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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups
G. M. Fel'dman
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Translations of Mathematical Monographs
1993; 223 pp; hardcover
Volume: 116
ISBN-10: 0-8218-4593-4
ISBN-13: 978-0-8218-4593-6
List Price: US$96
Member Price: US$76.80
Order Code: MMONO/116
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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.

Reviews

"There is no question that it was a wise decision by the editors of the Mathematical Monographs Translations series to accept this piece of work for publication and make it accessible to a broad community of mathematicians working in structural probability theory. Moreover, the handy monograph of a few more than 200 pages advertises a most interesting aspect of probability theory to all analysts who want to see abstract harmonic analysis at work."

-- Bulletin of the AMS

Table of Contents

  • Auxiliary results
  • Arithmetic of distributions
  • Characterization problems
  • 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
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