Progress in Pure and Applied Discrete Mathematics, Vol. 1: Probabilistic Methods in Discrete Mathematics: Proceedings of the Third International Petrozavodsk Conference, Petrozavodsk, Russia, May 12–15, 1992 [Reprint 2020 ed.] 9783112318980, 9783112307847

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PROBABILISTIC METHODS IN DISCRETE MATHEMATICS

PROGRESS IN PURE AND APPLIED DISCRETE MATHEMATICS, Vol. 1

PROBABILISTIC METHODS IN DISCRETE MATHEMATICS Proceedings of the Third International Petrozavodsk Conference Petrozavodsk, Russia, May 12 - 15, 1992

Editors V. F. Kolchin, V. Ya. Kozlov, Yu. L. Pavlov, and Yu. V. Prokhorov

///VSP/// Utrecht, The Netherlands Tokyo, Japan

TVP Science Publishers Moscow, Russia

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

© 1993 TVP Science Publishers/VSP BV

First published in 1993 ISBN 90-6764-158-8 UDC 519.1, 519.2, 519.7

All right 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, photo-copying, recording or otherwise, without the prior permission of the copyright owner.

Printed in Russia by Novosti, Moscow

CONTENTS Preface

ix

A Survey of the Early History of the Theory of Random Mappings B. Harris

1

On Goncharov's Works in the Field of Combinatorics V. F. Kolchin

23

Probability Distributions in Prime Number Theory S. W. Golomb

28

Polynomial and Polynomial-Like Allocations: Recent Developments V. G. Mikhailov

40

Properties of Random Permutations with Constraints on the Maximum Cycle Length A. A. Grusho

60

Stochastic Properties of Systems of Random Linear Equations over Finite Algebraic Structures I. N. Kovalenko and A. A. Levitskaya

64

On the Number of Solutions of Systems of Pseudo-Boolean Random Equations G. V. Balakin

71

Problems of Security in Information Processing Systems A. F. Ronzhin

99

Decomposable Statistics Based on the Spacings and Spacing-Frequencies of Higher Orders A. A. Ashurkov

128

On the Allocation Processes with Given Distributions of Frequency Vectors S. I. Chechota

133

vi

Statistical Problems Concerned with Estimation of the Composition of /^-Radioactive Wastes P. V. Chistyakov, V. P. Chistyakov, V.I. Dorogov, andV. I. Handogin

143

On Transient Phenomena for Branching Migration Processes E. E. Dyakonova

148

Asymptotic Behaviour of Waiting Times in a Scheme of Particle Allocation N. J. Enatskaya and E. R. Khakimullin

155

Limit Theorems for U-Statistics of Dependent Random Variables A.B. Gortchakov

186

Limit Theorems for Decomposable Statistics of Dependent Random Variables

A. B. Gortchakov

200

Estimating Finite Stratified Populations G. I. Ivchenko and S. A. Khonov

212

Asymptotic Expansions for Permutation Tests with Several Samples Sh. Ismatullaev and Sh. Mirakhmedov

226

On Approximation of Weighted Sums of Random Variables with Two-Dimensional Indices K. A. Jafarov

240

On a Lower Bound for the Isoperimetric Number of Cubic Graphs A. V. Kostochka and L. S. Melnikov

251

A Risk Function in Global Optimization Problem

S. V. Kovalev and V. V. Mazalov

266

On an Existence Theorem for Decomposable Statistics

E.M.Kudlaev The Property of Phase Transitions in Random Graphs A. V. Lapshin On Sums of Values of the Legendre Symbol over Components in a Scheme of Generation of Random Vectors Yu. I. Maksimov

271 287 298

vii

On the Strong Law of Large Numbers for Sums of Pairwise Independent Random Variables A.I. Martikainen

305

On a Special Class of Functions Associated with the Complexity of String Correction Problem

S. S. Martynov

313

Operator Moments of Random Euclidean Mappings V. M. Maxim ov

316

Percolation Method in a System of Numerous Particles

M. V. Menshikov and K. D. Pelikh Models of p-Percolation M. V. Menshikov and S. A. Zuev On Estimation of the Coefficient of Asymptotic Relative Efficiency of Discrete-Analogue Algorithms to Detect Signals in a Non-Gaussian Noise T. Yu. Morozova and A. N. Trunov Random Mappings and Trees: Relations between Asymptotics of Moments of Certain Characteristics L. R. Mutafchiev

329 337 348 353

On a Nonlinear Generalization of the Occupancy Problem G. V. Proskurin

366

Weak Convergence of Certain Multiple Sums of Dependent Random Variables I. Rakhimov

376

On Problems of Renewal System Theory for Finite Time Interval A. A. Rogov and V. I. Tchernetskii

386

On the Generating Functions of Coloured Partitions and Limit Theorems for Sums of Indicators M. N. Rokhlin '

395

Probabilistic Methods in the Theory of Approximation of Discrete Functions B. V. Ryazanov

403

Chernoff's Inequality for Sum of Independent Random Vectors N.P. Salikhov

413

viii

Some Limit Theorems in Batch Allocation Scheme with Random Levels Defined by an Allocation Scheme M. Savchuk

