Characterization of Distributions by the Method of Intensively Monotone Operators
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1088 Ashot V. Kakosyan Leo B. Klebanov Joseph A. Melamed
Characterization of Distributions by the Method of Intensively Monotone Operators
Springer-Verlag Berlin Heidelberg New York Tokyo 1984.
Authors
Leo Borisovich Klebanov Leningrad Civil Engineering Institute 2 Krasnoarmeiskaya 4, Leningrad, USSR Ashot Vazrikievich Kakosyan Joseph Aleksandrovich Melamed Department of Probability Theory and Mathematical Statistics Tbilisi Mathematical Institute of the Georgian Academy of Sciences Plekhanov Ave. 150 a TbilisH2, 380012, USSR
Consulting Editor
I.A. Ibragimov Steklov Mathematical Institute Fontanka 27, Leningrad D-111, USSR
AMS Subject Classification (1980): 62ElO, 62H05; 47H05, 47HlO, 47H15,62N05 ISBN 3-540-13857-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13857-9 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
To the memory of our colleague and friend S.T.Th&rttyan
PRE F ACE
The problems of characterization of distributions are today a section of probability theory and mathematical statistics rather substantial in in
size and variety of accumulated facts. There are four monographs
(Kagan, Linnik, Rao (1973);
Galambos, Kotz (1978); Mathai, Pederzoli
(1977); Azlarov, Volodin (1982)) specially devoted to characterization problems. Besides, whole chapters in a number of monographs and a considerable number of papers are devoted to these problems. However, the majority of works in this area are focused not on the methods of proof of appropriate results, but on obtaining concrete characterization theorems. The aim of this monograph is to propose
a general method of proving
characterization theorems and to illustrate its potentialities conformably to some specific results. The method makes substantial use of a new concept
that of an intensively monotone operator. It is found that
uniqueness theorems for functional equations containing intensively monotone operators can be proved easily enough. But obtaining various characterization theorems is often reduced to the proof of uniqueness of solutions of functional equations and as a consequence the method of intensively monotone operators is applicable to problems of characterization of distributions. Of course, by far not all characterization problems are reduced to equations with intensively monotone operators. Problems of such a kind are not considered in this book and therefore it makes no claim to any complete elucidation of characterization problems. On the other ha
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