Contributions to Probability. A Collection of Papers Dedicated to Eugene Lukacs

312 Pages·1981·19.441 MB·English

by  J. Gani, Vijay K. Rohatgi

EUGENE LUKACS
Contributions to Probability
A Collection of Papers Dedicated to Eugene Lukacs
Edited by
J. GANI
Division of Mathematics and Statistics
CSIRO
Canberra City, Australia
V. K. ROHATGI
Department of Mathematics and Statistics
Bowling Green State University
Bowling Green, Ohio
(^) 11 9 8
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Library of Congress Cataloging in Publication Data
Main entry under title:
Contributions to probability.
Includes bibliographical references
1. Probabilities—Addresses, essays, lectures.
2. Lukacs, Eugene. I. Lukacs, Eugene. II. Gani,
Joseph Mark. III. Rohatgi, V. K. , Date
QA273.18.C68 519.2 80-26103
ISBN 0-12-274460-8
PRINTED IN THE UNITED STATES OF AMERICA
81 82 83 84 9 8 7 6 5 4 3 2 1
List of Contributors
Numbers in parentheses indicate the pages on which the authors' contributions begin.
J. ACZEL (191) Faculty of Mathematics, University of Waterloo, Waterloo, On­
tario, Canada N2L 3G1
HARALD BERGSTROM (65) Chalmers Institute of Technology and University of
Goteborg, S-402220 Goteborg, Sweden
P. L. BUTZER (77) Lehrstuhl A fur Mathematik, Aachen University of Technol­
ogy, 51 Aachen, Federal Republic of Germany
M. CSORGO (143) Department of Mathematics, Carleton University, Ottawa,
Canada K1S 5B6
SANDOR CSORGO (215) Bolyai Institute, Szeged University, H-6720 Szeged,
Aradi vertanuk tere 1 Hungary
ROGER CUPPENS (101) U.E.R. de Mathematique, Universite Paul Sabatier,
Toulouse, France
M. DEISTLER (231) Institut fur Okonometrie, Technische Hochschule Wien,
Wien, Austria
DANIEL DUGUE (247) Institut de Statistique des Universites de Paris, Paris,
France
PAUL ERDOS (3) Mathematical Institute of the Hungarian Academy of Sciences,
Budapest v, Realtanoda u. 13-15, Hungary
xiii
xiv List of Contributors
CARL-GUSTAV ESSEEN (115) Department of Mathematics, Uppsala University,
S-752 38 Uppsala, Sweden
J. GANI (151) Division of Mathematics and Statistics, CSIRO, P.O. Box 1965,
Canberra City, A.C.T. 2601 Australia
B. GYIRES (255) Department of Mathematics, Kossuth Lajos University,
H-4010 Debrecen, Hungary
L. HAHN (77) Lehrstuhl A fur Mathematik, Aachen University of Technology,
51 Aachen, Federal Republic of Germany
B. JESIAK (29) Karl-Marx University, Leipzig, German Democratic Republic
D. A. KAPPOS (11) Lykalbttou Street 29, Athens 135 Greece
TATSUO KAWATA (157) Department of Mathematics, Keio University,
Yokohama, Japan
R. G. LAHA (123) Department of Mathematics and Statistics, Bowling Green
State University, Bowling Green, Ohio 43403
MICHEL METIVIER (173) Ecole Poly technique, Centre de Mathematiques
Appliquees, Palaiseau, France
R. MORTON (151) Division of Mathematics and Statistics, CSIRO, P.O. Box
1965, Canberra City, A.C.T. 2601, Australia
MELVYN B. NATHANSON (3) Department of Mathematics, Southern Illinois Uni­
versity, Carbondale, Illinois 62901
I. OLKIN (201) Department of Statistics, Stanford University, Stanford, Califor­
nia 94305
MADAN L. PURI (267) Department of Mathematics, Indiana University,
Bloomington, Indiana 47401
P. REVESZ (143) Mathematical Institute of the Hungarian Academy of Sciences,
Budapest v, Realtanoda u. 13-15, Hungary
M. TH. ROECKERATH (77) Lehrstuhl A fur Mathematik, Aachen University of
Technology, 51 Aachen, Federal Republic of Germany
V. K. ROHATGI (123) Department of Mathematics and Statistics, Bowling Green
State University, Bowling Green, Ohio 43403
M. ROSENBLATT (139) Department of Mathematics, University of California,
San Diego, California 92093 and Australian National University,
Canberra, Australia
H.-J. ROSSBERG (29) Karl-Marx University, Leipzig, German Democratic Re­
public
List of Contributors XV
L. A. SHEPP (201) Bell Laboratories, Murray Hill, New Jersey 07974
G. SIEGEL (29) Karl-Marx University, Leipzig, German Democratic Republic
LAJOS TAKACS (49) Department of Mathematics and Statistics, Case Western
Reserve University, Cleveland, Ohio 44106
G. TINTNER (231) Institut fur Okonometrie, Technische Hochschule Wien,
Wien, Austria
LANH T. TRAN (267) Department of Mathematics, Indiana University,
Bloomington, Indiana 47401
ISTVAN VINCZE (207) Mathematical Institute of the Hungarian Academy of Sci­
ences, Budapest v, Realtanoda u. 13-15, Hungary
V. M. ZOLOTAREV (283) Steklov Mathematical Institute of the Academy of Sci­
ences, Vavilov Str. 42, Moscow 117966, USSR
Preface
Professor Eugene Lukacs celebrates his seventy-fifth birthday on August 14,
1981. Eugene has served the statistical community for over thirty years through
his research, monographs and textbooks, and many other activities. These are
documented elsewhere in this volume. His colleagues and friends welcome this
opportunity to honor him with this collection of papers dedicated to him.
The papers included here reflect Eugene's broad range of research interests.
The contributions have been divided into two major parts, Part I: Probability and
Part II: Applications of Probability. Part I consists of papers in probability
theory, limit theorems, and stochastic processes, while Part II includes informa­
tion theory and statistical theory.
A large number of individuals have helped make this volume possible. We
thank the authors for their contributions. The following colleagues were kind
enough to help with the refereeing process.
Robert Adler E. J. Hannan
Josef Blass William Hudson
