Frontiers in Pure and Applied Probability: Vol. 1 Proceedings of the Third Finnish-Soviet Symposium on Probability Theory and Mathematical Statistics, Turku, Finland, August 13–16, 1991 [Vol. 2 didn't appear. Reprint 2020 ed.] 9783112314203, 9783112303047

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Frontiers in Pure and Applied Probability

FRONTIERS IN PURE AND APPLIED PROBABILITY Volume 1 Proceedings of the Third FinnishSoviet Symposium on Probability Theory and Mathematical Statistics Turku, Finland, August 13 - 16, 1991

Editors

Niemi H., Högnas G., Shiryaev A.N., and Melnikov A.V.

///VSP///

Utrecht, The Netherlands Tokyo, Japan

TVP

Science Publishers Moscow, Russia

VSPBV P.O. Box 346 3700 AH Zeist The Netherlands TVP Science Publishers Vavilov st. 42 117966 Moscow GSP-1 Russia

© 1993 VSP BV / TVP Science Publishers

First published in 1993 ISBN 90-6764-156-1 NUGI 811

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

Printed in Russia by Novosti, Moscow

CONTENTS Preface Renovation, Regeneration, and Coupling in Multiple-Server Queues in Continuous Time S. Asmussen and S. G. Foss On Functionals of Branching Brownian Motion A. N. Borodin and P. H. Salminen Convergence Rates in Transient Phenomena for Branching Processes K. A. Borovkov The Wick Product H. Gjessing, H. Holden, T. Lindstr0m, B. J. Ub0e, and T.-S. Zhang

Oksendal,

Modelling VARMAX-Processes by Extended Sample Autocorrelation and Linear Regression Techniques R. Hoglund and R. Ostermark A Brief Account of the Theory of Homogeneous Gaussian Diffusions in Finite Dimensions M. Jacobsen A Note on First Passages in Branching Brownian Motions I. Kaj and P. Salminen Random Graphs with Marked Cycles V. F. Kolchin Local Limit Theorem on Convergence of Markov Chains to Diffusion Processes V. D. Konakov A Note on the Stationary Modification of the Loglikelihood Function in Asymptotic Continuous Time System Identification T. Koski Convergence of Filtered Experiments to the Experiment Generated by a Semimartingale D. 0. Kramkov

Martingale Approach to the Procedures of Stochastic Approximation A. V. Melnikov and A. E. Rodkina

165

On a General Class of Stochastic Approximation Algorithms A. V. Melnikov, A. E. Rodkina, and E. Valkeila

183

Stochastic Approximations in the Maximum Likelihood Inference for Markov Random Fields A. Penttinen

197

The Edgeworth Expansion in R l Yu. V. Prokhorov

206

An Improved Nonuniform Convergence Rate Estimate in the Central Limit Theorem in R* V. V. Sazonov

209

Asymptotic Properties of the Maximum Likelihood Estimators under Random Normalization for a First Order Autoregressive Model A. N. Shiryaev and V. G. Spokoiny

223

Infinite Systems of Dissimilar Particles with Random Interactions A. V. Skorokhod

228

On Sequential Detection of a Small Disorder V. G. Spokoiny

238

Branching Processes and Random Trees V.A. Vatutin

256

On the Marcinkiewicz Weak Laws of Large Numbers in Banach Spaces A.I. Volodin

269

Estimating Functions and Efficiency in a Filtered Model W. Wefelmeyer

287

List of Contributors

296

PREFACE During last decade Finland and Soviet Union were constantly extending their scientific collaboration, largely in the form of contacts between Academies of Sciences of these countries. Intensification of the existing scientific contacts became possible in the framework of the Perspective Program of Collaboration in Sciences and Humanities, which included in particular concentration of joint efforts in two traditional to our countries branches of mathematics, namely probability theory and mathematical statistics. The problem of establishing more close relations between Soviet and Finnish specialists in probability theory was repeatedly discussed, and the both parties agreeded in necessity of periodically organizing (with 2-3 year intervals) joint symposia.

