Probability Theory and Mathematical Statistics: Vol. 2 [Reprint 2020 ed.] 9783112319024, 9783112307892

Table of contents :
CONTENTS
WEAK CONVERGENCE OF SEMI-MARKOV RANDOM EVOLUTIONS (MARTINGALE APPROACH)
CONVERGENCE OF MOMENTS OF RANDOMLY INDEXED RANDOM SEQUENCES
ON THE RATE OF CONVERGENCE OF THE DISTRIBUTIONS OF SEMIMARTINGALES TO THE DISTRIBUTION OF STABLE PROCESS
ANTICIPATING STOCHASTIC DIFFERENTIAL EQUATIONS
SECOND ORDER MINIMAX ESTIMATION: THE NUISANCE PARAMETERS AND HYPOELLIPTIC OPERATORS
LIMIT THEOREMS FOR THE RIEMANN ZETA-FUNCTION IN THE COMPLEX SPACE
STOCHASTIC INTEGRALS WITH RESPECT TO SEMIMARTINGALE MEASURES AND CHANGE OF THE FILTRATION
LIMIT DISTRIBUTIONS OF MEASURES OF SOJOURNS OF VECTOR GAUSSIAN FIELDS WITH STRONG DEPENDENCE
LIMIT THEOREMS FOR ONE-DIMENSIONAL RANDOM WALKS IN RANDOM ENVIRONMENTS
ON THE NORM DISTRIBUTION OF GAUSSIAN AND OTHER STABLE VECTORS
ON SET CONVOLUTIONS AND INTEGRAL-SUM KERNEL OPERATORS
LARGE DEVIATION FOR MARTINGALES WITH INDEPENDENT AND HOMOGENEOUS INCREMENTS
LAWS OF THE ITERATED LOGARITHM FOR I.I.D. CASE
DISTRIBUTION OF VALUES OF ADDITIVE ARITHMETIC FUNCTIONS
STATISTICAL INFERENCES FOR SEMIMARTINGALE REGRESSION MODELS
ON THE MARTINGALE PROBLEM ASSOCIATED WITH INTEGRO-DIFFERENTIAL OPERATORS
PREFERRED DIRECTIONS ON THE HYPERSPHERE
TWO-PARAMETER STRONG MARTINGALES: INEQUALITIES FOR QUADRATIC VARIATION AND SOME DECOMPOSITIONS
EXACT RESOLUTION THRESHOLDS FOR CLOSE FREQUENCIES
ASYMPTOTIC PROPERTIES OF TWO STATISTICAL DECISION RULES BASED ON VALUES OF STUDENTIZED RANGE
ON A NEW APPROACH TO THE STUDY OF THE DISTRIBUTION OF A NORM OF A RANDOM ELEMENT IN HILBERT SPACE
RECORDS FOR NONIDENTICALLY DISTRIBUTED RANDOM VARIABLES
LOCAL CHERNOFF EFFICIENCY OF LINEAR RANK TESTS
EXTENDED VECTOR STOCHASTIC INTEGRAL IN SOBOLEV SPACES OF WIENER FUNCTIONALS
WEIGHTED MULTIVARIATE EMPIRICAL PROCESSES AS RANDOM MEASURES
REMARKS ON THE RANDOM MULTIPLICATIVE ERGODIC THEOREM
PROBABILITY DISTRIBUTIONS AND BOREL MEASURES UNIQUELY DETERMINED BY THEIR RESTRICTIONS TO A HALF-SPACE
DISCRETE TIME QUANTUM STOCHASTIC FLOWS, MARKOV CHAINS AND CHAOS EXPANSIONS
THE ZERO-LEVEL SET OF THE BROWNIAN BRIDGE AS A MODEL OF DISORDERED PHYSICAL SYSTEMS
SOME INEQUALITIES FOR MOMENTS OF SUMS OF INDEPENDENT RANDOM VARIABLES
SOME MINIMAX ESTIMATION PROBLEMS
LARGE DEVIATIONS OF TRAJECTORIES IN THE CENTRAL LIMIT THEOREM FOR RANDOM PROCESSES: SOME NEW RESULTS
HOMOGENIZATION AND VANISHING VISCOSITY
SOME REMARKS ON ASYMPTOTIC EFFICIENCY IN BAHADUR SENSE OF MAXIMUM LIKELIHOOD ESTIMATOR
ON DISTRIBUTION OF SUPREMUM TYPE FUNCTIONALS OF NONPARAMETRIC ESTIMATES FOR SPECTRAL DENSITY
A CENTRAL LIMIT THEOREM FOR QUADRATIC FORMS IN INDEPENDENT RANDOM VARIABLES
SUBORDINATION DEPENDING ON A PARAMETER
GENERAL LEMMAS ON THE DISTRIBUTION DENSITY FOR LARGE DEVIATIONS OF A RANDOM VECTOR
SPEED OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM IN HILBERT SPACE UNDER WEAKENED MOMENT CONDITIONS
STATISTICAL FUNCTIONALS, .STOPPING TIMES AND ASYMPTOTIC MINIMUM RISK PROPERTY
THE CENTRAL LIMIT THEOREM FOR FINITELY DEPENDENT RANDOM VARIABLES
ON HARMONIC FUNCTIONS OF RANDOM WALKS ON GROUPS
EXPONENTS OF OPERATOR STABLE DISTRIBUTIONS IN BANACH SPACES
INFINITE DIMENSIONAL DISTRIBUTIONS AND SECOND QUANTIZATION IN STRING THEORIES
DIFFERENTIAL CALCULUS ON THE CONFIGURATION SPACE: SURFACES, SURFACE MEASURES, GAUSS-OSTROGRADSKII FORMULA
THE LOCAL THEOREMS FOR ADDITIVE ARITHMETIC FUNCTIONS
QUADRATIC FORMS WITH LONG-RANGE DEPENDENCE
BACKWARD LIMITS AND INHOMOGENEOUS REGENERATION
ON THE NORMAL APPROXIMATION OF SUMS OF WEAKLY DEPENDENT HILBERT—VALUED RANDOM VARIABLES
ON DISTRIBUTION OF ADDITIVE ARITHMETICAL FUNCTIONS ON THE SET OF SHIFTED PRIMES
A NEW LOOK ON SEMIMARTINGALE DISTRIBUTIONS WITH SEVERAL EXAMPLES
CENTRAL LIMIT THEOREM FOR DEPENDENT RANDOM VARIABLES
RÉNYI DISTANCES OF SOME DIFFUSION PROCESSES
ESTIMATES FOR THE SMOOTH FUNCTIONAL IN THE CLASSICAL CASE
ON LARGE DEVIATIONS IN AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS. 1
ON LOGARITHMIC REFINING TERMS IN LIMIT THEOREMS TAKING INTO ACCOUNT LARGE DEVIATIONS OF SUMS OF INDEPENDENT RANDOM VARIABLES
ON STOCHASTIC CONVOLUTIONS MODELS
SOME PROPERTIES OF NON-HOMOGENEOUS TRANSIENT RENEWAL PROCESSES
SOME LARGE DEVIATION RESULTS FOR U—STATISTICS
OSIPOV'S THEOREM REVISITED
GIBBS MEASURES FOR THE CURIE—WEISS RANDOM FIELD ISING MODEL
ON THE APPROXIMATION OF CONVOLUTIONS BY INFINITELY DIVISIBLE DISTRIBUTIONS
BANACH SPACE OF MEASURES

