Heyde Theorem on Locally Compact Abelian Groups with the Connected Component of Zero of Dimension 1

Abstract

Let X be a locally compact Abelian group with the connected component of zero of dimension 1. Let $\xi _1$ and $\xi _2$ be independent random variables with values in X with nonvanishing characteristic functions. We prove that if a topological automorphism $\alpha$ of the group X satisfies the condition ${\text {Ker}(I+\alpha )=\{0\}}$ and the conditional distribution of the linear form ${L_2 = \xi _1 + \alpha \xi _2}$ given ${L_1 = \xi _1 + \xi _2}$ is symmetric, then the distributions of $\xi _j$ are convolutions of Gaussian distributions on X and distributions supported in the subgroup $\{x\in X:2x=0\}$. This result can be viewed as a generalization of the well-known Heyde theorem on the characterization of the Gaussian distribution on the real line.