On a characterization of probability distributions on locally compact abelian groups

Summary In one of his recent papers, I. J.Kotlarski has proved the following result. IfX₁,X₂,X₃ are three independent real random variables and if the characteristic function of the pair (Z₁,Z₂) whereZ₁=X₁-X₂,Z₂=X₁-X₃ does not vanish, then the distribution of (Z₁,Z₂) determines the distributions ofX₁,X₂,X₃ up to a change of location. He extended this result to random variables taking values in a Hilbert space in another paper. Our aim in this paper is to extend the above result to probability distributions on locally compact abelian groups.