Generalization of the Heyde theorem to finite Abelian groups and groups of the form RxG, where G is a finite Abelian group

TL;DR

This paper extends the Heyde theorem, originally characterizing Gaussian distributions on the real line, to a broader class of locally compact Abelian groups, including finite Abelian groups and groups of the form R x G.

Contribution

It generalizes the Heyde theorem to finite Abelian groups and groups of the form R x G without restrictions on the coefficients of the linear forms.

Findings

Characterization of distributions on Abelian groups via symmetry of conditional distributions.

No restrictions on coefficients of linear forms in the group setting.

Extension of Heyde theorem to new classes of groups.

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. We study analogues of this theorem for some locally compact Abelian groups. We consider linear forms of two independent random variables with values in a locally compact Abelian group X. We assume that the characteristic functions of these independent random variables do not vanish. Unlike most previous works, we do not impose any restrictions on coefficients of the linear forms. They are arbitrary topological automorphisms of X.