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, fn: X→X a homomorphism fn(x)=nx, and X(n)=Im fn. Denote by I(X) the set of idempotent distributions on X and by Γ(X) the set of Gaussian distributions on X. Consider linear statistics L1=α1(ξ1)+α2(ξ2) and L2=β1(ξ1)+β2(ξ2), where ξj are independent random variables taking on values in X and with distributions μj, and αj, βj∈Aut(X). The following results are obtained. Let X be a totally disconnected group. Then the independence of L1 and L2 implies that μ1, μ2∈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 ξj taking on values in X and with distributions μj, and αj, βj∈Aut(X) such that L1 and L2 are independent, whereas μ1, μ2∉Γ(X) * I(X).