Probability Theory and Applications

Mathematics and Its Applications

Managing Editor:

M. HAZEWINKEL

Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Editorial Board:

F. CALOGERO, Universita degli Studi di Roma, Italy Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R. M. NN AT, Universite de Paris VII, Paris, France A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands G.-C. ROTA, MJ.T., Cambridge, Mass., U.SA.

Volume 80

Probability Theory and Applications Essays to the Memory of J6zsef Mogyor6di

Editedby'

Janos Galambos Department of Mathematics, Temple University, Philadelphia, Pennsylvania, U.S.A.

and

Imre Katai Computer Center, Eotvos Lorand University, Budapest, Hungary

technical editing by Lasz16 Lakatos

SPRINGER SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data

Probability theory and applications : essays to the memory of Jozsef Mogyorodi / edited by Janos Galambos and Imre Katai.

p. cm. -- (Mathematics and its applications ; v. 80) Includes bibliographical references and indexes. ISBN 978-94-010-5252-8 ISBN 978-94-011-2817-9 (eBook) DOI 10.1007/978-94-011-2817-9 1. Probabilities. 1. Mogyorodi, J. II. Galambos, Janos, 1940-III. Katai, 1. IV. Series: Mathematics and its applications

(Kluwer Academic Publ ishers) : v. 80. QA273.18.P758 1992 519.2--dc20 92-25135

ISBN 978-94-010-5252-8

Printed on acid-free pa per

AII Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 Softcover reprint ofthe hardcover Ist edition 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

SERIES EDITOR'S PREFACE

"Et moi, ... , si j'avait su comment en revenir, je n'y serais point all~.'

lu1esVeme

1be series is divergent; therefore we may be able to do something with it

O. Heaviside

One service mathematics bas rendered the human race. It bas put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded nonsense'~

Eric T. Bell

Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari­ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci­ences.

Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser­vice topology has rendered mathematical physics ... '; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'etre of this series.

This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote

"Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci­ences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experi­mental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large­scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics."

By and large, all this still applies today. It is still true that at first sight mathematics seems rather frag­mented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available.

If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need.aIl kinds of combina­torics, algebra, probability, and so on.

In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra

v

vi

mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines.

There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All three have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wonnhole' paths. There is no telling where all this is leading - fortunately.

Thus the original scope of the series. which for various (sound) reasons now comprises five subseries: white (Japan), yellow (anna), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with:

a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another; inftuences which the results, problems and concepts of one field of enquiIy have, and have had, on the development of another.

The shortest path between two truths in the real

domain passes through the complex domain.

I. Hadamard

La physique lit> IlOUS donne pas seuIement

I'occasion de r6s0udre des prob~ ... elle

IlOUS fait presseDlir Ia solutioa.

H.PoiDcarE

Bussum, 1992

Never Iea4 books, for DO ODe ever mums tbem;

the 0DIy books I have in my IibraIy are books

!bat otbc:r folk: have lcIIlme.

ADatoIe JIJaDce

The function of aD expert is DOt to be~ rigbt

1baD otbcr people, bIIt to be wnmg for more

sophisIicated ~IIS.

David Butler

Michiel Hazewinkel

TABLE OF CONTENTS

Series Editor's Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . v

J6zsef Mogyor6di in Memoriam ............................................ ·.· ix

List of Publications ........................................................... xi

Preface ...................................................................... xv

Random walk processes and their various applications Lajos Takacs .............................................................. 1

Duality of the Burkholder-Davis-Gundy inequality and the generalized Fefferman-Garsia inequality II

N.L. Bassily .............................................................. 33

Martingale Hardy spaces with continuous time F. Weisz ................................................................. 47

Construction of optimal Hankel approximations in the guise of stochastic processes

Gy. Michaletzky .......................................................... 77

Levy's random domains on the plain M. Arata ................................................................. 99

On the infinite divisibility of polynomials in infinitely divisible random variables

V.K. Rohatgi and G.J. Szekely .......................................... 103

Random sample sizes: limit theorems and characterizations J. Galambos ............................................................. 107

