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Functional Equations and Characterization Problems on Locally Compact Abelian Groups
Gennadiy Feldman, Institute for Low Temperature Physics and Engineering, Kharkov, Ukraine
A publication of the European Mathematical Society.
cover
EMS Tracts in Mathematics
2008; 268 pp; hardcover
Volume: 5
ISBN-10: 3-03719-045-0
ISBN-13: 978-3-03719-045-6
List Price: US$78
Member Price: US$62.40
Order Code: EMSTM/5
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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.

Table of Contents

  • Preliminaries
  • Gaussian distributions on locally compact Abelian groups
  • The Kac-Bernstein theorem for locally compact Abelian groups
  • The Skitovich-Darmois theorem for locally compact Abelian groups (the characteristic functions of random variables do not vanish)
  • The Skitovich-Darmois theorem for locally compact Abelian groups (the general case)
  • The Heyde theorem for locally compact Abelian groups
  • Appendix. The Kac-Bernstein and Skitovich-Darmois functional equations on locally compact Abelian groups
  • Comments and unsolved problems
  • Bibliography
  • Symbol index
  • Subject index
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