Heyde theorem for locally compact Abelian groups containing no subgroups topologically isomorphic to the 2-dimensional torus

TL;DR

This paper extends the Heyde theorem to certain locally compact Abelian groups, characterizing Gaussian distributions via symmetry conditions, and identifies the limitations of this extension on the 2-dimensional torus.

Contribution

It proves a Heyde-type characterization theorem for groups without 2-torus subgroups, introducing conditions under which distributions are Gaussian convolutions and distributions supported on a specific subgroup.

Findings

The theorem holds for groups with no 2-torus subgroups.

Distributions are Gaussian convolutions and supported on a subgroup generated by elements of order 2.

The theorem does not hold for the 2-dimensional torus.

Abstract

We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let $X$ be a second countable locally compact Abelian group containing no subgroups topologically isomorphic to the 2-dimensional torus. Let $G$ be the subgroup of $X$ generated by all elements of $X$ of order $2$ and let $\alpha$ be a topological automorphism of the group $X$ such that ${\rm Ker}(I+\alpha)=\{0\}$. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ and distributions $\mu_1$ and $\mu_2$ with nonvanishing characteristic functions. If the conditional distribution of the linear form $L_2 = \xi_1 + \alpha\xi_2$ given $L_1 = \xi_1 +\xi_2$ is symmetric, then $\mu_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in $G$. We also prove that this theorem is false if $X$ is the 2-dimensional torus.