# Characterization theorems for $Q$-independent random variables with values in a locally compact Abelian group

## TL;DR

This paper extends classical characterization theorems to the setting of $Q$-independent random variables on locally compact Abelian groups, providing new group analogues of well-known probabilistic theorems.

## Contribution

It introduces the concept of $Q$-independence for random variables on locally compact Abelian groups and proves their analogues of classical theorems like Cramér and Skitovich-Darmois.

## Key findings

- Group analogues of classical theorems established
- Functional equations on the character group solved
- Characterization results for $Q$-independent variables obtained

## Abstract

Let $X$ be a locally compact Abelian group, $Y$ be its character group. Following A. Kagan and G. Sz\'ekely we introduce a notion of $Q$-independence for random variables with values in $X$. We prove group analogues of the Cram\'er, Kac-Bernstein, Skitovich-Darmois and Heyde theorems for $Q$-independent random variables with values in $X$. The proofs of these theorems are reduced to solving some functional equations on the group $Y$.