The classical Skitovich-Darmois theorem states ([14], [1], see also [11, Ch. 3]): Let ξ i , i = 1, 2, . . . , n, n ≥ 2, be independent random variables, and α i , β i be nonzero numbers. Suppose that the linear formsare independent. Then all random variables ξ i are Gaussian.The Skitovich-Darmois theorem was generalized into various algebraic structures, in particular, into locally compact Abelian groups([3]-[9],[10], [12]). In these researches the random variables take values in a locally compact Abelian group X, coefficients of the linear forms are topological automorphisms of X, and the number of the linear forms are two. In the article we continue these researches and study the Skitovich-Darmois theorem in the case when random variables take values in different classes of locally compact Abelian groups but the number of linear forms more than 2.Throughout the article X is a second countable locally compact Abelian group. Let Aut(X) be the group of topological automorphisms of X, Z(k) = {0, 1, 2 . . . , k − 1} be the group of residue modulo k. Let x ∈ X. Denote by E x the degenerate distribution, concentrated at the point x. If K is a compact subgroup of X, denote by m K the Haar distribution on K. Denote by I(X) the set of shifts of such distributions, i.e. the distributions of the form m K * E x , where K is a compact subgroup of X, x ∈ X.

Let X be a finite Abelian group, ξ i , i = 1, 2, . . . , n, n ≥ 2, be independent random variables with values in X and distributions µ i . Let α ij , i, j = 1, 2, . . . , n, be automorphisms of X. We prove that the independence of n linear forms L j = n i=1 α ij ξ i implies that all µ i are shifts of the Haar distributions on some subgroups of the group X. This theorem is an analogue of the Skitovich-Darmois theorem for finite Abelian groups.

We prove a theorem which characterizes the Haar distribution on a compact Abelian group by the independence of n linear forms of n independent random variables. The coefficients of the forms are continuous endomorphisms of the group.