Independent random variables on Abelian groups with independent the sum and difference

TL;DR

This paper characterizes the distributions of independent random variables on certain Abelian groups where the sum and difference are independent, extending understanding of such distributions in the context of group theory.

Contribution

It provides a complete description of the distributions of independent variables with independent sum and difference on Abelian groups with specific structural properties.

Findings

Characterization of distributions on Abelian groups with finite 2-torsion component.

Extension of classical results to locally compact Abelian groups.

Conditions under which sum and difference independence implies specific distribution forms.

Abstract

Let X be a second countable locally compact Abelian group. Let $\xi_1, \xi_2$ be independent random variables with values in the group X and distributions $\mu_1, \mu_2$ such that the sum $\xi_1+\xi_2$ and the difference $\xi_1-\xi_2$ are independent. Assuming that the connected component of zero of the group $X$ contains a finite number elements of order 2 we describe the possible distributions $\mu_k$.