q`2K XRXkk G2i Pn - Q M/ fn #2 /2}M2/ b #Qp2X amTTQb2 i?i i?2`2 2tBbib  MQM@M2;iBp2 K2bm`#H2 7mM+iBQM f 7`QK (X, X ) iQ (R, B) bm+? i?i Q@X2X V ar. fn → f X h?2M Pn → P - r?2`2 P Bb i?2 T`Q##BHBiv /2}M2/ #v P (A) = A f dQA ∈ XX h?2 T`QQ7 Bb  /B`2+i +QMb2[m2M+2 Q7 a+?2zûǶb H2KK, G2KK XRXkj G2i f M/ fn - n ≥ 1- #2 Q@BMi2;`#H2 MQM@M2;iBp2 `2H 7mM+iBQMb   7`QK (X, X ) iQ (R, B)rBi? lim f = f Q@X2X M/ lim f dQ = X f dQX n↑∞ n n↑∞ n X  h?2M limn↑∞ X |fn − f | dQ = 0X w2`Q@QM2 Grb h?2 x2`Q@QM2 Hrb T`2b2Mi2/ BM i?Bb b2+iBQM +QM+2`M bvKTiQiB+ 2p2Mib M/ ;Bp2 +QM/BiBQMb mM/2` r?B+? bm+? 2p2Mib `2 i`BpBH Ui?i Bb- rBi? T`Q##BHBiv 0 Q` 1VX .2}MBiBQM XRXk9 G2i {Xn }n≥1 #2  b2[m2M+2 Q7 `M/QK p`B#H2b- M/ /2}M2 FnX := σ(X1 , . . . , Xn )X h?2 σ@}2H/ T X := ∩n≥1 σ(Xn , Xn+1 , . . .) Bb +HH2/ i?2 iBH σ@}2H/ Q7 i?Bb b2[m2M+2X h?2Q`2K XRXk8 h?2 iBH σ@}2H/ Q7  b2[m2M+2 {Xn }n≥1 Q7 BM/2T2M/2Mi `M/QK p`B#H2b Bb i`BpBH- i?i Bb- B7 A ∈ T X - i?2M P (A) = 0 Q` 1X G2i (S, S) #2 bQK2 K2bm`#H2 bT+2 M/ H2i μ #2  T`Q##BHBiv K2bm`2 QM BiX q2 b?HH rQ`F QM i?2 +MQMB+H K2bm`#H2 bT+2 Q7 S@pHm2/ `M/QK b2[m2M+2b(Ω, F) := (S N , S ⊗N ) , 2M/Qr2/ rBi? i?2 T`Q##BHBiv K2bm`2

8ke

SS1L.As P := μ⊗∞ .

AM T`iB+mH`- M 2H2K2Mi Q7 Ω ?b i?2 7Q`K ω = {x = (x1 , x2 , . . .) ∈ S N } . 6Q` 2+? n ≥ 1- i?2 b2[m2M+2 {Xn }n≥1 /2}M2/ #v Xn (ω) := xn Bb BB/- M/ i?2 +QKKQM T`Q##BHBiv /Bbi`B#miBQM Q7 i?2 Xn Ƕb Bb μX LQi2 i?i Xn (πω) = Xπ(n) (ω)X .2}MBiBQM XRXke UV  }MBi2 T2`KmiiBQM Q7 N Bb  T2`KmiiBQM π bm+? i?i π(i) = i 7Q` HH #mi  }MBi2 MmK#2` Q7 BM/B+2b i ≥ 1X U#V M 2p2Mi A ∈ F bm+? π −1 A = A Bb +HH2/ 2t+?M;2#H2X U+V h?2 σ@}2H/ E +QMbBbiBM; Q7 i?2 +QHH2+iBQM Q7 2t+?M;2#H2 2p2Mib Bb +HH2/ i?2 2t+?M;2#H2 σ@}2H/X h?2Q`2K XRXkd h?2 2p2Mib Q7 i?2 2t+?M;2#H2 σ@}2H/ E `2 i`BpBH- i?i Bb- 7Q` Mv A ∈ E- P (A) = 0 Q` 1X

Xk aiQ+?biB+ S`Q+2bb2b >BbiQ`B2b AM i?2 7QHHQrBM;- i?2 BM/2t b2i T Bb 2Bi?2` QM2 Q7 i?2 7QHHQrBM;, R- R+ - N- ZX .2}MBiBQM XkXR G2i (Ω, F) #2  K2bm`#H2 bT+2X h?2 7KBHv {Ft }t∈T Q7 bm#@ σ@}2H/b Q7 F Bb +HH2/  ?BbiQ`v QM (Ω, F) B7 7Q` HH s, t ∈ T bm+? i?i s ≤ tFs ⊆ Ft . AM Qi?2` rQ`/b-  ?BbiQ`v Bb  MQM@/2+`2bBM; 7KBHv Q7 bm#@σ@}2H/b Q7 F BM/2t2/ #v TX Ai Bb HbQ +HH2/  }Hi`iBQMX h?2 σ@}2H/ F∞ := ∨t∈T Ft Bb- #v /2}MBiBQM- i?2 bKHH2bi σ@}2H/ i?i +QMiBMb Ft 7Q` HH t ∈ TX G2i {X(t)}t∈T #2 biQ+?biB+ T`Q+2bb /2}M2/ QM (Ω, F)X h?2 ?BbiQ`v {FtX }t∈T /2@ }M2/ #v FtX = σ(X(s) ; s ≤ t) Bb +HH2/ i?2 BMi2`MH ?BbiQ`v Q7 {X(t)}t∈T X Mv ?BbiQ`v {Ft }t∈T bm+? i?i Ft ⊇ FtX ,