428

Runs in Finite Markov Chains L. Va. Savelyev

437

How Often is Addition Equal to Bitwise Addition? V. I. Sherstnev

451

Boolean Functions and Approximation of Vectors

List of contributors

V. M. Sidelaikov

and 0. V. Sipacheva

454 470

Limit Theorems for Random .A-Permutations A. L. Yakymiv

459

PREFACE This volume contains the Proceedings of the Third Petrozavodsk Conference "Probabilistic Methods in Discrete Mathematics" which was held at 12-15 May 1992 in Petrozavodsk in Russia. The conference was organized by the Steklov Mathematical Institute of the Russia Academy of Sciences, the Karelian Scientific Centre of Russia Academy of Sciences and the Karelian State University. T h e pioneering works carried out by V. L. Goncharov, published in 40's, are apparently the first where combinatorial objects were systematically investigated by modern probabilistic methods. These works contain deep results on random permutations and runs in random (0, l)-sequences. During the last three decades the interest in probabilistic problems of combinatorics has grown steadily b o t h in our country and abroad. In 1983 the First Ail-Union conference "Probabilistic Methods in Discrete Mathematics" and in 1988 the Second conference were held in Petrozavodsk. The themes of the Petrozavodsk conferences cover almost all areas of probabilistic combinatorics including probabilistic problems of discrete mathematics, statistical problems in discrete mathematics and graph theory. The participants of the two first conferences came up with the proposal to extend the topics of the conference and give more attention to actual applied probabilistic problems of discrete mathematics. In line with these suggestions, for the first time the section "Mathematical methods of information security" was organized along with the traditional sections "Probabilistic problems of combinatorics", "Statistical methods in combinatorics", "Random graphs". During the three working days of the III Petrozavodsk conference, 12 plenary and 33 sectional lectures were read. At the first plenary session the lectures were presented by B. Harris (Madison), V. F. Kolchin (Moscow), V. A. Vatutin (Moscow), Yu. L. Pavlov (Petrozavodsk); at the second plenary session the lectures were read by S. Golomb (Los-Angeles), V. G. Mikhailov (Moscow), A. A. Grusho (Moscow), V. A. Plaksin (Petrozavodsk); in the final day the conference heard the plenary lectures by I. N. Kovalenko and A. A. Levitskaya (Kiev), G. V. Balakin (Moscow), A. F. Ronzhin (Moscow). The plenary lectures prepared for publication are presented in the Proceedings in the above mentioned order. All sectional reports are arranged in alphabetic order of authors names. The papers presented in the Proceedings reflect the current state of the art in probabilistic combinatorics and contain the information which will be interesting to those who work in theoretical and applied areas of discrete mathematics. The Organizing Committee has the pleasant duty to thank the sponsors of the III Petrozavodsk conference without whose financial support the conference could not be held. These are T V P Science Publishers, Russian National Committee of Bernoulli Society, SCB Progress. The organizing duties were taken u p by the small venture Elmus (Petrozavodsk).

X

We axe deeply indebted to TVP Science Publishers who agreed to publish the Proceedings, we wish also to thank their highly skilled personal who carried out all their work at a top level. The Editors are grateful to those who submitted their papers on time, to the referees who carried out their tasks thoroughly. We especially value the efforts of Elena Dyakonova who coordinated ail the editorial activity. We wish also to express our gratitude to all participants of the Conference and invite all of them to take part in the next Petrozavodsk conference which will be held in 1996. Vice-Chairman of the Organizing Committee Professor

V. F. Kolchin

Probabilistic Methods in Discrete Mathematics, pp. 1 - 2 2 V. F . Kolchin et al. (Eds.) 1993 T V P / V S P

A SURVEY OF THE EARLY HISTORY OF THE THEORY OF RANDOM MAPPINGS B. HARRIS Department of Statistics, University of Wisconsin, 1210 Dayton Street, Madison, Wisconsin 53706, U.S.A. ABSTRACT The early history (prior to 1965) of the theory of random mappings is surveyed. The paper describes the types of questions that were studied during that period and the results that were obtained. In particular, some unpublished works with significant results are described. The appendices to the paper contain a catalog of the various results that were obtained by type of result and by reference. 1. I N T R O D U C T I O N A N D S U M M A R Y In this report, the early history (prior to 1965) of the theory of random mappings is studied. In this report, the types of questions studied at that time and the methodology that was utilized are examined. In particular, a substantial fraction of the research described is unpublished and it is hoped that this study will serve the purpose of recognizing these contributions to the theory and establishing their place in the history of the subject. The theory of random mappings, as described in this paper, concerns the following types of questions. Let Xn be a set with |3£„| = n. Let J„ be a subset of the mappings of X„ into X„. Let P j n be a probability measure on the subsets of J„. With no loss of generality, it will be convenient to let Xn = { 1 , 2 , . . . ,n}. Let a € Jn• Then since a is random, various characteristics of a are random variables and the presence of a in specified subsets of J„ are random events. The theory of random mappings is concerned with determining the probability distributions and characteristics (such as moments) of these random variables and the probabilities of these random events. A substantial proportioruof the principal results of the theory deals with the asymptotic behaviour of these quantities as n —» oo. In the early history, P j n has been generally chosen as the uniform probability measure on J„, that is P j n = | J n | - 1 . The following representation of the mapping a e Jn is frequently employed. Let x,y € Xn. If a ( x ) = y, then a directed arc is drawn from x to y. This procedure is continued for each element x € Xn resulting in the representation of a by a directed graph (with labelled vertices) with the property that exactly one edge emanates from each vertex. This representation of a mapping as a directed graph appears to have been first utilized by A. Suschkewitsch (Suschkewitsch, 1928). © T V P Sci. Publ. 1993

2

B.