Harry Cohn John Kellermeier
Roger Cuppens Fred Mc Morris
Daryl Daley F. Moritz
Joseph Diestel Thomas O'Connor
Humphrey Fong Ramesh Srivastva
Istvan Vincze
xvii
xviii Preface
We are greatly indebted to them for their kind assistance. We should also like to
express our gratitude to Mary Chambers, Cynthia Patterson, and Linda Shellen-
berger for their help in typing. The work of the second editor was supported by
the National Science Foundation under Grant No. MCS 78-01338.
Eugene Lukacs
This volume celebrates the 75th birthday of Eugene Lukacs, mathematician,
teacher, and research worker in probability and mathematical statistics. Eugene's
contributions defy easy classification, but a brief account of his life and work
may help the reader to place the man and his work in perspective.
Eugene Lukacs was born in Szombathely, Hungary, on August 14, 1906. His
elementary and secondary education was completed in Vienna. After graduating
from the Realgymnasium he studied mathematics, physics, and descriptive
geometry at the University and the Institute of Technology in Vienna. In 1929 he
was awarded his Lehramtsprufung or teacher's certificate in mathematics and
descriptive geometry. He then taught for some time at a secondary school and at
an institute for adult education. In 1930 he took the degree of Doctor of
Philosophy majoring in mathematics, with a minor in physics, and in 1931 qual­
ified as an actuary. He married Elizabeth Weisz in 1935; she has been a constant
support and source of inspiration to him ever since. Eugene pursued a successful
career in Vienna as an actuary until 1938 when Hitler invaded Austria. His
future as a professional of Jewish faith in Vienna was thereby closed, and in 1939
he and Lisl moved to the United States, where he was naturalized in 1945.
Soon after his arrival in the U.S., he was influenced by Abraham Wald and
decided to study mathematical statistics, attending Wald's lectures at Columbia
University, New York. He promptly proceeded to make lasting contributions to
this newly acquired discipline. He taught school in Baltimore during 1940-
1942, was appointed an Assistant Professor at Illinois College, Jacksonville, in
1942, an Associate Professor at Berea College, Kentucky, in 1944, and a full
Professor at Our Lady of Cincinnati College in 1945.
xix
XX Eugene Lukacs
In 1948 he was appointed to head the Statistics Branch of the Assessment
Division, U.S. Naval Ordnance Test Station at China Lake, California, and in
1950 he joined the National Bureau of Standards, Washington, D. C., as Head of
the Statistics Branch.
Since 1951, Eugene had been lecturing part-time at the graduate school of
American University, Washington, D.C., where he continued to teach until
1957. In 1955 he was appointed to a professorship at the Catholic University of
America, Washington, D. C. He served as Director of the Statistical Laboratory
there until 1972, when he moved with a group of colleagues to the Bowling
Green State University, Ohio. At Bowling Green, he remained actively on the
faculty until 1975.
Eugene has had a wide range of visiting appointments in various universities,
including the Sorbonne (1961-1962 and 1966), the Swiss Federal Institute of
Technology, Zurich (1961-1962), the Institute of Technology, Vienna (1965-
1966 and 1970-1971), Hull (1971 and 1976), Sheffield (1974), the Institute for
Analysis, Vienna (1975), the Institute of Statistics, Vienna (1976-1977), and
Erlangen- Niirnberg (1977-1978).
He has also served the mathematical community in various capacities, as an
Editorial Collaborator, Journal of the American Statistical Association (1951-
1955 and 1961 -1963), and as a member of the editorial committee of the Annals
of Mathematical Statistics (1958-1964). He was an Associate Editor of the An­
nals of Mathematical Statistics (1968-1970). He also served on the Council of
the Institute of Mathematical Statistics (1972-1973), participated in the visiting
lectureship program of the Mathematical Association of America, and served
twice as the IMS delegate to the Mathematics Division of the National Research
Council. He has been on the Editorial Board of the Journal of Multivariate
Analysis since 1970, coeditor of the Probability and Mathematical Statistics
series of Academic Press since 1969, and a member of the Translation Commit­
tee of the American Mathematical Society for several years.
He was elected to a fellowship of the Institute of Mathematical Statistics in
1957, the American Association for the Advancement of Science in 1969, and
the American Statistical Association in 1969. He was also elected to membership
in the International Statistical Institute in 1963 and to the Austrian Academy of
Sciences in 1973. He was the first University Professor appointed at Bowling
Green State University in 1973.
His research began in mathematics, more particularly in geometry and nomog-
raphy in 1930. He contributed to the theory of insurance risk over a period of
twenty years starting in 1936. He began his lifelong work in statistics in 1942,
following Wald's advice to study the subject. He has written on a variety of
topics, including characteristic functions, stochastic processes (particularly the
Wiener process), characterization of distributions, statistical inference, analytic
probability theory, stable processes, and the robustness of characterization
theorems; this last continues to be his current research interest. Few workers in