On a Characterization Theorem for Connected Locally Compact Abelian Groups

摘要

According to the classical Skitovich-Darmois theorem the Gaussian measure on the real line is characterized by the independence of two linear forms of n independent random variables. A similar result was proved by Heyde, where the independence of linear forms is replaced by the symmetry of the conditional distribution of one of them given the other. We prove an analogue of Heyde's theorem for linear forms of two independent random variables taking values in a connected locally compact Abelian group of dimension 1 containing no elements of order 2. Coefficients of the linear forms are topological automorphisms of the group. The proof is based on the description of all solutions of a functional equation in the class of characteristic functions (Fourier transforms) of probability measures. In contrast to the previous investigations we do not impose any restrictions on coefficients of the linear forms. We also investigate the role of elements of order 2 in Heyde's theorem for the mentioned class of groups.