Characterization of probability distributions on some locally compact Abelian groups containing an element of order 2

TL;DR

This paper extends Heyde's theorem to certain locally compact Abelian groups with elements of order 2, characterizing new classes of Gaussian distributions through symmetry conditions of linear forms.

Contribution

It generalizes Heyde's theorem to groups with elements of order 2, revealing new classes of distributions characterized by symmetry of linear forms.

Findings

Generalization of Heyde's theorem to groups $ eal imes F$ with elements of order 2

Identification of new classes of probability distributions in this setting

Linear forms with arbitrary topological automorphisms characterize these distributions

Abstract

The well-known Heyde theorem characterizes the Gaussian distributions on the real line by the symmetry of the conditional distribution of one linear form of independent random variables given another. We generalize this theorem to groups of the form $\mathbb{R}\times F$, where $F$ is a finite Abelian group such that its 2-component is isomorphic to the additive group of the integers modulo $2$. In so doing, coefficients of the linear forms are arbitrary topological automorphisms of the group. Previously, a similar result was proved in the case when the group $F$ contains no elements of order 2. The presence of an element of order 2 in $F$ leads to the fact that a new class of probability distributions is characterized