Skitovich-Darmois theorem for finite Abelian groups - Mazur, I. P. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
Let X be a finite Abelian group, let ξ_i, i = 1, 2, . . . , n, n ≥ 2, be independent random variables with values in X and distributions μ_i, and let α_ij, i, j = 1, 2, . . . , n, be automorphisms of X. We prove that the independence of n linear forms L_j = _i=1^n 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 analog of the Skitovich – Darmois theorem for finite Abelian groups.