Probability Theory and Mathematical Statistics: Vol. 1 [Reprint 2020 ed.] 9783112319000, 9783112307878

Table of contents :
CONTENTS
Preface
Quantum stochastic calculus
Glivenko-Cantelli convergence for weighted empirical and quantile processes of U-statistic structure
On symmetry properties and nonparametric estimates of the ʋ-th order spectral density of a stationary random process
Large deviations for first-passage times
Quasi-average approach to the description of the limit states of the n-vector Curie-Weiss ferromagnet
Inequalities at the Markov approximation of lumped processes
Remarks on limit theorems for sequences of random variables with random index
Continuity of local time for Markov processes with stationary independent increments
Detection and diagnosis of changes in the A.R. part of an A.R.M. A. model with nonstationary unknown M. A. coefficients
Nonparametric estimation of distribution functions based on incomplete data
Diffusion processes on the group T ∞ and elliptic equations of infinitely many variables
Lower estimates of the convergence rate in the CLT in Banach spaces
Weak limits of probability measures on metric Schauderspaces
Optimal consumption and investment in a stochastic model
Large deviations from classical paths and the classical limit of quantum stochastic flows
Limit theorems for multicomponent hierarchical models
On limit theorems for random vectors controlled by a Markov chain
Upper and lower estimates of the convergence rate in the invariance principle for empirical measures
The contraction principle for C0-summing operators and SLLN for weighed sums
Limit theorems for stochastic inventory models
Limit theorems under weak dependence conditions
Non commutative integration and probability on von Neumann algebras
On optimal controls in the problem of locally absolutely continuous change of measure (compact sets of decisions)
Non-uniform estimates of the remainder term in limit theorems with a stable limit law
Multiple stable stochastic integrals
A generalization of p-type spaces
Smooth measures on infinite dimensional manifolds and forward Kolmogorov equation
The structure of distributions of convex functionals
Wiener germs applied to the tails of m-estimators
On the asymptotic behaviour of the linear stochastic heat equation solutions
Some universal donsker classes of functions
Duplicates in mixed sequences and a frequency duplication principle. Methods and applications
Bootstrapping empirical measures indexed by Vapnik-Chervonenkis classes of sets
A remark on the Central Limit Theorem for random measures and processes
Elliptic law and elements of G-analysis
Mathematical aspects on the variation of air pollutant concentrations
Weak solutions of the stochastic evolution and invariance principles
Optimal stopping of a Markov chain with vector-valued gain function
Some strong laws of large numbers in Banach spaces with regular norms
Optimality in estimation for stochastic processes under both fixed and large sample conditions
On urn schemes imbedded in birth processes
Limiting distributions and mean-values of complex-valued multiplicative functions
Asymptotically minimax testing of nonparametric hypotheses

Citation preview

PROBABILITY THEORY AND MATHEMATICAL STATISTICS

Volume I PROBABILITY THEORY AND MATHEMATICAL STATISTICS

Proceedings of the Fourth Vilnius Conference

Vilnius, USSR, 24-29 June 1985

Edited

by

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

WWNU SCIENCE PRESS Hi I Jtrecht, The Netherlands

V N U Science Press B V P.O. Box 2093 3500 GB Utrecht The Netherlands

© 1987 V N U Science Press B V First published 1987 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 owner.

CIP-DATA KONINKLUKE BIBLIOTHEEK, DEN HAAG Probability theory and mathematical statistics: proceedings of the Fourth Vilnius Conference: Vilnius, USSR, 24-29 June 1985/ed. by Yu. V. Prohorov . . . [et al.] — Utrecht: VNU Science Press. — 111. ISBN 90-6764-067-0 (vol. I) ISBN 90-6764-069-7 (set) SISO 517 UDC 519.2(063) Subject headings: probability theory/mathematical statistics.

Printed in Great Britain by J. W. Arrowsmith Ltd., Bristol

CONTENTS Preface Quantum stochastic calculus L. Accardi Glivenko-Cantelli convergence for weighted empirical and quantile processes of U-statistic structure M. Aerts, P. Janssen and D. Mason On symmetry properties and nonparametric estimates of the v-th order spectral density of a stationary random process A.G. Alekseev Large deviations for first-passage times A.K. AleSkeviiiene Quasi-average approach to the description of the limit states of the n-vector Curie-Weiss ferromagnet N. Angelescuand V.A. Zagrebnov Inequalities at the Markov approximation of lumped processes V. V. Anisimov Remarks on limit theorems for sequences of random variables with random index T.A. Azlarov, A.A. Dzamirzaev and A.G. Abdullaev Continuity of local time for Markov processes with stationary independent increments M. T. Barlow Detection and diagnosis of changes in the A.R. part of an A.R.M. A. model with nonstationary unknown M. A. coefficients M. Basseville, A. Benveniste, G. Moustakides and A. Rougee Nonparametric estimation of distribution functions based on incomplete data Yu. K. Belyayev Diffusion processes on the group many variables A.D. Bendikov and I. V. Pavlov

