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An Analogue of Heyde's Theorem for Discrete Torsion Abelian Groups with Cyclic $p$-Components

Gennadiy Feldman

Abstract

According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of $n$ independent random variables given another. In the article, we prove an analogue of this theorem for two independent random variables taking values in a discrete torsion Abelian group $X$ with cyclic $p$-components. In doing so, we do not impose any restrictions on coefficients of the linear forms and the characteristic functions of random variables. The proof uses methods of abstract harmonic analysis and is based on the solution some functional equation on the character group of the group $X$.

An Analogue of Heyde's Theorem for Discrete Torsion Abelian Groups with Cyclic $p$-Components

Abstract

According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the article, we prove an analogue of this theorem for two independent random variables taking values in a discrete torsion Abelian group with cyclic -components. In doing so, we do not impose any restrictions on coefficients of the linear forms and the characteristic functions of random variables. The proof uses methods of abstract harmonic analysis and is based on the solution some functional equation on the character group of the group .
Paper Structure (3 sections, 13 theorems, 93 equations)

This paper contains 3 sections, 13 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Heyde theorem for the direct product of the group of real numbers and a discrete torsion Abelian group with cyclic $p$-components

Key Result

Theorem 2.1

Let $X$ be a discrete torsion Abelian group with cyclic $p$-components containing no elements of order $2$. Let $\alpha$ be an automorphism of the group $X$. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ and distributions $\mu_1$ and $\mu_2$. Assume that the conditional

Theorems & Definitions (16)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3: Rima, see also Febooknew
  • Lemma 2.4: Febooknew
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Corollary 2.8
  • Corollary 2.9
  • proof
  • ...and 6 more