# Book Details

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Preface | Table of Contents | MARC record | Metadata XML | e-Book PDF (2421 KB)*Gennadiy Feldman (B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences, Kharkov, Ukraine )*

#### Functional Equations and Characterization Problems on Locally Compact Abelian Groups

ISBN print 978-3-03719-045-6, ISBN online 978-3-03719-545-1DOI 10.4171/045

April 2008, 268 pages, hardcover, 17.0 x 24.0 cm.

58.00 Euro

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.

*Keywords: *Locally compact Abelian group, Gaussian distribution, characterization problems of mathematical statistics, functional equation

#### Further Information

Review in Zentralblatt MATH 1156.60003