t ∈ T,

Bb +HH2/  ?BbiQ`v Q7 {X(t)}t∈T X PM2 HbQ bvb, i?2 biQ+?biB+ T`Q+2bb {X(t)}t∈T Bb /Ti2/ iQ i?2 ?BbiQ`v {Ft }t∈T - Q` Ft @/Ti2/X .2}MBiBQM XkXk G2i T = R Q` R+ X .2}M2 7Q` HH t ∈ T Ft+ := ∩s>t Fs . h?2 ?BbiQ`v {Ft }t∈T Bb +HH2/ `B;?i@+QMiBMmQmb B7 7Q` HH t ∈ T- Ft = Ft+ X

XkX ahP*>ahA* S_P*1aa1a

8kd

J2bm`#BHBiv Ai Bb Q7i2M +QMp2MB2Mi iQ pB2r  biQ+?biB+ T`Q+2bb b  KTTBM; X : T×Ω → E/2}M2/ #v (t, ω) → X(t, ω)X h?Bb QT2Mb i?2 rv iQ p`BQmb K2bm`#BHBiv +QM+2TibX .2}MBiBQM XkXj h?2 biQ+?biB+ T`Q+2bb {X(t)}t∈R Bb bB/ iQ #2 K2bm`#H2 Bz i?2 KTTBM; 7`QK R × Ω BMiQ E /2}M2/ #v (t, ω) → X(t, ω) Bb K2bm`#H2 rBi? `2bT2+i iQ B ⊗ F M/ EX h?Bb BKTHB2b BM T`iB+mH` i?i 7Q` Mv ω ∈ Ω i?2 KTTBM; t → X(t, ω) Bb K2bm`#H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B(R) M/ EX HbQ- B7 E = R M/ B7 X(t) Bb MQM@M2;iBp2- QM2 +M /2}M2 i?2 G2#2b;m2 BMi2;`H  X(t, ω)dt R

7Q` 2+? ω ∈ Ω- M/ HbQ TTHv hQM2HHBǶb i?2Q`2K iQ Q#iBM 

 E X(t)dt = E [X(t)] dt . R

R

"v 6m#BMBǶb i?2Q`2K- i?2 Hbi 2[mHBiv HbQ ?QH/b i`m2 7Q` K2bm`#H2 biQ+?biB+ T`Q+2bb2b Q7 `#Bi``v bB;M biBb7vBM;  E [|X(t)|] dt < ∞ . R

.2}MBiBQM XkX9  biQ+?biB+ T`Q+2bb {X(t)}t∈R+ iFBM; Bib pHm2b BM i?2 K2@ bm`#H2 bT+2 (E, E) Bb bB/ iQ #2 Ft @T`Q;`2bbBp2Hv K2bm`#H2 Bz 7Q` HH t ∈ R+ i?2 KTTBM; (s, ω) → X(s, ω) 7`QK [0, t] × Ω BMiQ E Bb K2bm`#H2 rBi? `2bT2+i iQ i?2 σ@}2H/b B([0, t]) ⊗ Ft M/ EX *H2`Hv- {X(t)}t∈R+ Bb i?2M Ft @/Ti2/ M/ K2bm`#H2X h?2Q`2K XkX8 G2i {X(t)}t∈R+ #2  biQ+?biB+ T`Q+2bb- iFBM; Bib pHm2b BM  iQTQHQ;B+H bT+2 E U2M/Qr2/ rBi? Bib "Q`2H σ@}2H/ E = B(E)V- /Ti2/ iQ {Ft }t∈R+ M/ `B;?i@+QMiBMmQmb U`2bTX H27i@+QMiBMmQmbV- i?2M {X(t)}t∈R+ Bb Ft @T`Q;`2bbBp2Hv K2bm`#H2X h?2Q`2K XkXe A7 i?2 MQM@M2;iBp2 biQ+?biB+ T`Q+2bb {X(t)}t∈R  + Bb Ft @T`Q;`2bb@ Bp2Hv K2bm`#H2- i?2M- 7Q` 2+? t ∈ R+ - i?2 `M/QK p`B#H2 (0,t] X(s, ω) ds Bb Ft @K2bm`#H2X aiQTTBM; hBK2b  T`BM+BTH MQiBQM BM i?2 i?2Q`v Q7 biQ+?biB+ T`Q+2bb2b Bb i?i Q7 biQTTBM; U`2bTX QTiBQMHV iBK2X AM i?Bb bm#b2+iBQM- i?2 BM/2t b2i Bb T Bb N- Q` R+ - M/ r2 /2}M2 T = T ∪ {+∞}X ¯ `M/QK p`B#H2 τ Bb .2}MBiBQM XkXd G2i {Ft }t∈T #2  ?BbiQ`vX  T@pHm2/ +HH2/ M Ft @biQTTBM; U`2bTX QTiBQMHV iBK2 Bz 7Q` HH t ∈ T{τ ≤ t} ⊂ Ft

