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

TL;DR

This paper extends Heyde's theorem to discrete torsion Abelian groups with cyclic p-components, characterizing distributions via symmetry of conditional linear forms without restrictions on coefficients or characteristic functions.

Contribution

It provides an analogue of Heyde's theorem for discrete torsion Abelian groups with cyclic p-components, using harmonic analysis and functional equations.

Findings

Characterization of distributions on specific Abelian groups.

No restrictions on coefficients or characteristic functions.

Proof based on harmonic analysis and functional equations.

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 $X$.