In above mentioned Program this idea was formally

realised as one of the forms of our collaboration. The first symposium took place in Helsinki and Lahti, August 31-September 5, 1987, and 26 scientists participated in it. As all following symposia, it was conducted under the aegis of the Soviet-Finnish Commission on Scientific and Technical Collaboration, and concrete organization work was done by Helsinki University. Scientific meetings were held in Mukkula-Conference-Hall in Lahti and were devoted to such topics as martingales, Markov processes, limit theorems in statistics and stochastic analysis. The second symposium was organized in Leningrad, October 16-20, 1989, in Leningrad department of Steklov Mathematical Institute of the USSR Academy of Sciences; more then 30 scientists participated in it. The topics discussed included such branches of modern probability theory and mathematical statistics as martingales and their applications in statistics of stochastic processes, limit theorems in statistics, stochastic analysis. The third symposium took place in Abo Academy in Turku, August 13-16, 1991, with participation of about 50 scientists, and among them there were specialists

Vili from several Scandinavian find Baltic countries. The symposium had gathered a number of famous specialists in probability theory and mathematical statistics. Accordingly, a wide range of actual topics in probability theory, diffusion and branching processes, and stochastic analysis was discussed in the reports of its participants. As expected, these symposia that have been prepared up till now by A.N. Shiryaev and A.V. Melnikov (Mathematical Institute of Russian Academy of Sciences) and H. Niemi and E. Valkeila (Helsinki University) axe becoming more and more representative in the level of reports, the number of participants and countries involved. So it was agreed that the future Russian-Finnish symposia will be held with participation of specialists from Scandinavian and Baltic countries on permanent basis. The proceedings of the first two symposia were published in Acad. Sci. Fennicae. Mathematica

in 1989 and 1992.

The present volume is a collection of papers presented by the participants of the third symposium. It encompasses the whole range of the topics discussed and is a tangible result of our collaboration. The editors hope that it will find readers in other countries as well. It is a good opportunity for the editors to thank University of Helsinki, Âbo Academy, Academy of Finland, and Russia Academy of Sciences for their significant support of the symposium. TVP Science Publishers agreed to publish the Proceedings of the Third Symposium in a special volume and realized it in rather short time with high quality. And we are grateful to TVP Science Publishers for this work.

H. Niemi

A.N.

Shiryaev

G.

A.V.

Melnikov

Hógnas

Frontiers ill P u r e and Appl. P r o b a b . 1, pp. 1 - 6 H. Niemi et al. (Eds.) 1993 T V P / V S P

RENOVATION, REGENERATION, A N D COUPLING IN MULTIPLE-SERVER QUEUES IN CONTINUOUS TIME S. ASMUSSEN and S. G. FOSS Institute of Electronic Systems, Fr. Bajersv. 7, DK 9220 Aalborg, Denmark Novosibirsk State University, Pirogov Street 2, 630090 Novosibirsk, Russia ABSTRACT This paper demonstrates some different, though related, approaches for obtaining ergodicity results for queues on a GI/GI/p example, where the weak convergence of the workload- and queue length vectors in continuous time to their stationary distributions is derived under minimal assumptions. The spectrum of methods incorporate concepts like renovating events, regeneration (in a broad sense) and e-coupling.

I. I N T R O D U C T I O N We consider the GI/GI/p queue with service times Uo,U\,... and interarrival times To, T i , . . . with common distribution A(dt). We denote by Wn = (Wn\ • • •, whP^) the Kiefer-Wolfowitz vector of workloads at the p stations just before the arrival of the nth customer (n = 0 , 1 , . . . ) , with the components ordered in ascending order, and by Vt = ^ V / 1 ' , . . . , the (ordered) vector of workloads at time t. We allow general initial conditions so that Vo is not necessarily the zero vector and that customer 0 arrives at time r(0) = D, where D needs not be zero (the nth customer arrives at time r(n) = D + To + • • • + T n _i). The connection between these two processes in the customer time scale and the physical time scale, respectively, is Wn = VD+To+...+Tn_^.