Aging solutions of ~ertain renewal type equations A. Kovats and T. Mari .................................................. 125

Extensions of some univariate Bonferroni-type inequalities to multivariate setting

J. Galambos and Min- Young Lee ........................................ 143

Univariate and multivariate Bonferroni-type inequalities: methods for proof and questions of optimality

J. Galambos and Yuan Xu . .............................................. 155

Sharp Bonferroni-type inequalities in explicit forms Masaaki Sibuya ......................................................... 165

viii TABLE OF CONTENTS

Analytical representation of limit distributions for a class of random symmetric polynomials

L. Szeidl and V.M. Zolotarev ............................................ 195

Tail behavior in Wicksell's corpuscle problem H. Dress and R.-D. Reiss ................................................ 205

Universal contractive projections and a.e. convergence F. Schipp ............................................................... 221

Pointwise Bahadur-Kiefer-type theorems (1) P. Deheuvels ............................................................ 235

Laws of small numbers: some applications to conditional curve estimation M. Falk and F. Marohn ................................................. 257

Design of statistical lifetime models by functional equations E. Castillo, A. Fernandez-Cantelli and R. Ruiz-Cobo ..................... 279

Another approach to the ergodic distribution in the MIGII system

L. Lakatos and V. eerie ................................................. 289

A new method in probabilistic number theory K.-H. Indlekofer ........................................................ 299

Distribution of Q-additive functions 1. Ktf.tai ................................................................. 309

Number systems and fractal geometry K.-H. Indlekofer, 1. Ktf.tai and P. Racsko ................................ 319

On sequences of solid type Z. Daroczy, A. Jarai and T. Szn.bo ...................................... 335

Author Index ............................................................... 343

Subject Index ............................................................... 347

JUDef Mogyor6di in Memoriam

1933 -1990

We lost a friend, a scientist and an educator when Professor J6zsef Mogyor6di unexpectedly died during a scientific visit to Holland. He was 57 and he was at the prime of his life: he travelled, lectured, published, but, above all, worked with much devotion and energy to make the Department of Probability Theory and Mathematical Statistics at the L. Eotvos University, which he headed at the time of his death, the prime example of university education of Hungary. Hundreds of secondary school teachers, and several young academics all over the world, remember him with affection not just as a scientist and an extremely good lecturer but also as one of those very few professors who would sit down with students and would converse with them with joy about the arts, the great museums of the world, and history. He was a great ambassador for Hungary: he was fluent in English, French and Russian, and his pleasant personality came across in all of these languages as well as in Hungarian.

Professor Mogyor6di was born in Nagyoroszi, Hungary, in 1933. He received his elementary and secondary education there. He entered the L. Eotvos University, Budapest, in 1952, and studied mathematics. He graduated with distinction in 1957, receiving a degree comparable with a M.Sc. In his Dissertation for this degree, which he wrote under the supervision of Lajos Takacs, Mogyor6di worked on stochastic processes modelling the movements of neutrons in nuclear reactors. His early publications belong to this field as well. These works naturally led him to the study of limit theorems with random sample sizes, a field in which he found his mentor in Alfred Renyi. Mogyor6di's deep interest in both teaching' and research convinced Renyi to move Mogyor6di to the University where he remained until his death. Soon after this move, Mogyor6di defended his Thesis for a degree which is comparable with a Ph.D. in the West. Mter just a few years of teaching, Mogyor6di's work was redirected by his recognition of the need of the University for a faculty in the computer sciences. He retrained himself, and within a short period of time he became the first head of the newly established Department of Computer Sciences. His results were quick: the department soon became a recognized center, and Mogyor6di's ability to lead and to organize earned him respect among the members of the university's governing body. This led to his appointment as the head of the Department of Probability Theory and Mathematical Statistics at Renyi's death in 1970. He cleverly exploited the scientific reputation of the department, built by Renyi, and developed an international advanced

ix

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degree program with the support of UNESCO. At the same time, the university decided on expanding the department for which Mogyor6di was the most suitable person. He hired young talented mathematicians who were interested in both theory and practice, and who understood that undergraduate teaching was a significant part of their duties. Mogyor6di's own devotion to teaching was an excellent example of his leadership qualities. Mogyor6di' s return to research proved even more fruitful than before. His work on martingales in abstract spaces, in particular his study of maximal inequalities and their duals, placed him among the leaders in the world.