and elliptic equations of infinitely

vi

Contents

Lower estimates of the convergence rate in the CLT in Banach spaces V. Bentkus

171

Weak limits of probability measures on metric Schauderspaces H. Bergstrom

189

Optimal T. Bjork consumption and investment in a stochastic model

199

Large deviations from classical paths and the classical limit of quantum stochastic flows Ph. Blanchard, Ph. Combe, M. Sirugeand M. Siruge-Collin

203

Limit theorems for multicomponent hierarchical models P. M. Bleher

211

On limit theorems for random vectors controlled by a Markov chain I. V. Bokuchava, Z.A. Kvatadzeand T.L. Shervashidze

231

Upper and lower estimates of the convergence rate in the invariance principle for empirical measures I.S. Borisov

251

The contraction principle for Q,-summing operators and SLLN for weighed sums V. V. Buldygin and S. A. Solntsev

269

Limit theorems for stochastic inventory models E. V. Bulinskaya

291

LimitBulinskii theorems under weak dependence conditions A.V. Non commutative integration and probability on von Neumann algebras C. Cecchini

307 327

On optimal controls in the problem of locally absolutely continuous change of measure (compact sets of decisions) R.J. Chitashvili and M. G. Mania

331

Non-uniform estimates of the remainder term in limit theorems with a stable limit law G. Christoph

357

Multiple stable stochastic integrals Z. Ciesielski

363

Contents

A generalization of p-type spaces A.N. Cuprunov

375

Smooth measures on infinite dimensional manifolds and forward Kolmogorov equation Yu. L. Dalecky

385

The structure of distributions of convex functionals Yu.A. Davydov

405

Wiener germs applied to the tails of M-estimators H. Dinges On the asymptotic behaviour of the linear stochastic heat equation solutions

411

A. Ya. Dorogovtsev, S. D. Ivasishen and A. G. Kukush

419

Some universal Donsker classes of functions R.M. Dudley

433

Duplicates in mixed sequences and a frequency duplication principle. Methods and applications A.T. Fomenko

439

Bootstrapping empirical measures indexed by Vapnik-Chervonenkis classes of sets P. Gaenssler

467

A remark on the Central Limit Theorem for random measures and processes E. GineandJ. Zinn

483

Elliptic law and elements of G-analysis V.L. Girko

489

Mathematical aspects on the variation of air pollutant concentrations J. Grandell

509

Weak solutions of the stochastic evolution and invariance principles B. Grigelionis and R. Mikulevicius

513

Optimal stopping of a Markov chain with vector-valued gain function U.S. Gugerli

523

Some strong laws of large numbers in Banach spaces with regular norms B. Heinkel

529

viii

Contents

Optimality in estimation for stochastic processes under both fixed and large sample conditions C. C. Heyde On urn schemes imbedded in birth processes L. Hoist Limiting distributions and mean-values of complex-valued multiplicative functions K.-H. Indlekofer Asymptotically minimax testing of nonparametric hypotheses Yu.I. Ingster

PREFACE The International Vilnius Conferences on Probability Theory and Mathematical Statistics have been held every four years since 1973 and have become traditional. However, their proceedings were never published. Only a limited number of copies of the abstracts of contributions were available, published locally. The present publication makes it possible for a wide international scientific community to become familiar with the invited lectures delivered at the 4th Vilnius Conference which was held from 24-29 June, 1985. The main topics of the conference were (controlled) random processes and fields, limit theorems of probability theory, and asymptotic methods of statistics. The organizing committee of the conference invited all the 45- and 30-minute speakers to present a paper to the Proceedings. We included in the Proceedings all of the papers received. There were 453 participants at the conference of whom 150 were invited speakers. We hope that the 4th Vilnius Conference and its Proceedings will give rise to many important and interesting investigations in the future. Yu. Prohorov

ix

Prob. Theory and Math. Stat., Vol.1, pp. 1-21 Prohorov et al. (eds) 1986 VNU Science Press

Q U A N T U M STOCHASTIC CALCULUS Luigi Accardi Princeton University D e p a r t m e n t of Statistics Princeton, New Jersey 08544

1.)