Harris

This representation provides a convenient method for visualizing the various characteristics of interest, including those defined below. Fix x € Xn and consider the set S a (x) = {a:,a(a:),a 2 (®),...}. Sa(x) is called the set of successors of x under a. Similarly, let Pa(x) = {i, a - 1 ( x ) , a~2(x),...}. Pa(x) is the set of predecessors of x under a. That is, Sa(x) is the set of elements in X„ which "can be reached" from x in the directed graph representing a, and Pa(x) is the set of elements in Xn from which x "can be reached" in the directed graph representing a . Since Xn is a finite set, there is a least positive integer m such that for some r > 0, a m + r ( s ) = .¥)• i f * } m \ „ f E < max — > = u„ n J

(Golomb 1959)

joint distribution of

et

al.,

(Goncharov, 1944)

least probable configuration

C] = n, Cj = 0, j £ 1

(Goncharov, 1944)

most probable configuration

ci = 1, c „ _ i = 1, Cj = 0, j ^ l , n-1

(Goncharov, 1944)

History of Theory of Random

Mappings

9

T a b l e 3. Generating functions for probabilities or expected values of characteristics of random parmutations Characteristic (c,

Coefficient of Function

c„)

References

£ 4 i=i 3

(Goncharov, 1944)

i °° f x 1 cycle partitions of S„ indexed by cj, 1 < < n

t

exp < ^ —r- > (j=i 3 )

(Goncharov, 1944)

Ka

tmx"

exp j t ¿ j y | = (1 - x)'

(Goncharov, 1944)

Ka

tm

(n^-'tit+lJ-.-ii + n - l )

(Goncharov, 1944; Greenwood, 1953; Riordan, 1962)

Ka

xn

-î-j log m (l — e ) ~ 1 m!

Cm

tm

£ (j)"1 ( — ; =0 *

n

(Goncharov, 1944)

) '

(Goncharov, 1944)

B. Harris

10

APPENDIX 1.2 RANDOM PERMUTATIONS WITH NO FIXED POINTS T a b l e 1. Components Characteristics

Formulae n

1 P { * a = k}

Un

{Ka}

(-1 yat-j,n-j (» ~ j ) \ j \

n

, D " - . , ~logn + 0(l) s(n — «)! .=2

U n

Var{| Ka)

~ ( 2 7 - 1) log n + 0 ( 1 )

P(I*-(»)I = J}

( » - D ! A.-, (n-j)'.Dn '

1 n' e ~ 2n

P{|*0(x)| = » - 1 } P{|*.(*)| = n - 2 }

(Harris, 1960)

i=0

1 E

References

(Harris, 1960) (Harris, 1960)

„ 3

(Harris, 1960) (Harris, 1960)

i > 2

(Harris, 1960)

e

(Harris, 1960)

~ 3n

T a b l e 2. Length of cycles Characteristics joint distribution of (C2,...,C„)

Formulae n! Dn n O - c , !) ;= 2

References n

Jci -

»

n

(Harris, 1960)

T a b l e 3. Generating functions for probabilities or expected values of characteristics of random parmutations with no fixed points Characteristic

Coefficient of

(ci

rCl1 . • • xrnc*

Ka

Cn)

tmXn

Function

References

(Harris, 1960) n!

e~tx

D„ (1 - z)>

(Harris, 1960)

History of Theory of Random

Mappings

11

APPENDIX 2.1 RANDOM MAPPINGS T a b l e 1. Components Characteristics

Formulae

P{*«=j}

References (Katz, 1955)

r

¡1=)

kk'

L

•••kk'

t;>0, £ * ; = » I

n

(Harris, 1960)

* ¿.Ai©' ^ n

E{*a}

(Rubin 1954)

(n — I

n! E , (n - m)! m=l '

Sitgreaves,

(Kruskal, 1954; Lenard, 1964; Riordan, 1962)

mnm

v

and

~ i ( l o g 2 n + 7 ) + o(l) (n-1)! (n-j)!(j-l)!

j"(n-j)"-i

(Rubin 1954)

and

Sitgreaves,

fru-'y-j' asymptotic density of

P { a is connected} = P { t f a = 1} = P { * . ( « ) = n}

^(1 - t ) ~ 1 / 2 ,

nn

2—é

m=0

(Lenard, 1964; Rubin and Sitgreaves, 1954)

0< t < 1

m\

tr \ 1/2 \2n)

(Harris, 1960; Lenard, 1964; Riordan, 1962)

^EinpHF). So > 1, ki >0,

n—1 S0 + J2k