His connection with UNESCO was not accidental to his chairmanship at the University. He established contacts with third world scientists when in the late 60s he participated in postgraduate education in Algeria and in Mali. Much closer home, his organizational skills were partly responsible for the initiation of the very popular sequence of Pannonian Symposia on Mathematical Statistics.

We join his family in their sorrow, in particular, his wife Klan. and their two sons, J6zsef, Jr. and Zoltan. They became a part of the mathematical community of Budapest as their home was always open to mathematicians. We also join the family in keeping the memory of J6ska, as he was known to all of us, alive. To this end we present this volume.

Janos Galambos Imre Katai

LIST OF PUBLICATIONS

Jozsef Mogyorodi 1933-1990

1. Az atommagreaktorokban vegbemeno neutronlassitas - folyamattal kapcsolatos val6sziniisegszamitasi problemakr61, MTA Mat.Kut.lnt.Kozl., 1(1956),337-348. (~emeth <iezaval)

2. Problems of neutron movements in atomic reactors according to the theory of probability, Proceedings of the Second UN International Conference on the Peaceful Uses of Atomic Energy, Geneve, 1958, voLl6, pp. 406-408. P /1711, Hungary

3. ~eutronok atommagreaktorokban val6 mozgasanak val6sziniisegszamitasi prob­lemru, MTA Mat.Kut.Int.Kozl., 3(1958), 237-250 o.

4. On limiting distributions for the sums of a random number of independent random variables, MTA Mat.Kut.lnt.Kozl., 6(1961), 365-371 o.

5. A central limit theorem for the sums of a random number of independent random variables, MTA Mat.Kut.lnt.Kozl., 7(1962),409-424 o.

6. A neutronlassitas folyamataban felIepo iitk6zesszam atlagar61 es sz6rasar61, MTA Mat.Kut.lnt.Kozl., 9(1964), 733-741 o.

7. On a consequence of a mixing theorem of A.Renyi, MTA Mat. Kut. Int. Kozi. , 9(1964), pp. 263-268.

8. On the law of large numbers for the sum of a random number of independent random variables, Annales Univ.Sci.Bud.Sect.Math., 8(1965), pp. 33-38.

9. A remark on stable sequences of random variables and a limit distribu­tion theorem for a random sum of independent random variables, Acta Math.Acad.Sci. Hungar. , 17(1966), pp. 401-409.

10. Some remarks concerning the stable sequences of random variables, Publ. Math. Debrecen, 14(1967), pp. 227-238. (with I.Katai)

11. VeIetlen elemszamu rendezett minta maximaIis tagjanak hatareloszlasar61, MTA III. Oszt.Kozl., 17(1967), 75-83 o.

12. A Kolmogorov egyenlotlensegr81, MTA III.Oszt.Kozl., 17(1967), 113-121 o .

.xi

xii J6ZSEF MOGYOR6DI

13. A central limit theorem for the sums of a random number of random variables, Annales Univ.Sci.Bud.Sed.Math., 10(1967), pp. 171-182.

14. Va16szinusegi valtoz6k veletlen tagszamu osszegeinek hatareloszlasar61, kan­didatusi ertekezes, Budapest, 1967.

15. Veletlen id8kozonkent mukod8 kesziiIek elettartamanak hatareloszlasarol, MTA III.Oszt.Kozl., 17(1967),421-433 o. (Tomko Jozseffel)

16. Limit distributions for sequences of random variables with random indices, Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes, pp. 463-470, Academia, Prague, 1967.

