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

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