(1.1)

We assume throughout that p = EZ7/(pET*) < 1. Then the following result is well-known (see, e.g., (Borovkov, 1984; Asmussen, 1987)). THEOREM 1.1. Without conditions beyond p < 1, there exists a random vector W such that Wn —• W in total variation as n —> oo.

(Actually, it is even known that {Wn} is a Harris chain, see (Chariot et al., 1978).) From this and analogues with single-server queues, one expects the following result to be true:

©

T V P Sci. P u b l . 1993

2

S. Asmussen

and

S. G. Foss

If p < 1 a n d A is nonlattice, then there exists a random vector V such that Vt —• V as t —• oo. THEOREM

1.2.

In fact, it is sometimes claimed in the literature that this can be proved exactly as for single-server queues. In our opinion, the problem is, however, more delicate than it may appear at first glance, and it seems that Theorem 1.2 has only been proved with some additional assumptions like regeneration in the idle state (see the discussion in §2) or A being spread-out as in (Asmussen, 1987, Chapter XI). Therefore, our purpose here is therefore of two kinds: to provide rigorous proofs without assumptions beyond those of Theorem 1.2 and to give some general discussion of the problem which we believe has a wider methodological interest. In this respect, the aim is the same as in (Asmussen, 1992; Foss and Kalashnikov, 1991). In fact, §5 of the present note and (Foss and Kalashnikov, 1991), which exploit the method of renovating events originating from the Russian literature (e.g., (Borovkov, 1984; Foss, 1986)) and discuss its relation to the concept of regeneration, are parallel outcomes of discussions at a January 1990 conference in Karpacz, Poland; similarly, §4 provides one of the main applications of the method of e-coupling as discussed in (Asmussen, 1992) (and originating from (Lindvall, 1977)). 2. E X I S T E N C E OF LIMITS IN DISCRETE TIME The problem in question is easy if P(U < T) > 0: The system then empties infinitely often (i.o.), and the instants of customers entering an the empty system serve as regeneration points (both in discrete and continuous time) in the sense that the process is split up into i.i.d. cycles. Unfortunately, P(U < T) > 0 is not automatic from p < 1 as in the single-server case, and what can be inferred is only that P(U 0. These points are discussed, e.g., in (Whitt, 1972). In discrete time it turns out, however, that the relation P ( U < p T ) > 0 can be exploited in a more subtle way to prove Theorem 1. The idea goes back at least to (Gnedenko and Kovalenko, 1968) (see also (Borovkov, 1978)) and is used, e.g., in (Borovkov, 1984) and (Asmussen, 1987; Chapter XI) in slightly different ways. An outline of the argument is as follows. Choose u+,i_ with u + < pt_, P(U < u+) > 0, P ( T > i_) > 0, let K, L be large numbers to be defined later on (with L integer), and define e = pt- — u+, n+L

An = {W™ < K} p| {Uj < u+}

n {Tj > t.}.

(2.1)

j=n

If An occurs, then at least the amount e of work is cleared at each service station between each arrival j = n,... ,n + L and the following one. Thus, if L > pK/e, then the Wn+i, Wn+L+i, ••• are completely independent of the value of Wn. Equivalent ly, Wn+k

= iPk(Tn,...,Tn+k-1,Un,...,Un+k-1),

k>L,

(2.2)

which in the terminology of (Borovkov, 1984; Foss, 1986) means that the event An is renovating on { n , . . . , n + L}. Note that this line of thought is similar to the