Citation preview

Volume II

Probability Theory and Mathematical Statistics Proceedings of the Fifth Vilnius Conference June 25-July 1,1989 Edited by

B. Grigelionis, Yu. V. Prohorov, V. V. Sazonov and V. Statulevicius

MOKSLAS Vilnius, Lithuania

///VSP111

Utrecht, The Netherlands

VSP BV Post Box 346 3700 AH Zeist The Netherlands

MOKSLAS Zvaigzdziu 23 Vilnius Lithuania

©1990 VSP B V / I M I Lithuanian Ac. Sei.

First published in 1990

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the copyright owners.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Probability theory and mathematical statistics: Proceedings of the Fifth Vilnius Conference: Vilnius, Lithuania, June 25 - July 1, 1989 ed. by B. Grigelionis... [et al.\ - Utrecht: VSP BV/ Vilnius: Mokslas. ISBN 90-6764-129-4 (vol.II) ISBN 90-6764-130-8 (set) SISO 517 UDC 519.2(063) Subject headings: probability theory/ mathematical statistics.

Typeset in Lithuania by Baltic Amadeus / Publishing Service Group of IMI, Vilnius Printed in Lithuania by Spindulys, Kaunas

CONTENTS Weak convergence of semi-Markov random evolutions (martingale approach) V.S. Koroljuk and A.V. Swishchuk Convergence of moments of randomly indexed random sequences V.M. Kruglov and V.Yu. Korolev Oh the rate of convergence of the distributions of semimartingales to the distribution of stable process K. Kubilius Anticipating stochastic differential equations H.-H. Kuo and J. Potthoff Second order minimax estimation: the nuisance parameters and hypoelliptic operators Z.M. Landsman and B.Ya. Levit Limit theorems for the Rieman zeta-function in the complex space A. Laurincikas Stochastic integrals with respect to semimartingale measures and change of the filtration V.A. Lebedev Limit distributions of measures of sojourns of vector Gaussian fields with strong dependence N.N. Leonenko Limit theorems for one—dimensional random walks in random environments A.V. Letchikov On the norm distribution of Gaussian and other stable vectors M.A. Lifshits On set convolutions and integral-sum kernel operators J.M. Lindsay Large deviation for martingales with independent and homogeneous increments R.S. Liptser and A.N. Shiryaev Laws of the iterated logarithm for i.i.d. case A.I. Martikainen Distribution of values of additive arithmetic functions E. Manstavicius Statistical inferences for semimartingale regression models A.V. Melnikov and A.A. Novikov

iv

Contents

On the martingale problem associated with integro-differential operators R. Mikulevicius and H. Pragarauskas