(`2bTX- {τ < t} ⊂ Ft ) .

8k3

SS1L.As

_2K`F XkX3 AM i?Bb `2K`F- i?2 BM/2t b2i Bb R+ X  biQTTBM; iBK2 Bb M QTiBQMH iBK2 #2+mb2 1 {T < t} = ∪n {T ≤ t − } ∈ Ft . n A7 i?2 }Hi`iBQM {Ft }t∈T Bb `B;?i@+QMiBMmQmb- M QTiBQMH iBK2 Bb  biQTTBM; iBK2#2+mb2 1 {T ≤ t} = ∩n {T < t + } ∈ Ft+ . n h?2Q`2K XkXN G2i {Tn }n∈N #2  b2[m2M+2 Q7 Ft @QTiBQMH iBK2b /2+`2bBM; iQ i?2 `M/QK p`B#H2 T X h?2M T Bb M Ft @QTiBQMH iBK2X h?2Q`2K XkXRy G2i {Ft }t∈R+ #2  `B;?i@+QMiBMmQmb ?BbiQ`v- M/ H2i τ #2 M Ft @ biQTTBM; iBK2X .2}M2 7Q` 2+? n ≥ 1 M TT`QtBKiBQM τ (n) Q7 i?2 biQTTBM; iBK2 τ #v, ⎧ ⎪ B7 τ (ω) = 0 ⎨0 k τ (n, ω) = k+1 B7 ≤ τ (ω) < k+1 n 2 2n 2n ⎪ ⎩ +∞ B7 τ (ω) = ∞. h?2M τ (n) Bb M Ft @biQTTBM;@iBK2 /2+`2bBM; iQ τ b n ↑ ∞X h?2Q`2K XkXRR G2i {X(t)}t∈R+ #2  `B;?i@+QMiBMmQmb U `2bTX H27i@+QMiBMmQmbV `2H@pHm2/ T`Q+2bb /Ti2/ iQ {Ft }t≥0 - M/ c #2  ;Bp2M `2H MmK#2`X h?2 `M/QK iBK2 τ := inf{t | X(t) ≥ c} Bb M Ft @biQTTBM; iBK2X AM i?2 H27i@+QMiBMmQmb +b2X(τ ) ≤ cX h?2Q`2K XkXRk G2i T = N Q` R+ X G2i {Ft }t∈T #2  ?BbiQ`vX G2i τ #2 M Ft @ biQTTBM; iBK2X h?2 +QHH2+iBQM Q7 2p2Mib Fτ = {A ∈ F∞ | A ∩ {τ ≤ t} ∈ Ft , 7Q` HH t ∈ T} Bb  σ@}2H/- M/ τ Bb Fτ @K2bm`#H2X G2i {X(t)}t∈T #2 M 1@pHm2/ Ft @/Ti2/ biQ+?biB+ T`Q+2bb- M/ H2i τ #2  }MBi2 Ft @biQTTBM; iBK2X A7 r2 /2}M2 X(τ ) #v X(τ )(ω) = X(τ (ω), ω)X h?2M- B7 T = N U `2bTX- T = R+ M/ {X(t)}t∈R+ Bb Ft @ T`Q;`2bbBp2Hv K2bm`#H2V X(τ )1{τ t} UA ∈ Ft - t ∈ R+ V Bb /2MQi2/ #v Fτ − X .2}MBiBQM XkXR9  biQ+?biB+ T`Q+2bb {Y (t)}t≥0 rBi? pHm2b BM M `#Bi``v K2bm`#H2 bT+2 (E, E) bm+? i?i 7Q` HH t ≥ 0 M/ HH ω ∈ Ω- i?2`2 2tBbib  TQbBiBp2 MmK#2` ε(t, ω) bm+? i?i Y (t + h, ω) = Y (t, ω) 7Q` HH h ∈ [t, t + ε(t, ω)]- Bb bB/ iQ ?p2  HQ+H bi2T bi`m+im`2X

h?2Q`2K XkXR8 G2i {Y (t)}t≥0 #2  biQ+?biB+ T`Q+2bb rBi? pHm2b BM M `#Bi``v K2bm`#H2 bT+2 (E, E) M/ rBi?  HQ+H bi2T bi`m+im`2X G2i FtY := σ(Y (s) ; 0 ≤ s ≤ t) .