Renovation,

Regeneration,

and

3

Coupling

splitting argument for Harris chains (Nummelin, 1978; Athreya and Ney, 1978), the set SK = {u; : < K} serves as a splitting set from which there is a uniform probability of having the "minorant measure" after L steps. Moreover, it was shown in (Borovkov and Foss, 1992) that these two concepts, positive recurrence in the Harris sense and the existence of a stationary sequence of renovating events, are equivalent in the Markov case. Total variation convergence of Wn can now be deduced, for example, along the following lines. First show that if K is large enough, then P(j4„ i.o.) = 1 and then let c*(0) = inf{n > 0: An}, a(k + 1) = inf{n > a(k) + L: An}. Then {a(fc)} is a renewal process, and ,Wa^)+L+1» • • • is independent of { a ( 0 ) , . . . , with distribution not depending on k. This means that j Wn J is regenerative (in a wide sense, cf. (Asmussen, 1987, Chapter V)) with respect to the renewal process { T A ^ ) + L - I

and,

therefore, may depend on r(a(k) + L). To overcome this difficulty, we need a modification of our definition of renovating events. Let w+,i- be as before, but choose now t+ > t- with P( 0 and define n+L Bn

-

{ w ^

< K}

fl

{Uj


0(k) + L : Bn}, then

is regenerative (in the wide sense) w.r.t. {T(/?(fc))} ajid {T(/?(fc))} is

readily checked to be a renewal process. Thus to obtain weak convergence of w e have only to check that {r(0(k))} and thereby is nonlattice and has finite mean interarrival time. The first of these statements is a general property of thinning of a nonlattice renewal process (Asmussen, 1987, p.187). For the second, according to Wald's identity it is sufficient to check that {/?(&)} has finite mean interarrival time. However, from the positive recurrence in discrete time we know that the limiting proportion of time < WN > visits SK is strictly positive. Hence

4

S. Asmussen

and

S. G. Foss

by the geometric trials argument, the limiting proportion of instants of the form is also positive. 4. e - C O U P L I N G Our second proof of Theorem 1.2 proceeds by e-coupling, in the form of the following result from (Asmussen, 1992). Let {Zt}0 0 one can find versions of {Zt}, {Z[} defined on a common probability space and a.s. finite random times D = De, T — Te such that |D|(r/) £ E),

in [a, b] the function

u is the

unique

on ( a , b ) \ { y i , y 2 ) - • • ,J/n}, (2.9) on (a, b).

To get (2.8) consider u(x) = È I ( e x p ( - i 4 0 0 ) ; r < /?) + È , ( e x p ( - i 4 o o ) ; r > /3) = ù0(x) + È I ( e x p ( - A ^ - (Aoo - Ap));

T > ¡3)

= Ù0(x) + Èz ^exp(—A^) ¿ p n i È ^ i e x p i - A o o ) ) ) " ; r >

fij

OO = tio(i) + j

dtae~ai

0

= ù0(x) + J

J

È f ( e x p ( - A t ) ; ut €

dy)G(Ù(y))

R

I T * _ i ; Nt ± iV t _}

and introduce a sequence of functions via

12

A. N. Borodin

and

P. H.

Salminen

Ù (0) (x) = 0, «(*) = È I ( e x p ( - y l 0 0 ) ; NTk

= 0).

Using the convention "if Nt — 0, then Nt+, = 0 for all s > 0 " , and (2.7) we have v,(k\x) | ù(x) for every x as k —» oo. Furthermore, we have for k > 2

ù^\x)

=

= È . i e x p C - A o o ) ; NTl

UQ(X) + p0Ka(x,R)

= 0, Nn

= 0 ) + E I ( e x p ( - ^ l o o ) ; NTl

00 r ^ / Ka(x,dy)Y^pnEyn(exp(-A00);

+

R

"

± 0, NTk

NTk_1

= 0)

= 0 ) ,

=1

where yn = ( y , . . . , y ) (E R " . Denote by u>^ the subtree starting at the ith coordinate in y„. Then, taking into account that

{ X o = y

n

;

N t ^

=

0 } C {X0

=

y„,

N

T k

_l(J

1

)) =

0,...,NTk_l(u;(n))

=

0},

we have

E ^ e x p i - A o o ) ; 7V T t _, = 0) = E s „ ^exp

K

on

(a, b).

Let

St := {a
0,

for

x < 0.