INTRODUCTION

Hudson and P a r t h a s a r a t h y have introduced t h e Fock stochastic calculus over L2[R+, dt) which is a beautiful generalization of the classical stochastic calculus t h a t already found several applications in different questions both of mathematics and of physics . However , since there are uncountably many inequivalent representations of the C C R and infinitely many choices for the Hilbert spaces on which they are based , one might suspect t h a t there are infinitely many "stochastic calculi" and t h a t for each of these one should develop an ad hoc machinery based on the particular properties of the representation chosen . Since in the theory of classical stochastic processes there is a general theory of stochastic integration which does not depend on the particular features of the integrator process [11], [13] , [17], [19], [20], [22], it is n a t u r a l to ask oneself if also in the theory of quantum stochastic processes it is possible t o develop a general theory of stochastic integration which is in some sense canonical i.e. , independent of the particular features of the integrator processes . Such a theory has been developed recently ¡5j , ¡6], [7] and the goal of the present paper is to outline some of the main ideas of this representation free stochastic calculus. It will be clear from what follows t h a t the q u a n t u m stochastic calculus we are going t o describe is t h e n a t u r a l translation , in a quantum probabilistic context , of the classical stochastic calculus for general stochastic processes and all t h e basic constructions and techniques introduced in the representation free calculus reduce t o the classical ones when t h e algebras considered are commutative i.e. the algebras generated by the random variables of some classical stochastic process (a property which is not shared by Hudson and Parthasarathy 's Fock stochastic calculus ). Moreover the q u a n t u m Levy representation theorem (cf. Section (7) ) shows the relation between the representation free stochastic calculus and the Fock stochastic calculus , or the "universally invariant" stochastic calculus introduced by Hudson and Lindsay [16] is roughly speaking the same as the relations between these calculi and the usual Wiener process : as the latter calculi can be looked at as "lumping together in a noncommutative way" a "bunch" of classical Wiener processes , the representation free stochastic calculus can be looked at as the result of the same procedure applied t o a "bunch" of Husdson and P a r t h a s a r a t h y (or universally invariant ) processes . T h e results I am going t o describe have been obtained in collaboration with A. Calzolari (Sect. (7) ) , J.Quaegebeur (Sects. (5) ,(6),(10) ) and K . R . P a r t h a s a r a t h y (Sections (3) , (9) ) T h e present exposition is aimed at presenting to an audience of classical probabilists t h e main ideas of the q u a n t u m stochastic calculus with emphasis on the strict analogy between t h e classical and t h e q u a n t u m theory and on the computational and algorithmic aspects, rather t h a n on detailed proofs (which would be impossible within t h e prescribed limitations of space 1

2

Quantum stochastic

calculus

) . In particular, when I had to choose between a rigorous definition or notation and a clear and intuitive one I did not hesitate in favoring the latter. Therefore many technical details have been skipped in the exposition and for these the reader is referred to the bibliography at the end of the paper . 2.) C L A S S I C A L AND Q U A N T U M S T O C H A S T I C C A L C U L U S A quick , intuitive idea of the basic facts about quantum stochastic processes can be obtained by translating in abstract Hilbert space language the theory of classical square integrable stochastic processes . Thus, for example , starting from a probability space (ÍÍ, 7, P), the associated space L 2 (íí, 7, P) of complex valued , square integrable functions is replaced by an abstract complex separable Hilbert space M : L

2

(

t

l

,

?

,

P

)

—

(

1

)

The constant function 1 on fi is replaced by a unit vector $ in )t ||«|| = 1

UL2(n,J,P)-**eMThe algebra L°°(fi, 7, P) , which acts on

L2(fi,

(2)

7, P) by multiplication

/ : teL2(il,7,P)^

f(eL°°(n,7,P)

is replaced by an arbitrary sub-algebra A of the algebra of all bounded operators on )t : L°°(il,7,P)—>ACB{X)

(3)

So that the condition that the set L°°(fi, 7, P)

1 = {/ £l°° n I 2 ( f i , 7, P)}

is dense in L'{fi, 7, P) becomes that the set A $ = {a$

: aiA)

is dense in )i . This property is usually expressed by saying that $ is cyclic for A . If moreover = 0 a = 0 then we say that $ is separating for A . The integral , associated to the probability measure P , is replaced by the expectation functional (or "state" in the quantum mechanical terminology) associated to the vector $ through the formula : f / d P - ^ ( a ) = QtD[X)

L.