XkX ahP*>ahA* S_P*1aa1a

8kN

h?2MUBV i?2 ?BbiQ`v {FtY }t≥0 Bb `B;?i@+QMiBMmQmb- M/ UBBV 7Q` Mv Ft @biQTBM; iBK2 T FTY = σ(Y (s ∧ T ) ; s ≥ 0) . Y Y S`QQ7X UBV Ai bm{+2b iQ T`Qp2 i?i B7 A ∈ Ft+2 −k 7Q` HH k ∈ N- i?2M A ∈ Ft X LQr #v i?2 bbmK2/ HQ+H bi2T bi`m+im`2- /2MQiBM; #v Q i?2 b2i Q7 `iBQMH MmK#2`b,

1A = Φk (Y (s), s ∈ (Q ∩ [0, t + 2−k ]) ∪ {t + 2−k }) . .2}M2 Φk (Y (s), s ∈ Q ∩ [0, t]) := Φk (Y  (s), s ∈ (Q ∩ [0, t + 2−k ]) ∪ {t + 2−k }) , r?2`2 G2i M/

Y  (s) = Y (s)1[0,t) (s) + Y (t)1[t,t+2−k ] (s) . ξk := Φk (Y (s), s ∈ Q ∩ [0, t]) , Ωk := {ω ; Y (t + δ, ω) = Y (t, ω) 7Q` HH δ ∈ [0, 2−k ]} .

h?2M ξk Bb FtY @K2bm`#H2- Ωk ↑ Ω M/ 1A∪Ωk = ξk 1Ωk X Ai 7QHHQrb i?i 1A = lim inf k ξk - M/ i?2`27Q`2 A ∈ FtY X UBBV h?2 BM+HmbBQM ⊇ Bb +H2`X h?2 `2p2`b2 BM+HmbBQM rBHH }`bi #2 T`Qp2/ 7Q` M FtY @ biQTTBM; iBK2 iFBM;  /2MmK2`#H2 b2i Q7 pHm2b 0 ≤ a0 < a1 < . . . ≤ +∞X Mv 2p2Mi A ∈ FTY +M #2 /2+QKTQb2/ b A = ∪p≥0 Ap r?2`2 Ap := A ∩ {T = ap } Bb BM FaYp X b #Qp2- r2 +M r`Bi2 1Ap = ϕp (Y (t) ; t ∈ Q ∩ [0, ap ]) . LQr- bBM+2 T = ap QM Ap - t ∧ T (ω) = t B7 t ∈ [0, ap ]X h?2`27Q`2- 7Q` HH p ≥ 01Ap = ϕp (Y (t ∧ T ) ; t ∈ Q ∩ [0, ap ]) M/ BM T`iB+mH` Ap ∈ σ(Y (s ∧ T ) ; s ≥ 0)X *QMbB/2` MQr i?2 ;2M2`H +b2X 6Q` HH k ≥ 1- i?2 iBK2  p2−k 1{(p−1)2−k ≤T t} ∈ ∨n≥1 Fa− 1 = Fa− ⊆ Fa . n n 

XjX J_hAL:G1a

8jR

h?2Q`2K XkXky G2i X = {X(t)}t≥0 M/ Y = {Y (t)}t≥0 #2 irQ biQ+?biB+ T`Q@ +2bb2b /Ti2/ iQ i?2 `B;?i@+QMiBMmQmb ?BbiQ`v {Ft }t≥0 - M/ bm+? i?i 7Q` Mv Ft @ biQTTBM; iBK2 S E X(S)1{S t} = {Tn > t} ∩ {Z0 ∈ L0 } ∩

 n−1 

 1{[0,s]} (Ti ) 1L (Zi ) = k

i=1

Bb +H2`Hv BM σ (T0 , Z0 , T1 , Z1 , . . . , Tn−1 , Zn−1 , Tn )X U+V FT∞ − = ∨n FTn − = σ (Tn , Zn ; n ≥ 0) = ∨n FTn = FT∞ X



