Skitovich-Darmois theorem for finite Abelian groups

Abstract

Let $X$ be a finite Abelian group, let $\xi_i,\; i = 1, 2, . . . , n,\; n ≥ 2$, be independent random variables with values in $X$ and distributions $\mu_i$, and let $\alpha_{ij},\; i, j = 1, 2, . . . , n$, be automorphisms of $X$. We prove that the independence of n linear forms $L_j = \sum_{i=1}^{n} \alpha_{ij} \xi_i$ implies that all $\mu_i$ 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.