Probability Theory and Mathematical Statistics: Vol 1 Proceeding 1989 9067641286, 9067641308

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Probability Theory and Mathematical Statistics: Vol 1 Proceeding 1989
 9067641286, 9067641308

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Volume I

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

Edited by

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

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

MOKSLAS Vilnius, Lithuania


Utrecht, The Netherlands

VSP BV Post Box 346 3700 AH Zeist The Netherlands

MOKSLAS Zvaigzdziu 23 Vilnius Lithuania

©1990 VSP BV/IMI 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.

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CIP-DATA KONINKLIJKE BIBLIOTHEEK, D E N HAAG Probability theory and mathematical statistics: Proceedings of the Fifth Vilnius Conference: Vilnius, Lithuania, J u n e 25 - J u l y 1, 1989 ed. by B . Grigelionis... [et al.] - Utrecht: V S P B V / Vilnius: Mokslas. I S B N 90-6764-128-6 (vol.1) I S B N 90-6764-130-8 (set) S I S O 517 U D C 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


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The low density limit and the quantum Poisson process L. Accardi and Lu Yun Gang

vii 1

Systems assisting in gathering and analysis of statistical data and adjustable to subject domains S.A. Aivazyan, Yu.N. Blagoveschensky and L.D. Meshalkin


On the maximum of sums of random variables and the supremum for stable processes A.K. Aleskeviciene


On the necessity of Cramer's condition in local limit theorems N.N. Amosova


Markov random graphs and polygonal fields with Y-shaped nodes T. AraJc and D. Surgailis


Probability measures on quantum logics and a non-commutative Choquet theory Sh.A. Ayupov


Convolution semigroups of instruments in quantum probability A. Barchielli and G. Lupieri


Stochastic posterior equations for quantum nonlinear filtering V.P. Belavkin


Diffusions and parabolic equations in principal bundles Ya.I. Belopolskaya


On Markov processes associated with a projective sequence of harmonic spaces A.D. Bendikov


Estimation of the tail of the spectral distribution by means of high level sojourn times S.M. Berman


Renormalization of Dyson's vector-valued hierarchical model at low temperatures P.M. Bleher and P. Major


Analitic functionals of stochastic processes and infinite dimensional oscillatory integrals V.I. Bogachev




Exponential inequalities for the distributions of von Mises and (/-statistics I.S. Borisov


Ergodicity and stability of Markov chains and of their generalizations. Multidimensional chains A.A. Borovkov


Event and time averages: stationary case P. Bremaud


The stationary case and the non-

On the convergence of sums of independent random vectors normed by matrices V.V. Buldygin and S.A. Solntsev


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C L T for families of integral functionals arising in solving the multidimensional Burgers Equation

A.V. Bulinskii


The theory of dynamical semigroups and its applications A.M. Chebotarev


An approximation of integer-valued measures by generalized Poisson measures V. Cekanavicius


Lim inf results for the standardized empirical distribution function E. Csaki


The smooth measures in Banach spaces and their smooth mappings Yu.L. Daletsky and V.R. Steblovskaya


Local invariance principle for i.i.d. random variables Yu.A. Davydov


Conditionally positive definite functions on the coefficient algebra of a nonabelian group L.V. Denisov and A.S. Holevo


On the integrability of the maximum and the local properties of Gaussian fields V.A. Dmitrovskii


On large deviation probabilities for the maximum likelihood estimators K. Dzhaparidze and E. Valkeila


Functional integrals, a variational method and some problems of stochastical physics G.V. ESmov




The law of the iterated logarithm and order statistics

V.A. Egorov


Martingales in non-life insurance

P. Embrechts


Asymptotic minimaxity of usual goodness of fit tests M.S. Ermakcrv Les fonctions aléatoires à valeurs dans les espaces lusiniens et leurs modifications régulières X. Fernique



Critical branching caused by interaction with point catalysts K. Fleischmann


Statistical analysis and dating of the observations on which Ptolomy's "Almagest" star catalogue is based A.T. Fomenko, V.V. Kalashnikov and G.V. Nosovskii


Global asymptotics of solutions of the first and second boundary value problems for Kolmogorov equation with small diffusion S.M. Frolovichev


Hydrodynamic limit for Ginzburg—Landau type continuum model T. Funaki


Almost everywhere limit convergence of subsequences of powers V.F. Gaposhkin and J.M. Rosenblatt


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On shot noise processes with long range dependence L. Giraitis and D. Surgailis


G-estimates of eigen-values of covaxiance matrices V.L. Girko


On large deviations for trimmed sums of independent random variables V.V. Godovan'chuk and V.V. Vinogradov


Random operator functions defined by stochastic differential equations N.Yu. Goncharuk and Yu.L. Daletsky


Hellinger integrals and Hellinger processes for solutions of martingale problems B. Grigelionis


On connection between the theory of branching processes and queueing theory S.A. Grishechkin


Estimation of MANOVA eigenvalues A.K. Gupta




On joint distributions of random elements with given mutual conditional distributions B.M. Gurevich


Improvement and generalization of Cramer—Rao inequality for a filtered space A.A. Gushchin


Limit theorems for record times A. Gut


On statistics of continuous Markov processes: s e m i - Markov approach B.P. Harlamov


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Quantum stochastic flows R.L. Hudson


