# On a group analogue of the Heyde theorem

## TL;DR

This paper extends Heyde's theorem, which characterizes Gaussian distributions via symmetry conditions, to the setting of locally compact Abelian groups with integer coefficients in linear forms.

## Contribution

It introduces a group-theoretic analogue of Heyde's theorem, broadening the understanding of distribution characterization in more general algebraic structures.

## Key findings

- Established a characterization of Gaussian distributions on locally compact Abelian groups.
- Demonstrated the conditions under which symmetry of conditional distributions implies Gaussianity.
- Extended classical results to a more general algebraic setting.

## Abstract

Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analog of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.