On Heyde's theorem for locally compact Abelian groups containing elements of order 2

TL;DR

This paper extends Heyde's theorem to certain locally compact Abelian groups with elements of order 2, showing that non-Gaussian distributions can also be characterized by symmetry conditions.

Contribution

It demonstrates that in groups containing elements of order 2, a broad class of distributions, including non-Gaussian ones, can be characterized similarly to Gaussian distributions via symmetry conditions.

Findings

Characterization of distributions on groups with elements of order 2

Extension of Heyde's theorem beyond Gaussian distributions

Use of topological automorphisms as coefficients in linear forms

Abstract

According to the well-known Heyde theorem the class of Gaussian distributions on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this can lead to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form given the other. In so doing coefficients of linear forms are topological automorphisms of X.