h?2 7QHHQrBM; `2bmHib 72im`2 i?2 bvK#QHB+ +QMp2MiBQM T∞+1 := +∞X h?2Q`2K X9Xk G2i S #2  }MBi2 Ft @biQTTBM; iBK2X h?2M- 7Q` HH n ∈ N ∪ {+∞}FS ∩ {Tn ≤ S < Tn+1 } = FTn ∩ {Tn ≤ S < Tn+1 } , M/ i?2`2 2tBbib  b2[m2M+2 {Rn }n∈N∪{+∞} Q7 MQM@M2;iBp2 FTn @K2bm`#H2 `M/QK p`B#H2b bm+? i?i 7Q` HH n ∈ N ∪ {+∞}S ∧ Tn+1 = (Tn + Rn ) ∧ Tn+1 QM {S ≥ Tn } . S`QQ7X Ai bm{+2b iQ b?Qr i?i QM2 +M bbQ+Bi2 rBi? Mv ;2M2`iQ` U Q7 FS  ;2M2`iQ` V Q7 FTn bm+? i?i U ∩ {Tn ≤ S < Tn+1 } = V ∩ {Tn ≤ S < Tn+1 } , M/ pB+2 p2`bX h?2 Q#b2`piBQM i?i 7Q` HH k ∈ N- HH t ≥ 0- HH L ∈ K{N ((0, t ∧ S] × L) = k}∩{Tn ≤ S < Tn+1 } = {N ((0, t ∧ Tn ] × L) = k} ∩ {Tn ≤ S < Tn+1 } T2`KBib iQ /Q i?Bb- mbBM; UBBV Q7 h?2Q`2K X9XRX h?2 `M/QK p`B#H2 S Bb FS @K2bm`#H2 M/ FS = σ(Z0 ) ∨ σ (NZ ((0, t ∧ S] × L) ; t ≥ 0, L ∈ K) . "v i?2 `2bmHi Dmbi T`Qp2/- i?2`2 2tBbib KTTBM;b ψn Un ≥ 0V M/ ψ∞ bm+? i?i S 1{Tn ≤S a M/ HH ω ∈ Ω Tna (ω) := a1A (ω) + n1A (ω) . h?Bb Bb  biQTTBM; iBK2X HbQ {(t, ω) ; t ≤ Tna (ω)} = A × [a, ∞) + A × [n, ∞) M/ i?2`27Q`2- B7 a ≤ b < n{(t, ω) ; t ≤ Tna (ω)} − {(t, ω) ; t ≤ Tnb (ω)} = A × [a, b) . h?Bb #2BM; i`m2 7Q` HH n- b M22/ MQi #2 +QMbi`BM2/ iQ #2 }MBi2X



h?2Q`2K X9X9 h?2 T`Q+2bb {X(t)}t≥0 Bb Ft @T`2/B+i#H2 B7 M/ QMHv X(0) Bb F0 @ K2bm`#H2 M/ QM R+ \{0} f (n) (t, ω) 1{Tn (ω)T∞ } , URkXNdV X(t, ω) := n≥0

r?2`2 7Q` 2+? n ∈ N ∪ {+∞}- i?2 KTTBM; (t, ω) → f (n) (t, ω) Bb FTn ⊗ B(R+ )@ K2bm`#H2X S`QQ7X am{+B2M+v Bb H27i b M 2t2`+Bb2X 6Q` M2+2bbBiv- Bi bm{+2b #v h?2Q`2K X9Xj iQ T`Qp2 i?2 `2T`2b2MiiBQM 7Q` T`Q+2bb2b Q7 i?2 7Q`K X(t) = 1{t≤S} - r?2`2 S Bb  }MBi2 Ft @biQTTBM; iBK2X "mi i?Bb BM im`M Bb  +QMb2[m2M+2 Q7 h?2Q`2K X9XjX  h?2Q`2K X9X8 G2i A ∈ FTn − 7Q` bQK2 n ≥ 1X h?2`2 2tBbib M Ft @T`2/B+i#H2 T`Q+2bb {X(t)}t≥0 bm+? i?i X(Tn ) = 1A B7 Tn < ∞- M/ X(t) = 0 B7 t ∈ (Tn−1 , Tn ]X S`QQ7X "v UBBB-V Q7 h?2Q`2K X9XR- i?2`2 2tBbib  K2bm`#H2 7mM+iBQM f bm+? i?i 1A (ω) = f (T0 (ω), Z0 (ω), . . . , Tn−1 (ω), Zn−1 (ω), Tn (ω)) . hF2 X(t) := 1{Tn−1 0 ,

loc. r?2`2 n≥1 qn = ∞ M/ n≥1 pn < ∞X h?2M- #v i?2 TQbBiBpBiv +QM/BiBQM- Q  P X >Qr2p2` Q M/ P `2 KmimHHv bBM;mH` bBM+2 Q(Xn → 0) = 0 M/ P (Xn → 0) = 1X h?2Q`2K X8Xk {Ln }n≥1 +QMp2`;2b Q@HKQbi@bm`2Hv M/ P @HKQbi@bm`2Hv iQ bQK2 `M/QK p`B#H2 L∞ M/ dQ = 1{L∞ =∞} dQ + L∞ dP , r?2`2 P (L∞ = ∞) = 0X h?2Q`2K X8Xj Q  P ⇔ EP [L∞ ] = 1 ⇔ Q(L∞ < ∞) = 1 , Q ⊥ P ⇔ EP [L∞ ] = 0 ⇔ Q(L∞ = ∞) = 1 .

