A characterization of Gaussian measures on locally compact Abelian groups

Let ξ and η be independent random variables having equal variance. In order that ξ + η and ξ - η be independent, it is necessary and sufficient that ξ and η have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when ξ and η take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of ξ + η and ξ - η, functions of ξ and η are considered which satisfy the associativity equation