On the Skitovich-Darmois Theorem for Compact Abelian Groups

Let X be a separable compact Abelian group, Aut(X) the group of topological automorphisms of X, f n: X (longrightarrow) X a homomorphism f n(x)=nx, and X (n)=Im f n. Denote by I(X) the set of idempotent distributions on X and by (gamma) (X) the set of Gaussian distributions on X. Consider linear statistics L 1=(alpha)1 ((xi) 1)+(alpha) 2 ((xi) 2) and L 2=(beta)1 ((xi) 1) +(beta) 2 ((xi) 2), where (xi)j are independent random variables taking on values in X and with distributions (mu)j, and (alpha)j, (beta) j (element of) Aut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L 1 and L 2 implies that (mu)1, (mu) 2 (element of) I(X) if and only if X possesses the property: for each prime p the factor-group X/X (p) is finite. If X is connected, then there exist independent random variables (xi)j taking on values in X and with distributions (mu)j, and (alpha)j, (beta) j (element of) Aut(X) such that L 1 and L 2 are independent, whereas (mu)1, (mu) 2 (not element of) (gamma) (X) * I(X).