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 theindependence of n linear forms $L_j = \sum_{i=1}^{n} α_{ij} ξ_i$ 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.