Characterization of distributions of $Q$-independent random variables on locally compact Abelian groups

TL;DR

This paper extends classical characterization theorems to the setting of $Q$-independent random variables on second countable locally compact Abelian groups, using functional equations on the character group.

Contribution

It provides group analogues of key characterization theorems for $Q$-independent variables, expanding their applicability to a broader algebraic context.

Findings

Established group versions of Skitovich--Darmois, Heyde, and Kac--Bernstein theorems.

Reduced proofs to solving functional equations on the character group.

Enhanced understanding of $Q$-independent variables in harmonic analysis.

Abstract

Let $X$ be a second countable locally compact Abelian group. We prove some group analogues of the Skitovich--Darmois, Heyde and Kac--Bernstein characterisation theorems for $Q$-independent random variables taking values in the group $X$. The proofs of these theorems are reduced to solving some functional equations on the character group of the group $X$.