An Analogue of Heyde's Theorem for a Certain Class of Compact Totally Disconnected Abelian Groups and p-quasicyclic Groups

TL;DR

This paper extends Heyde's theorem to certain compact totally disconnected Abelian groups and p-quasicyclic groups, characterizing distributions via symmetry of conditional linear forms without restrictions on coefficients or characteristic functions.

Contribution

It provides a new group analogue of Heyde's theorem applicable to a broader class of groups without restrictions on automorphisms or characteristic functions.

Findings

Characterization of distributions on specific groups via symmetry conditions

No restrictions on coefficients or characteristic functions in the theorem

Application of harmonic analysis methods to prove the result

Abstract

According to 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 article, we study an analogue of this theorem for two independent random variables taking values either in a compact totally disconnected Abelian group of a certain class, which includes finite cyclic groups and groups of p-adic integers, or in a p-quasicyclic group. In contrast to previous works devoted to group analogues of Heyde's theorem, we do not impose any restrictions on either coefficients of linear forms (they can be arbitrary topological automorphisms of the group) or the characteristic functions of random variables. For the proof we use methods of abstract harmonic analysis.