Random variables with an invariant random shift in compact metrisable abelian groups

TL;DR

This paper characterizes when the sum of independent random variables in compact metrisable abelian groups has the same distribution as one of the variables, linking it to subgroup invariance and Haar measure.

Contribution

It provides a necessary and sufficient condition for the invariance of distribution under addition in such groups, revealing new structural insights.

Findings

$X+Y$ has the same distribution as $X$ iff $Y$ is almost surely in a subgroup and $X$ is invariant under that subgroup.

If $X+Y$ has the same distribution as $X$, then $X+Y$ and $Y$ are independent.

The distribution of $X$ is Haar measure if $Y$ has positive probability in every open set.

Abstract

The main result of this paper states that for independent random variables $X, Y$ taking values in a compact metrisable abelian group, $X + Y$ has the same distribution as $X$, if and only if there exists a compact subgroup $A$ such that $P(Y\in A)=1$ and $X + a$ has the same distribution as $X$ for all $a\in A$. As a conclusion from the above it is shown that for independent random variables $X, Y$ such that $X+Y$ has the same distribution as $X$, $X+Y$ and $Y$ are also independent. It becomes also apparent that the distribution of $X$ is the Haar measure (uniform distribution) if for each open set $U$, $P(Y\in U) > 0$.