On a characterization of probability distributions on locally compact abelian groups

Abstract

In one of his recent papers, I. J. Kotlarski has proved the following result. If X₁, X₂, X₃ are three independent real random variables and if the characteristic function of the pair (Z₁, Z₂) where Z₁ = X₁ - X₂, Z₂ = X₁ - X₃ does not vanish, then the distribution of (Z₁, Z₂) determines the distributions of X₁, 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.