To solve this consider first the case x < 0. Setting v := 1 — u we see that t), satisfies J

= 0 introduce t :— u — B, where B is as in (4.3). Using the fact that B solves the equation in (4.2), we see that t satisfies

f t" + at2 + 2 ( a B - (a + y))t = 0, \ lraixjoo t(x) - limxioo t'(x) = 0.

(4.5)

The problem (4.5) is solved analogously to (4.4). The result is

y / A - C s + y / A - (y/A - C

2

-

VA)e~Dx

(4.6)

where A := 3y/(a + 7 / a ) 2 - 1, D = y/2aA/3, and C2 is an unknown constant such that 0,

P , { i n f X, > 0} = Px{Loo

= 0} = 1 - ( l + y ^ a

4.4. Total length of a killed critical branching Brownian tree Let X be a killed B B M such that particles hitting zero are immediately at that time removed from the process to a cemetery point A isolated from R , where they are trapped. Consider the critical case p0 = p2 = 1/2 and assume that XQ = x > 0. Let loo be the functional introduced in (4.1). By Theorem 2.2 the function

Functionals

of Branching

Brownian

21

Motion

û(x) := E x (exp(—7/00)) is the unique solution of the problem ' i û " - ( a + 7)û + a(|û 2 + |) = 0, ^ lim^oo û(x) = B, Hindoo û'(x) = 0, ¿(0) =

x > 0,

1,

0 < û < 1. From this we obtain the same expression for t := û — B as in (4.6) with C = 1 — 5 . 4.5. Total length of a killed supercritical branching Brownian tree Let X be as in Example 4.4 but now with P2 = 1. Then û(x) := E r (exp(—7/œ)) is the unique solution of the problem ' \û" - (a + 7 ) û + QÛ2 = 0, ( lim IÎOO û(x) = lim IÎOO û'(x) = 0, "(0) =

x

> 0,

1,

0 < û < 1. Proceeding as in Example 4.2 we obtain

Û(x)

= A ( I - (

where A = 3(a +

+ V/I + (Y/Â=Ï

-

V I ) EXP(-Y^

+ ^X) \

y)/(2a).

REFERENCES

Asmussen, S. and Hering, H. (1983). Branching

Processes. Birkhäuser, Boston, Basel, Stuttgart.

Borodin, A . N . (1989). Distributions of functionals of Brownian local time.I. Theory Probab.

Appi.

34, 385-401. Erdélyi, A., ed. (1954). Tables of Integral Transforms,.

Vol. I. McGraw-Hill, New York, Toronto,

London. Ikeda, N., Nagasawa, M., and Watanabe, S. (1968a, 1968b, 1969). Branching Markov processes, I, II, I I I . J. Math. Kyoto

Univ. 8, 233-278; 8, 365-410; 9, 95-160.

Sevast'yanov, B.A. (1958). Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries. Theory Probab. Appi. 3, 111-126. Watanabe, S. (1965). On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto

Univ. 4, 385-398.

Frontiers in Pure and Applied P r o b a b . 1, pp. 22 - 28 H. Niemi et al. ( E d s . ) 1993 T V P / V S P

C O N V E R G E N C E RATES I N TRANSIENT PHENOMENA F O R B R A N C H I N G PROCESSES K. A. B O R O V K O V Steklov Mathematical Institute, Vavilov Street 42, 117966 Moscow GSP-1, Russia ABSTRACT We consider the limiting behaviour of nonhomogeneous Markov branching processes in discrete time when the offspring means are uniformly close to unity. In this set up, 1) the uniform convergence (over classes of processes with uniformly square-integrable offspringnumbers) of conditional laws to the exponential one is proved; 2) bounds for the rate of this convergence are given (these bounds are uniform over classes of processes with a bounded third moment); 3) asymptotic expansions are constructed which are uniform over classes of processes with uniformly cube-integrable offspring numbers.

