Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups

✍ Scribed by G. M. Feldman

• Publisher: American Mathematical Society
• Year: 1993
• Tongue: English
• Leaves: 234
• Series: Translations of Mathematical Monographs, Vol. 116
• Category: Library

✦ Synopsis

This book studies the problem of the decomposition of a given random variable into a sum of independent random variables (components). Starting from the famous Cramér theorem, which says that all components of a normal random variable are also normal random variables, the central feature of the book is Fel'dman's use of powerful analytical techniques. In the algebraic case, one cannot directly use analytic methods because of the absence of a natural analytic structure on the dual group, which is the domain of characteristic functions. Nevertheless, the methods developed in this book allow one to apply analytic techniques in the algebraic setting. The first part of the book presents results on the arithmetic of probability distributions of random variables with values in a locally compact abelian group. The second part studies problems of characterization of a Gaussian distribution of a locally compact abelian group by the independence or identical distribution of its linear statistics.

Readership: Specialists in probability theory, mathematical statistics and functional analysis.

✦ Table of Contents

Cover

Translations of Mathematical Monographs 116

Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups

Copyright (c)1993 by the American Mathematical Society
ISBN 0-8218-4593-4
QA180.F4513 1993 512'.2-dc2O
LCCN 92-45025 CIP

Contents

Introduction

CHAPTER I Auxiliary Results
§1. Results on duality theory and on the structure of locally compact abelian groups
§2. Results on probability theory
§3. Results on function theory and on analytic properties of characteristic functions

CHAPTER II Arithmetic of Distributions
§4. Group analogs of the Khinchin factorization theorems
§5. Gaussian distribution
§6. Decomposition of a generalized Poisson distribution
§7. Group analogs of Linnik's theorems
§8. General theorems on distributions of class Io

CHAPTER III Characterization Problems
§9. Bernstein's characterization of Gaussian distribution
§10. Characterization of Gaussian distribution by independence of linear statistics
§11. Characterization of Gaussian distribution by identical distribution of a monomial and a linear form

APPENDIX 1 Group Analogs of the Marcinkiewicz Theorem and the Lukacs Theorem

APPENDIX 2 On Decomposition Stability of Distributions

APPENDIX 3 Structure of Infinitely Divisible Poisson Distributions

APPENDIX 4 On Distributions with Mutually Singular Powers
Unsolved problems

Comments
Section 4
Section 5
Section 6
Section 7
Section 8, Section 9, Section 10
Section 11, Appendix 1, Appendix 2, Appendix 3, Appendix 4

References

Notation

Subject Index

Author Index

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