The Heyde theorem on a group $\mathbb{R}^n\times D$, where $D$ is a discrete Abelian group

Abstract

Heyde proved that a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear statistic given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables $\xi_1$, $\xi_2$ with values in a group $X=\mathbb{R}^n\times D$, where $D$ is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear statistic $L_2 = \xi_1 + \delta\xi_2$ given $L_1 = \xi_1 + \xi_2$, where $\delta$ is a topological automorphism of $X$ such that ${Ker}(I+\delta)=\{0\}$.