We consider the limiting behaviour of conditional distributions of nonhomogeneous Markov branching processes in discrete time as time tends to infinity when the offspring means are close to unity uniformly over generations. The following three basic aspects are analysed here: the stability of transient phenomena under violation of time-homogenity, convergence rates in the integral limit theorem, and asymptotic expansions. Though all considerations are given for discrete-time processes, the results can be easily carried to the continuous-time case too by the standard embedding procedure. First we prove the uniform (over suitable classes of processes) convergence of conditional distribution functions (d.f.'s) of the number of particles to the exponential law (the homogeneous case see in (Nagaev and Muhamedhanova, 1966a; 1968; Sevast'yanov, 1971; Fahady et al., 1971)). Then we establish uniform (over some classes of processes too) bounds for the rate of this convergence. Earlier only the case of one critical process has been investigated, see (Nagaev and Muhamedhanova, 1966b), the estimate there has the form 0{n~ l log 2 n) when the third moment is finite. Asymptotic expansions are obtained here for Galton-Watson processes only; moreover, they are valid for the expectations of functions from a rather wide class that does not comprise, however, the indicators of intervals, and hence these expansions do not hold in the general case for d.f.'s. This part of paper answers a question posed in the author's paper (Borovkov, 1988): do the expansions derived there for Laplace transforms correspond to similar expansions

©

T V P Sci. Publ. 1993

Transient

Phenomena

for

Branching

23

Processes

for d.f.'s? Now we introduce the basic notation. Let Mi)>

j =0, l,...,n,

/i(0) = 1,

be a nonhomogeneous branching process,

be the generating function (g.f.) of the offspring distribution on the j t h step. Set Aj = fj( 1),

Bj = }•{ 1),

Cj =

/)"(1),

these are the left derivatives, as usual. Now let a = an = max\Aj - 1 j,

A(j, m) = AjAj+1

j & } . The notation M o (AC) will be used for the class of Galton-Watson processes, for which the g.f.'s / lie in AC. Finally, Ti^,, i[> 6 J~, will denote the class of functions of bounded variation H on the real line, Fourier-Stiltjes transforms h(t)

= J

eitzdH(x)

of which satisfy J

r > 0 .

W

Our first main result can be stated as follows. THEOREM

1. For

a n y ip £ F

and

A n = sup |F„(x) - G(x)\

b > 0,

0,

Q(n)T(n)/A(

1, n) -

1

X

uniformly

in A4(/C2,^(b))

a s n —> oo,

a->0.

Now we consider the case of branching process starting with a large initial number of particles fi/v(0) = N, and let FNin(x)

= p{nN(n)/T(n)


0

the compound Poisson law with parameter a and spectral measure F. The law II^[Gi] has the characteristic function exp(iA 0, c < oo. Below we single out the case, where Ai = l + 0 ( l + e i ) ,

M(fC3c(b))

|e,-|0,

e = |A-l| + 5 - 1 l o g 5

uniformly in as n —> oo, A —> 1. Remark 1. What we can say about the relation between the terms in the righthand side of (3)? It is easy to see that \A - 1| = 0 ( S ~ 1 ) for A < 1 + 0 ( n _ 1 ) , and hence the term S - 1 logS is the main one. Moreover, the "critical" value of A here is A - 1 + ( 1 + loglogn. We have \A- 1| = o ( s - 1 l o g s ) for h < 0, 5 _ 1 log S = o(\A - 1|) for h> 0. Remark 2. Theorem 2 enables us to obtain the corresponding results for the processes with large initial number of particles from Theorem 3. For GaltonWatson processes, e.g., we have Q(n) = 0(a"5-x)

= o(|A - 1| +

5-1)

by Theorem 1, and hence (3) and Theorem 3 imply that sup | F N , n ( x ) - II a [GI](X)| = o(\

( S " 1 log S + \A - 1|) log s )

uniformly in Mo (£3,0(6)). Now let us turn back to asymptotic expansions. In (Borovkov, 1988) it was shown that in the case of one critical Galton-Watson process with finite third moment the following representation holds true for the Laplace transform