Convergence, of the stochastic quantisation method I. A. Ignatiuk, V.A. Malyshev and V. Sidoravicius


Adaptive tests for minimax testing of nonparametic hypotheses Yu.I. Ingster


A survey of the metric theory of continued fractions, fifty years after Doeblin's 1940 paper M. Iosifesku


Boundary and entropy of random walks in random environment V.A. Kaimanovich


Non-linear law of large numbers and Gaussian fluctuation for s t a r like networks M.Y. Kelbert and Y.M. Suhov


Asymptotic solution of inverse Kolmogorov equation for diffusion processes with small diffusion V.M. Khametov


Asymptotic robustness of discriminant procedures for dependent and non-homogeneous observations Yu. Kharin and A. Medvedev


Characterization of distributions: problems, methods, applications L.B. Klebanov and A.A. Zinger


The entropy order of operators and limit theorems in Banach spaces V.I. Kolchinsky


Inverse problem for potentials of measures in Banach spaces A.L. Koidobski


Investigation of the convergence of functions of sample means and covariances to limit distributions T. Kollo


PREFACE T h e traditional Vilnius Conference on Probability Theory and Mathematical Statistics was held for the fifth time in 1989, June 26 - July 1. It was organized, as usual, by the Institute of Mathematics and Cybernetics of the Lithuanian Academy of Sciences, the Steklov Mathematical Institute of the USSR Academy of Sciences and the Vilnius University. T h e Organizing Committee was headed by the cochairmen Professor Yu.V. Prohorov (Moscow) and Professor V. Statulevicius (Vilnius). There were 744 participants at the Conference from 28 countries. The main directions of discussions covered asymptotic methods in probability and mathematical statistics, theory of Markov processes, stochastic analysis, martingales, stochastic physics and random fields. Q u a n t u m probability and computer statistics were included in the program for the first time. Two plenary lectures delivered by Professor A.A. Borovkov (USSR) and Professor K.R. Parthasarathy (India), as well as 446 talks, divided into 63 sectional sessions, were presented. Poster session included 172 communications. T h e present Proceedings in two volumes contain the papers received by the Organizing Committee b o t h from the invited speakers and authors of some selected short communications. We wish to thank most cordially all contributors to the success of the Vth Vilnius Conference and express hopes that these Proceedings will stimulate further progress in probability and mathematical statistics.

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

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Piob. Theory and Math. Stat., Vol. 1, pp. 1 - 2 7 B.Grigelion» tt al. (Eds.) 1990 VSP/Moksla*

THE LOW DENSITY LIMIT AND THE QUANTUM POISSON PROCESS L. ACCARDI and LU YUN GANG* Centro Matematico V.Volterra, Dipartimento di Matematica, Università' di Roma II, Italy ABSTRACT We prove that, in the low density limit (z2 = fugacity —» 0), the family of processes given by the collective Weyl operators and the collective coherent vectors (cf. a free Bose gas), converge to the Fock quantum Brownian motion over L 2(R, dt, K ) , where K is an appropriate Hilbert space (cf. Section 1). Moreover we prove that if we couple a nonrelativistic quantum system with a quadratic interaction (cf. formula (1.10) below) to a free Boson gas in the Fock state, then the matrix elements of the wave operator of the system at time t / z 2 in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation ruled by some pure number process (i.e., no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes). The explicit form of the equation is determined and the unitarity is proved.

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1. I N T R O D U C T I O N In the last two years our understanding of the weak coupling limit has drastically increased since it has been possible to acquire control of the limit not only of the reduced dynamics, but of the full quantum dynamics, both in the Schròdinger and in the Heisenberg form (Accardi et al., 1989, 1990a, 1990b; Accardi and Lu Yun Gang, 1989a, 1989b). The basic new idea in these papers was the introduction of some classes of "collective states" (collective coherent states or collective number states) and the study of the limit of the matrix elements of the Schrodinger as well as of the Heisenberg dynamics with respect to these collective states. Both in the classical and quantum case the results available in the low density case are much poorer than the corresponding ones for the weak coupling case: for example the existence of the Markovian semi-group has been proved only for a short time interval in a very special Fermion model (Diimcke, 1989a, 1989b). Let H0 and Hi be complex separable Hilbert spaces interpreted respectively as the system Hilbert space and the one particle reservoir Hilbert space. Let W { H




{ W ( f ) :



* On leave of absence from Beijing Normal University © L . A c c a r d i and Lu Yun Gang. 1990



The Low Density


and the Quantum



be the Weyl C*-algebra on H\\ let i f be a self-adjoint bounded below operator on Hi and z, /3 positive real numbers interpreted respectively as density of the reservoir particles and inverse temperature. Define Qz

:= (l + z2e~PH)

( l — z2e~/3H)~1

= coth{0H + p)


(with z2 = e*1) and suppose that, for each z in an interval [0, Z], Qz is a selfadjoint operator on a domain D, independent on z. Denote y'Q, the mean zero gauge invariant quasi-free state on W(H\) with covariance operator Q 2 , i.e. V Q . m f ) ) = exp and let

(/,«./>) ,

be the GNS - triple of (W))*Q.)

We Q, {Ji, the

(1.3) so that

= 0,

StQz = QzSt,

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V/ G H 1


where the equality is meaxit on D. This implies that the second quantization of St, denoted W(S