17. A valosztnusegszamttas elemei, MTESZ Geofizikai Tarsulat, Budapest, 1968.

18. On the distribution of digits, Publ.Math.Debrecen, 15(1968), pp. 57-68. (with I. Katai)

19. On mixing sequences of (j - algebras, Acta Sci.Math.Szeged, 29(1968), pp. 187-197.

20. On the density of certain sequences of integers, Publ.Math.Debrecen, 16(1969), pp. 17-23. (with I.Katai)

21. On the number of solutions of a diophantine system, Acta Math. Acad. Sci. Hun­gar., 20(1969), pp. 185-191. (with I.Katai)

22. Rekurrens folyamatok ritkitasarol, MTA III. Oszt.Kozl., 19(1969), 25-31 o.

23. 06 O~OH 3a,lJ;aQe B.B.rHe,lJ;eHKo, Acta Sci.Math.Szeged, 30(1969), CTp. 241-245. (coBMecTHo c A.IT. BmuIJIHCOM)

24. 06 aCHMllTOTHl.IeCKHX pa3JIO:>KeHHHX n-KpaTHbIX CBepTOK k-MepHbIx pacnpe­,lJ;eJIeHHH , Lietuvos Mat. Rinkinys, 10(1970), CTp. 433-443. (coBMecTHo C A.IT. BmuIJIHCOM)

25. Some remarks on the rarefaction of the renewal processes, Lietuvos Mat. Rinkinys, 11(1971), pp. 303-315.

26. A remark on limiting distributions for sums of a random number of independent random variables, Revue Roumaine de Mathematiques Pures et Appliquees, 16(1971), pp. 551-557.

27. Veletlen pontfolyamatok ritkitasarol, MTA III.Oszt.Kozl., 20(1971), 85-95 o. (Szantai Tamabsal)

28. On the rarefaction of the renewal processes, Selected translations in mathemat­ical statistics and probability, vol.10. Published for Institute of Mathematical Statistics by AMS, Providence, R.I., 1972.

LIST OF PUBLICATIONS xiii

29. Egy ritkitasi eljarasr61, MTA III. Oszt.Kozl., 20(1972), 407-418 o.

30. Val6szlnusegszam{tasi feladatgyujtemeny, Tankonyvkiad6, Budapest, 1972. (Bognar Janosneval, Prekopa Andnissal, Renyi Alfreddel es Szasz Domokossal)

31. On the rarefaction of renewal processes 1., Studia Sci.Math.Hungar., 7(1972}, pp. 285-291.

32. On the rarefaction of renewal processes 11., Studia Sci.Math.Hungar., 7(1972), pp. 293-305.

33. Correction to my paper "Some remarks on the rarefaction of the renewal processes" (Letter to the editor), Lietuvos Mat.Rinkinys, 13(1973), 189.

34. On the rarefaction of renewal processes IlL, Studia Sci.Math.Hungar., 8(1973), pp. 21-28.

35. On the rarefaction of renewal processes IV., Studia Sci.Math.Hungar., 8(1973), pp. 29-38.

36. On the rarefaction of renewal processes V., Studia Sci.Math.Hungar., 8(1973), pp. 193-198.

37. On the rarefaction of renewal processes Vr., Studia Sci.Math.Hungar., 8(1973), pp. 199-205.

38. ValOszlnusegszamztasi feladatgyujtemeny, masodik javitott kiadas, Tankonyvki­ad6, Budapest, 1975. (Bognar Janosneval, Prekopa Andrassal, Renyi Alfreddel es Szasz Domokossal)

39. Some inequalities for the maximum of partial sums of random variables, Math.Nachr., 70(1976), pp. 71-85.

40. Sur quelques inegalites de la theorie des probabilites, Annales Univ.Sci.Bud.Sect. Math., 19(1976), pp. 143-157. (avec M.Guisse)

41. On an inequality of H.P.Rosenthal, Periodica Math.Hungar., 8(1977), 3-4, pp. 275-279.

42. Remark on a theorem of J.Neveu, Annales Univ. Sci. Bud. Sect. Math., 21(1978), pp. 77-81.

43. A convergence theorem and a strong law of large numbers for martingales, Math.Nachr., 84(1978}, pp. 311-318. (with A..Somogyi)