168

Preferred directions on the hypersphere P. Miiasevic

176

Two-parameter strong martingales: variation and some decompositions

inequalities for quadratic

Yu.S. Mishura and A.A. Gushchin

181

Exact resolution thresholds for close frequencies G.M. Molchan

193

Asymptotic properties of two statistical decision rules based on values of studentized range A.V. Nagaev

207

On a new approach to the study of the distribution of a norm of a random element in Hilbert space S.V. Nagaev

214

Records for nonidentically distributed random variables V.B. Nevzorov

227

Local Ya.Yu.Chernoff Nikitin efficiency of linear rank tests

234

Extended vector stochastic integral in Sobolev spaces of Wiener functionals N.V. Norin

244

Weighted multivariate empirical processes as random measures R. Norvaisa.

257

Remarks on the random multiplicative ergodic theorem V.J. Oseledec

275

Probability distributions on Borel measures uniquely determined by their restrictions to a half-space I V . Ostrovskii and A.M. Ulanovskii

278

Discrete time quantum stochastic flows, Markov chains and chaos expansions K.R. Parthasarathy

288

The zero-level set of the Brownian bridge as a model of disordered physical systems D. Petritis

298

Some inequalities for moments of sums of independent random variables V.V. Petrov

309

Some mini max estimation problems I F . Pinelis

315

Contents

v

Large deviations of trajectories in the central limit theorem for random processes: some new results V.I Piterbarg

325

Homogenization and vanishing viscosity A.L. Pyatnickiy and S.M. Kozlov

330

Some remarks on asymptotic efficiency in Bahadur sense of maximum likelihood estimator M. Radavicius

340

On distribution of supremum type functional of nonparametric estimates for spectral density R. Rudzkis

349

A central limit theorem for quadratic forms in independent random variables Ju.M. Ryzhov

364

Subordination depending on a parameter K. Sato

372

General lemmas on the distribution density for large deviations of a random vector L. Saulis

383

Speed of convergence in the central limit theorem in Hilbert space under weakened moment conditions V. V. Sazonov and V. V. Ulyanov

394

Statistical functionals, stopping times and asymptotic minimum risk property P.K. Sen

411

The central limit theorem for finitely dependent random variables V.V. Shergin

424

On harmonic functions of random walks on groups M.G. Shur

432

Exponents of operator stable distributions in Banach spaces G. Siegel

437

Infinite dimensional distributions and second quantization in string theories O.G. Smolyanov

446

Differential calculus on the configuration space: surfaces, surface measures, Gauss-Ostrogradskii formula N. V. Smorodina

451

The local theorems for additive arithmetic functions G. Stepanauskas

460

vi

Contenta

Quadratic forms with long-range dependence N. Terrin and M.S. Taqqu

466

Backward limits and inhomogeneous regeneration H. Thoriason

474

On the normal approximation of sums of weakly dependent Hilbertvalued random variables A.N. Tikhomirov

482"

On distribution of additive arithmetical functions on the set of shiffted primes N.M. Timofeev

495

A new look on semimartingale distributions with several examples E.I. TroGmov

505

Central limit theorem for dependent random variables S.A. Utev

519

Renyi distances of some diffusion processes I. Vajda

529

Estimates for the smooth functional in the classical case G. Valiukevicius On large deviations in averaging principle for stochastic differential equations with periodic coefficients A.Yu. Veretennikov

535

542

On logarithmic refining terms in limit theorems taking into account large deviations of sums of independent random variables V.V. Vinogradov

552

On stochastic convolutions models V.E. Volkovich

563

Some properties of non-homogeneous transient renewal processes W. Wajda

569

Some large deviation results for U-statistics W. Wolf

579

Osioov's theorem revisited V.V. Yurinsky

585

Gibbs measures for the Curie-Weiss random field Ising model V.A. Zagrebnov, J.M.G. Amaro de Matos and A.E. Patrick

590

On the approximation of convolutions by infinitely divisible distributions A.Yu. Zaitsev

602

Banach space of measures Z.S. Zerakidze

609

P r o b . T h e o r y a n d M a t h . S t a t . , Vol. 2, p p . 1 - 9 B.Grigelionis et al. ( E d s . ) 1990 V S P / M o k s l a s

