# Functional Equations and Characterization Problems on Locally Compact Abelian Groups

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*Gennadiy Feldman*

A publication of the European Mathematical Society

This book deals with the characterization of probability
distributions. It is well known that both the sum and the difference of two
Gaussian independent random variables with equal variance are independent as
well. The converse statement was proved independently by M. Kac and S. N.
Bernstein. This result is a famous example of a characterization theorem. In
general, characterization problems in mathematical statistics are statements in
which the description of possible distributions of random variables follows
from properties of some functions in these variables.

In recent years, a great deal of attention has been focused upon
generalizing the classical characterization theorems to random variables with
values in various algebraic structures such as locally compact Abelian groups,
Lie groups, quantum groups, or symmetric spaces. The present book is aimed at
the generalization of some well-known characterization theorems to the case of
independent random variables taking values in a locally compact Abelian group
\(X\). The main attention is paid to the characterization of the
Gaussian and the idempotent distribution (group analogs of the Kac–Bernstein, Skitovich–Darmois, and Heyde theorems).
The solution of the corresponding problems is reduced to the solution of
some functional equations in the class of continuous positive definite
functions defined on the character group of \(X\). Group analogs of the
Cramér and Marcinkiewicz theorems are also studied.

The author is an expert in algebraic probability theory. His comprehensive
and self-contained monograph is addressed to mathematicians working in
probability theory on algebraic structures, abstract harmonic analysis, and
functional equations. The book concludes with comments and unsolved problems
that provide further stimulation for future research in the theory.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students interested in probability and analysis.