44. Val6szznusegszamitas, egyetemi jegyzet programoz6 matematikus hallgat6k reszere, Tankonyvkiad6, Budapest, 1978. (Bar6ti Gyorggyel, Bognar Janos­neval es Fejes T6th Gaborral)

45. On an inequality of Marcinkiewicz and Zygmund, Publ. Math. Debrecen, 26 (1979), pp. 267-274.

xiv J6ZSEF MOGYOR6DI

46. Martingalok Orlicz es a bellJle szarmaztaiott terekben, doktori ertekezes, Bu­dapest, 1980.

47. Duality of the maximal inequality for nonnegative submartingales and of the convexity inequality of Burkholder, Pannonian Symposium on Mathematical Statistics, Bad Tatzmannsdorf, 1979, Lecture Notes in Statistics, Springer, 8, 1981, pp. 169-173.

48. Maximal inequalities, convexity inequality and their duality 1., Analysis Math., 7(1981), pp. 131-140.

49. Maximal inequalities, convexity inequality and their duality 11., Analysis Math., 7(1981), pp. 185-197.

50. Decomposition of Doob of nonnegative submartingales, Annales Univ. Sci. Bud. Sect.Math., 24(1981), pp. 255-264.

51. On a concave function inequality for martingales, Annales Univ.Sci.Bud.Sect. Math., 24(1981), pp. 265-271.

52. On the generalization of the Fefferman-Garsia inequality, 3rd Working Confer­ence ofIFIP-WG. Stochastic Differential Systems, Lecture Notes in Control and Information Sciences, Springer, 36, 1981, pp. 85-97. (with S.Ishak)

53. On a problem of R.F.Gundy, Annales Univ. Sci. Bud. Sect. Math., 25(1982), pp. 273-278.

54. Valoszinusegszamitasi feladatgyujtemeny, harmadik javltott kiadas, Tankonyv­kiad6, Budapest, 1982. (Bognar Janosneval, Prekopa Andrassal, Renyi Alfreddal es Szasz Domokossal)

55. Valoszinusegszamitas I., egyetemi jegyzet matematikus 1. szakos hallgat6k reszere, Tankonyvkiad6, Budapest, 1982. (Somogyi Arpaddal)

56. Valoszinusegszamitas II., egyetemi jegyzet matematikus 1. szakos hallgat6k reszere, Tankonyvkiad6, Budapest, 1982. (Somogyi Arpaddal)

57. On the P¢-spaces and the generalization of Herz's and Fefi"erman's inequalities 1., Studia Sci.Math.Hungar., 17(1982), pp. 229-234. (with S.Ishak)

58. Maximal inequalities and the decomposition of Doob for nonnegative super­martingales, Annales Univ.Sci.Bud.Sect.Math., 26(1983), pp. 175-183.

59. Linearfunctionals on Hardy spaces, Annales Univ.Sci.Bud.Sect.Math., 26(1983), pp. 161-174.

60. Necessary and sufficient condition for the maximal inequality of convex Young functions, Acta Sci.Math.Szeged, 45(1983), pp. 325-332. (with T.M6ri)

LIST OF PUBLICATIONS xv

61. On the p",-spaces and the generalization of Herz's and Fefferman's inequalities 11., Studia Sci.Math.Hungar., 18(1983), pp. 205-210. (with S.Ishak)

62. On the p",-spaces and the generalization of Herz's and Fefferman's inequalities IlL, Studia Sci.Math.Hungar., 18(1983), pp. 211-219. (with S.Ishak)

63. On some problems for predictable random variables, Statistics and Probability, Proceedings of the Third Pannonian Symposium on Mathematical Statistics, Visegrad, 1982, Akademiai Kiad6 - Reidel Publ. Comp., 1984, pp. 221-230.

64. Statistics and Probability, Proceedings of the Third Pannonian Symposium on Mathematical Statistics, Visegrad, 1982, Akademiai Kiad6- Reidel Publ.Comp., 1984. (editor)

65. 06 o~oii np06JIeMe raH~, c6. "MemoOb1. peme'H,'UJ{ 3aoa"t MameMamu­"teC1CoiJ: gJU3U1CU u ux npOZpaMM'H,Oe 06ecne"te'H,ue" no~ pe~. JI.H. KOpOJIeBa 11: M.M. XanaeBa, H3~. Mry, MocKBa, 1984, CTp. 34-37.

