The Skitovich-Darmois theorem for finite Abelian groups

2011/06/21 by Mazur, