On a characterization theorem for probability distributions on discrete Abelian groups

Let $X$ be a countable discrete Abelian group containing no elements of order 2, $\alpha$ be an automorphism of $X$, and $\xi_1$ and $\xi_2$ be independent random variables with values in the group $X$ and having distributions $\mu_1$ and $\mu_2$. The main result of the present paper is as follows. The symmetry of the conditional distribution of the linear form $L_2 = \xi_1 + \alpha\xi_2$ given $L_1 = \xi_1 + \xi_2$ implies that $\mu_j$ are shifts of the Haar distribution of a finite subgroup of $X$ if and only if the automorphism $\alpha$ satisfies the condition $\operatorname{Ker}(I+\alpha)=\{0\}$. This theorem is an analogue, for discrete Abelian groups, of the well-known Heyde theorem, where a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variable given the other. We also prove some generalizations of this theorem.