Heyde characterization theorem for some classes of locally compact Abelian groups

TL;DR

This paper extends Heyde's theorem to certain classes of locally compact Abelian groups, using harmonic analysis to characterize when symmetry conditions imply Gaussianity of random variables.

Contribution

It generalizes Heyde's theorem to totally disconnected groups and groups of the form R^n times a totally disconnected group, expanding the theorem's applicability.

Findings

Extended Heyde's theorem to totally disconnected groups.

Proved the theorem for groups of the form R^n x G.

Used harmonic analysis to solve functional equations on character groups.

Abstract

Let $L_1$ and $L_2$ be linear forms of real-valued independent random variables. By Heyde's theorem, if the conditional distribution of $L_2$ given $L_1$ is symmetric, then the random variables are Gaussian. A number of papers are devoted to generalisation of Heyde's theorem to the case, where independent random variables take values in a locally compact Abelian group $X$. The article continues these studies. We consider the case, where $X$ is either a totally disconnected group or is of the form $\mathbb{R}^n\times G$, where $G$ is a totally disconnected group consisting of compact elements. The proof is based on the study of solutions of the Heyde functional equation on the character group of the original group. In so doing, we use methods of abstract harmonic analysis.