Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups

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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups is written by Gennadiĭ Mikhaĭlovich Felʹdman and published by American Mathematical Soc.. It's available with International Standard Book Number or ISBN identification 0821845934 (ISBN 10) and 9780821845936 (ISBN 13).

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 Cramer theorem, which says that all components of a normal random variable are also normal random variables, the central feature of the book is Feldman'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.