66. On the BMO-spaces with general Young function, Annales Univ.Sci.Bud.Sect. Math., 27(1985), pp. 215-227. (with N.L.Bassily)

67. On the K",-spaces with general Young function, Annales Univ. Sci. Bud. Sect. Math., 27(1985), pp. 205-214. (with N.L.Bassily)

68. Proceedings of the Fourth Pannonian Symposium on Mathematical Statistics, Vol. A. Probability and Statistical Decision Theory, Bad Tatzmannsdorf, 1983, Akademiai Kiad6 - Reidel Publ.Comp., 1985. (editor)

69. On the representation of L"'-mean oscillating random variables, Annales Univ.Sci. Bud. Sect.Math. , 30(1987), pp. 213-222. (with N.L.Bassily and S.Ishak)

70. On Wald-type inequalities, Annales Univ.Sci.Bud.Sect.Comput., 8(1987), pp. 5-24. (with N.L.Bassily and S.Ishak)

71. Probability theory and mathematical statistics with applications, Proceedings of the Fifth Pannonian Symposium on Mathematical Statistics, Visegrad, 1985, Akademiai Kiad6 - Reidel Publ.Comp., 1988. (editor)

72. On stable and mixing sequences of O"-fields, Annales Univ. Sci. Bud. Sect. Comput., 11(1991), 11-12. (with N.L.Bassily and S.Ishak)

73. Remarks on stopped random walks. To appear. (with N.L.Bassily and S.Ishak)

74. Exercises in discrete parameter martingale theory, manuscript (with co-authors)

75. Martingale maximum inequalities in Orlicz spaces, manuscript.

76. Valosz{nuseg (tankonyv), kezirat.

PREFACE

This volume is a collection of articles dedicated to the memory of Professor Jozsef Mogyorodi of L. Eotvos University, Budapest, Hungary. While the con­tributions to this volume represent diverse fields of probability theory and its applications, the volume became a remarkably coherent one, covering most fields investigated by Mogyorodi himself. By limitation of space, however, we could not make each topic represented here as widely covered as we would have liked, so we did not split the book into chapters. Yet, a system of ordering the contributions was possible, in which, apart from the first three articles, the scientific content provides the basis. As a matter of fact, we opened the volume with a contribution by Lajos Takacs who was the supervisor of the first major project carried out by the late Professor J. Mogyorodi. Then come contributions by two of his pupils, followed by those who worked with him, one time or another, in the Department of Probability and Mathematical Statistics of L. Eotvos University. These latter papers as well as those which follow are grouped by subject matter: starting with classical problems and going through more and more advanced topics. Finally, we end the volume with a variety of applications, which spread from engineering applications to probabilistic number theory. The last two papers of the volume are only marginally in probabilistic number theory. However, they do lay down the foundations for further metric results on representations of real numbers. This short description, together with the Subject Index, will guide the reader on the subject matter of the book.

All contributions have been reviewed. We wish to thank all those who so willingly read the manuscripts and sent their comments to us. But above all we are indebted to the authors of this volume for their high quality articles and for their cooperations in meeting deadlines.

On the technical side, unmeasurable assistance was given by Dr. Laszlo Lakatos as our Technical Editor and by Dr. Margit Kovacs as an advisor and supervisor for our wordprocessor specialists, Ms. Krisztina Bajerle and Ms. Ildiko Furka. All manuscripts were retyped on wordprocessor, and Ms. Bajerle and Ms. Furka did a remarkable job. We thank them both for their professionalism and patience. Finally, we are indebted to Dr. D.J. Larner, Publisher, Science and Technology Division, Kluwer Academic Publishers, for his kind support for this project, and to Ms. Margaret Deignan, Assistant to the Publisher, for her assistance and guidance in the publication process.

Janos Galambos (Philadelphia, PA) Imre Katai (Budapest, Hungary)

xvii