The Skitovich–Darmois theorem for discrete periodic Abelian groups

This paper gives a complete description of the periodic Abelian groups $X$ such that the independence of the linear statistics of independent random variables taking values in group $X$ (coefficients of the linear statistics are the automorphisms of group) implies that distributions of the random variables are shifts of the Haar distributions on compact subgroups of group $X$.
For the class of groups under consideration the given theorem is a group analogue of the well-known Skitovich–Darmois theorem on a characterization of the Gaussian distribution by the independence of linear statistics.