W E A K C O N V E R G E N C E OF S E M I - M A R K O V R A N D O M EVOLUTIONS (MARTINGALE A P P R O A C H ) V.S. KOROLJUK and A.V. SWISHCHUK Institute of Mathematics Ukr. Ac. Sci., Repina str. 3, 252601 Kiev, USSR ABSTRACT Two types of limit theorems for semi-Markov random evolutions (SMRE) in the series scheme are discussed: the averaging and the diffusion approximation. The weak convergence of SMRE is established by reducing it to a martingale problem for the infinitesimal operator of the limit evolution. Applications to storage theory, traffic processes and switching processes are considered. 1. SEMI-MARKOV SCHEME

RANDOM

EVOLUTION

IN

THE

SERIES

SMRE in the series scheme (with a small parameter e) is defined by the equation ' V'(t) =1+

V(a)T(x(s/e))ds o

"(t/e) + £ V'(eTk)[DM) = P{x„+i E A,

9n+1 ^ t \ x

n

- x},

n

rn — Yh Ok, v(t) = max{n : r n ^ f} is a counting process, r(t) = Tv(t) is a point fc=i process. Example 1. The storage process with transfer speed v(u, x) and jumps function a(x) (the output and the input respectively) is defined by the integral equation (Koroljuk and Swishchuk, 1985) u; =u0+

i / v(u:,x(s/e))ds o

© V . S . K o r o l j u k a n d A.V. Swishchuk. 1990

"(«/') + e £ o(i f c _i).

(2)

2

Semi-Markov

Random

Evolutions

Let Ut(x, uo) be the solution of the Cauchy problem ^

= v(Ut,x),

Uo(x,u0)

= u0,

xÇX.

(3)

Choose generators r ( x ) , x 6 X, on a Banach space B of functions in the following way r ( * M u ) = «,(«,

(4)

Define also linear operators D€(x), x g X, as follows D'{x)tp(u)

= >p(u + ea(x)).

(5)

Then SMRE (1) is defined by the expression V " ( * M « ) = v{W t ).

(6)

Example 2. Switching processes are defined by the equation dCt = dat(x(t/e))

+ ea(x(t/e))dv(t/e).

(7)

Here a 0) with transition probabilities P(x, A) = Q(x, A, +oe) is uniformly ergodic with stationary distribution p(A), A G X; the first two moments of sojourn times in states x are uniformly continuous and bounded on x : oo mi(z)

oo

= J tGz(dt),

m2(x) = J

o

t2Gx(dt)\

o

Gx(t) = Q(x,X,t). C2. De(x) =I + eD(x) + o(e) a s £ - » 0 , where D(x), x 6 X, are closed linear operators with domain BQ. C3. Operator T* = ± f p(dx)[mi(x)T(x) + D(x)] is closed, x m = J x

p(dx)m\(x).

C4. /p(dx)[mi(a;)||r(®)/|| + ||D(«)/||] < Cx < +oo, V/ 6 B0 C Dom(r a (at)). x Then SMRE V'(t) converges weakly as £ —* 0 to the averaged evolution V*(t), which is defined as the solution of the equation t V*(t) - / - j V(s)T*ds o

= 0.

(12)

Remark. If T* is a bounded operator, then limit evolution V*(t) = exp {ir*}. In the general case equation (12) is considered on functions from the domain of operators T 2 (x), x € X. 3. DIFFUSION APPROXIMATION (KOROLJUK A N D SWISHCHUK, 1989) Let instead of conditions C2—C4 the following conditions be satisfied C'2. D'(x) = I + eD(x) + e2D2(x) as e 0 ; here D(x),D2(x), x e X, are closed linear operators with domain Bo • C'3. (the balance condition) T*f = 0, V/ € SoC'4. for all / € So C Dom(r 4 (x)), J p ( ^ ) [ m 2 ( x ) | | r 2 ( i ) | | + m i ( i ) | | r ( a ! ) P i > ( i ) / | | + ||J3a(i)/||] < C2 < +oo. x

4

Semi-Markov

Random

Evolutions

If conditions C I , C'2—C'4 are satisfied, then SMRE V'(t/e) £ —• 0 to the solution W*(t) of the martingale problem

converges weakly as

t W*(t) - / - J

W*(s)L*ds

= M(t),

(13)

o Here M(t) is a -F ( -martingale, Ft = cr{x(s) : 0 < 5 < i}, = ± f m J

p{dx)L{x),

x

L(x)

= c(x)(Ro

+ m2(x)T2(x)/2

- I)c(x)

+

mi(x)T(x)PD(x)

+

D2(x),

c(x) = m ^ x ^ x ) + PD{x),

(14)

-1

where Ro = (7 — P + I I ) — II is the potential of the embedded Markov chain, P is the operator of transition probabilities, II is the stacionaxy projector.

4. APPLICATIONS OF THE LIMIT AVERAGING THEOREM Taking into account (5) for the storage processes (2) we obtain De{x)f{u)

=