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n 

(n ≥ 1)

Xi

i=1

Bb  MQM@M2;iBp2 K`iBM;H2 M/ Bi +QMp2`;2b HKQbi bm`2Hv iQ  }MBi2 `M/QK 1 ! p`B#H2 M∞ X G2i an := E[Xn2 ]X ULQi2 i?i ∞ n=1 an ≤ 1XV h?2Q`2K X8X9 h?2 7QHHQrBM; +QM/BiBQMb `2 2[mBpH2Mi, 1 !∞ ! 2 UBV ∞ n=1 an > 0- r?2`2 an = E[Xn ]X ULQi2 i?i n=1 an ≤ 1XV UBBV E[M∞ ] = 1X UBBBV Mn → M∞ BM L1 X UBpV {Mn }n≥0 Bb mMB7Q`KHv BMi2;`#H2X 1tKTH2 X8X8, EFmiMBǰb .B+?QiQKv h?2Q`2KX G2i {Xn }n≥1 #2  b2@ [m2M+2 Q7 `M/QK 2H2K2Mib rBi? pHm2b QM i?2 K2bm`#H2 bT+2 (E, E)X q2 Kv bmTTQb2 i?i Bi Bb i?2 +QQ`/BMi2 b2[m2M+2 Q7 i?2 +MQMB+H bT+2 (Ω, F) := (E N , E ⊗N )X G2i Q M/ P #2 irQ T`Q##BHBiv K2bm`2b QM (Ω, F) bm+? i?i i?2 b2@ [m2M+2 Bb BB/ `2HiBp2 iQ #Qi?X G2i QXn M/ PXn #2 i?2 `2bi`B+iBQMb Q7 Q M/ P `2bT2+iBp2Hv iQ σ(Xn ) M/ H2i Qn M/ Pn #2 i?2 `2bi`B+iBQMb Q7 Q M/ P `2bT2+iBp2Hv iQ Fn := σ(X1 , . . . , Xn )X q2 bbmK2 i?i 7Q` HH n ≥ 1- QXn  PXn M/ /2MQi2 dQ i?2 +Q``2bTQM/BM; _/QMĜLBFQ/ɷK /2`BpiBp2 dPXXn #v fn (Xn )X h?2M 7Q` HH n ≥ 1Qn  Pn M/ Ln =

dQn dPn

= Πni=1 fi (Xi )X aBM+2

n



{L∞ < ∞} = {log L∞ < ∞} =

n 

 log fi (Xi ) < ∞

i=1

Bb  iBH@2p2Mi Q7 i?2 b2[m2M+2- Bib T`Q##BHBiv Bb 0 Q` 1X h?2`27Q`2- i?2`2 `2 QMHv irQ TQbbB#BHBiB2b- 2Bi?2` Q  P Q` Q ⊥ P X 1tKTH2 X8Xe, EFmiMBǰb *QM/BiBQMX EFmiMBǶb i?2Q`2K Uh?2Q`2K X8X9V +M #2 TTHB2/ iQ i?2 bBimiBQM UMHQ;Qmb iQ i?i Q7 1tKTH2 X8X8 #Qp2V r?2`2 Ln =

n 

Zi ,

i=1

r?2`2 {Zn }n≥1 Bb  b2[m2M+2 Q7 BB/ MQM@M2;iBp2 `M/QK p`B#H2b Q7 K2M 1X "v i?Bb i?2Q`2K- i?2 +`Bi2`BQM Q7 #bQHmi2 +QMiBMmBiv Q7 Q rBi? `2bT2+i iQ P EP [L∞ ] = 1 ! 2 1 Q7 h?2Q`2K X8Xj Bb ∞ n=1 E[Zn ] > 0X "v i?2 bK2 `;mK2Mi b BM 1tKTH2 X8X8i?2 QMHv Hi2`MiBp2 iQ  P Bb Q ⊥ P - M/ i?2`27Q`2  M2+2bb`v M/ bm{+B2Mi 1 ! 2 +QM/BiBQM 7Q` i?2 Hii2` Bb ∞ n=1 E[Zn ] = 0X

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© Springer Nature Switzerland AG 2020 P. Brémaud, Point Process Calculus in Time and Space, Probability Theory and Stochastic Modelling 98, https://doi.org/10.1007/978-3-030-62753-9