Accardi

3

where, here and in the following , D ( X ) will denote the domain of the operator X in M . A classical stochastic process is a family of random variables (-Xt)- Similarly , in first approximation, a quantum stochastic process can be thought of simply as a family (Jf|) of pre-dosed operators on M . A deeper analysis should involve in the definition the notion of "stochastic equivalence", but we skip this point here (cf. [3] ) . For example, one could choose )l to be the space L 2 ( i l , J, P; H) of square integrable H-valued functions on Í1 , for some Hilbert space H , and A to be the algebra of all the operators on M, identified with the tensor product

The quantum stochastic processes corresponding to this choices coincide with the classical operator valued processes which are studied in the quantum theory of measurement [9], [14], and they include the operator valued processes studied in [25] . A filtration (?t\) of sub-(7algebras of 7 is uniquely determined by the corresponding filtration of algebras ¿ ° ° ( n , J, P) ( t > 0) ,and this will be replaced in the quantum model by a filtration ( ^ t j ) of sub-algebras of A : L°°in,?,P)— = closure

of

{a$

: ae.4,]}

If (Xt) is a classical stochastic process , the property of being adapted to the filtration Jt\ can be expressed in Hilbert space language by saying that for each t and for each vector £ in Mt| n D(Xt) also the vector Xt£ lies in . Phrased in this way the definition of adaptedness makes sense for any family ( X t ) of pre-closed operators on X and therefore for any quantum stochastic process . In the notations introduced above , for any Hilbert space )l we denote Pt\ -n — the orthogonal projection and for any

#4

filtration Et]

we denote

:A^At}

the ^conditional expectation from A onto At| which is a linear operator characterized by the properties Et] ( 0 . Remark. If M is such that the pair M, M+

is a L e v y pair in the sense of [6] then the conditions of the

theorem above are certainly satisfied with dA(t) where a is the conditional variance of M .

=

a(t)dt

Quantum

12

stochastic

calculus

8.)REPRESENTATIONS OF THE CCR OVER A COMPLEX HILBERT SPACE. The main source of examples of quantum stochastic processes is provided by the representations of the canonical commutations (or anticommutation) relations (CCR or CAR). In this section we briefly review the main notions concerning the representations of the CCR over a complex Hilbert space which should be considered as a real Hilbert space with an involution (given by the multiplication by the imaginary unit i). One can convince oneself that this is not a mathematical snobbery , but an essential intrinsic feature of the problem considered , by looking at the proof of the quantum Levy representation theorem [6] where in fact the covariance of the Levy pair defines canonically a space of complex valued functions with complex valued scalar product which is however a real and not a complex pre-Hilbert space . Let be a complex Hilbert space with scalar product denoted, ( , ) , antilinear in the first component. Denote a the symplectic form on H defined by the imaginary part of { , •) i.e.

(1)

Denote W„{H) the algebra generated by the set {W, :

feH}

with the relations W,Wa

= exp - . > ( / , g)WJ+g-

/, gtH

(2)

W(0) = 1

(3)

Because of the relations (2) , (3) , WU(H) coincides with the set of all (finite) linear combinations of the Wj •s UtH). It can be proved that there exists a unique C*-norm on W„(H) (i.e. a norm ||.|| satisfying the relation ]|z*z|| = ||z||2). By definition the Weyl algebra over H is the closure of W0[H) with respect to this C ' - n o r m . It will be denoted W(H). A representation of the W(H) is defined by : - a complex separable Hilbert space X - a *-honiomorphism 7r : W ( f f ) — *

— {the bounded linear operators on M}

(i.e. an algebraic homomorphism satisfying

TT(X')

=

IR(Z)'

UR ~>*(W(tf))

) such that for each f in H the map eB(X)

(4)

is continuous in the strong operator topology. Note that we include in the definition of representation of W(H) the conditions that that the map (4) is strongly continuous.

is separable and

Given a representation U, it of W(H) the field operators are defined , through S t o n e ' s theorem, by the condition w(W(tf))

= expttB(f)

(5)

and the creation and annihilation operators are defined respectively by =

-«B(/)]

a{f) = l\B(lf)+tB(f)}

(6)

(7)

The relations (2) for the W ( f ) ' s imply that \B{f),B(g)}

= 2iIm(f,g)

(8)

L. Accardi

13

or equivalently [ « ( / ) • « ( * ) ] = which is positive on positive elements and normalized (i.e.