The Heyde characterization theorem on compact totally disconnected and connected Abelian groups

TL;DR

This paper extends the Heyde theorem to compact totally disconnected Abelian groups, providing a complete characterization, but shows it fails on connected groups, using Fourier analysis of probability distributions.

Contribution

It offers a full description of groups where the Heyde theorem holds and demonstrates its failure on connected groups, advancing the understanding of distribution characterization on these groups.

Findings

Heyde theorem characterizes Gaussian distributions on real line.

Complete description of groups where the theorem holds.

Failure of the theorem on compact connected Abelian groups.

Abstract

By the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the case of two independent random variables we give a complete description of compact totally disconnected Abelian groups X, where an analogue of this theorem is valid. We also prove that even a weak analogue of the Heyde theorem fails on compact connected Abelian groups X. Coefficients of considered linear forms are topological automorphisms of X. The proofs are based on the study of solutions of a functional equation on the character group of the group X in the class of Fourier transforms of probability distributions.