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"A"GAP:_S>u

898

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89e

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"A"GAP:_S>u

89d

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893

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HX+X/X#X- R GTH+2 7mM+iBQMH ě Q7  SQBbbQM T`Q+2bb- 3k ě Q7  +Hmbi2` TQBMi T`Q+2bb- j9 ě Q7  TQBMi T`Q+2bb- Rd ;2M2`iQ`- ek ě Q7  `M/QK K2bm`2- Rd BM}MBi2bBKH ě- ek HB7Q9yR bi#H2 M/ +QMb2`piBp2 ě- ek GBM/H2v T`Q+2bb- 9kN :B##b HQ+H ě /Bbi`B#miBQM- k9N ě +?`+i2`BbiB+b Q7 J`FQp }2H/- k9N ě TQi2MiBH- k9N ě 2M2`;v- k8R ;`B/- 8 "`iH2ii bT2+i`mK Q7 i?2 biiBQM`v ě- HQ+H +?`+i2`BbiB+b- Rd9 HQ+HBxBM; b2[m2M+2- 8jR j9e HQ+HHv i?BMMBM; Q7 i?2 ě- Ryy ě }MBi2 K2bm`2- R ě BMi2;`#H2- Rdk ?`/@+Q`2 KQ/2H- kjj HQM;@`M;2 /2T2M/2M+2- j8k >rF2b T`Q+2bb GQiFĜoQHi2`` KQ/2H- Rkj HBM2` ě- 9dd GQvM2b KmHiBp`Bi2 ě- 9e9 ěǶ 2[miBQM- 9jy MQM@HBM2` ě- 8yy ěǶ b2[m2M+2- 9jy ?x`/ ěǶ p`B#H2- 9jy ě K2bm`2- 9y ě `i2- 9R M (E)- R ?BbiQ`v- 88 Mp (E)- k +QKTiB#H2 ě- kNy M(E)- k BMi2`MH Ĝ- Rdk Mp (E)- k ?TT- 8y μ(ϕ)- 3 biM/`/ ě- RNe Jf:AfRf∞- 99d- 98R JfJfEfy- 9yy BK#2//BM; i?2Q`2K K`F /B`2+i ě- kRR ě T`Q+2bb- k83 }`bi ě- kRR +QKTiB#H2 ě- k83 b2+QM/ ě- kRe- kke ě b2[m2M+2- 39 BMp2`b2 ě- kR9 K`F2/ TQBMi T`Q+2bb- e BMi2MbBiv ě +QKTiB#H2 rBi?  ~Qr- k83 ě K2bm`2- 9- dd ě rBi? BM/2T2M/2Mi BB/ K`Fb- d T`2/B+i#H2 ě- R3k biiBQM`v ě- k8N biQ+?biB+ Ĝ- Ree J`FQp }2H/- k9N biQ+?biB+ ě F2`M2H- Rdk K`iBM;H2- 8jR BMp2`bBQM 7Q`KmH ě +QMp2`;2M+2 i?2Q`2K- 8jk SHK ě- kdk- kNN ě `2T`2b2MiiBQM i?2Q`2K- RN8 HQ+H ě- 8jR CMQbbv /2MbBiv- kjy bm#ě- 8jR E2`biM bmT2`ě- 8jR ě TQBMi T`Q+2bb- 8yj K2bm`2 ěǶb +QM/BiBQM- 8yj /Bzmb2 ě- Ry E2bi2MǶb #QmM/- R9k MQM@iQKB+ ě- Ry

AL.1s `M/QK ě- k J2+F2 ěǶb n@i? Q`/2` K2bm`2- jkj ěǶb 7Q`KmH- ke9 ěǶb K2bm`2- jRy ěǶb `2/m+2/ K2bm`2- jk9 K2KQ`v BMi2`K2/Bi2 ě- j8k HQM; ě- j8k KQK2Mi K2bm`2- RR 7+iQ`BH ě- Rk }`bi ě Q7  TQBMi T`Q+2bb- RR b2+QM/ ě Q7  TQBMi T`Q+2bb- RR- jj8 KmimH BM7Q`KiBQM 7Q`KmH- jd3 N − εx - k8N N − x- k8N N L - Rdk NZ - d (N, Z)- d N (ϕ)- 3 M2`2bi M2B;?#Qm` /BbiM+2- jy8 M2B;?#Qm`b- k9y MQM@HiiB+2- Rjy QTiBQMH bKTHBM;- 8jk SHK ě /Bbi`B#miBQM- kej ě F2`M2H- jRR mi? Q`/2` ě- jkj ě T`Q##BHBiv- ke9- jRR- jkj T2M2i`#H2 bT?2`2b ě KQ/2H- kj8 T2`KM2MiH TQBMi T`Q+2bb- Rj T2`im`#iBQM MHvbBb- d3 T?b2 i`MbBiBQM- j8j TQBMi K2bm`2- k TQBMi T`Q+2bb- k- 9N #b2 ě- e /Bbi`B#miBQM Q7  ě- R9 }`bi@Q`/2` ě- e- RR HB7i2/ ě- Rdj K`F2/ ě- e #b2 Q7  ě- 39 }MBi2 ě- kjj biiBQM`v ě- k8N KmHiBp`Bi2 ě- Rd8 M@i? Q`/2` ě- RR b2+QM/@Q`/2` ě- RR b2H7@2t+BiBM; ě- R38

888 b2KB@*Qt ě- R38 bBKTH2 ě- j biiBQM`v ě- 9- k8N rB/2@b2Mb2 biiBQM`v ě- jje SQBbbQM T`Q+2bb +QKTQmM/ ě- 8y /Qm#Hv biQ+?biB+ ě- 8 ?QKQ;2M2Qmb ě- 8- 9N K`F2/ ě- 39 J2+F2Ƕb +?`+i2`BxiBQM Q7 ě- kNe KBt2/ ě- 8 biM/`/ ě- 8 SQHH+x2FĜE?BMi+?BM2 7Q`KmH- 99j TQr2` bT2+i`H K2bm`2 ě Q7  TQBMi T`Q+2bb- j9y ě Q7  biQ+?biB+ T`Q+2bb- jj9 ě Q7 i?2 bKTH2/ #`mb?- j3N T`2/B+i#H2 ě σ@}2H/- Red ě BMi2MbBiv- j39 ě T`Q+2bb- Red ě BM/2t2/ #v- Red ě T`QD2+iBQM- R3j ě bi`m+im`2- kNR T`22KTiBp2 `2bmK2- 9yR T`Q##BHBiv ;2M2`iBM; 7mM+iBQMH- kk T`Q+2bbQ` b?`BM;- 98j T`Q;`2bbBp2Hv K2bm`#H2- 8kd _/QMĜLBFQ/ɷK /2`BpiBp2- kdN `M/QK ě }2H/- k93 ě K2bm`2- k `2;2M2`iBp2 T`Q+2bb- R8R `2HiBp2Hv +QKT+i- R `2M2rH ě 2[miBQM- RRN /272+iBp2 ě- RRN 7mM/K2MiH ě- Rky KmHiBp`Bi2 ě- R8d ě 7mM+iBQM- RRd r2B;?i2/ ě- R9R ě K2bm`2- RRd ě T`Q+2bb- RR8- jjd ě i?2Q`2K 2H2K2Mi`v ě- RR3 ě iBK2- RR8 KmHiBp`Bi2 ě i?2Q`v- R8d `2+m``2Mi ě T`Q+2bb- RR3 i`MbB2Mi ě T`Q+2bb- RR3

88e `2bB/mH MHvbBb- j38 `2r`/ ě 7mM+iBQM- Rjd ě T`Q+2bb- Rjd _B2KMM Ƕb ě BMi2;`H- Rjy ě BMi2;`#H2- Rjy /B`2+i ěǶb BMi2;`H- Rjk /B`2+iHv ě BMi2;`#H2- Rjy- Rjk _BTH2vǶb 7mM+iBQM- jjN _BTH2vĜE2HHvǶb i?2Q`2K- k9R _vHH@L`/x2rbFBǶb 7Q`KmH- ke9 Sx (N )- k8N bKTH2 ě #`mb?- j3N ě +QK#- j33 ě Q7  TQBMi T`Q+2bb /Bbi`B#miBQM- dd ě b2[m2M+2- j33 `M/QKě- j33 bKTHBM;- R8 ě  SQBbbQM T`Q+2bb- dd 2t+i ě- N3 `M/QK ě- j33 b/2- 9N b2H7@2t+BiBM; TQBMi T`Q+2bb- R38 b2KB@*Qt TQBMi T`Q+2bb- R38 b2KB@J`FQp ě F2`M2H- R83 ě T`Q+2bb- R83 b2KB@K`iBM;H2 K2i?Q/- jee b2`pB+2 /Bb+BTHBM2- 9jj b?B7i- k8e b?Qi MQBb2 ě TQr2` bT2+i`H K2bm`2- j8e SQBbbQM ě- Ny bBM`- jR3 aHBpMvFǶb i?2Q`2K- RyN- RRy- kNj- jR8 aKBi?Ƕb `2;2M2`iBp2 7Q`KmH- R8e bKQQi?BM; 7Q`KmH- ReN- kjd bi#H2 bvKTiQiB+HHv ě- 8y9 biiBQM`v ě 7`K2rQ`F- k8e- k8N ě ;`B/- e ě K`F2/ TQBMi T`Q+2bb- k8N ě TQBMi T`Q+2bb- 9- k8N ě biQ+?biB+ T`Q+2bb- key rB/2@b2Mb2 ě ě TQBMi T`Q+2bb- jje

AL.1s biQ+?biB+ ;2QK2i`v- RyR aiQM2Ƕb /2+QKTQbBiBQM- R9N bi`2bb `2H2b2 KQ/2H- Rd8 m;K2Mi2/ ě- j33- 9e8 bi`QM; J`FQp T`QT2`iv Q7 ?TTb- 8k- kk9 bm#2tTQM2MiBH /Bbi`B#miBQM- R9k i?BMMBM;- j9- j8e i`{+ BMi2MbBiv- 9y8- 9kN oM GB2b?QmiĜ"//2H2v BM/2t- jy8 oQ`QMQś +2HH- kNN qiM#2 ěǶb SQBbbQM +?`+i2`BxiBQM- eR 2ti2MbBQM Q7 ě- ky3 bTiBH ěǶb i?2Q`2K- k9y r2F +QMp2`;2M+2- ke r?Bi2 MQBb2 :mbbBM ě- kR3 SQBbbQM ě- j9k rB/2@b2Mb2 biiBQM`v ě TQBMi T`Q+2bb- jje ě biQ+?biB+ T`Q+2bb- jjj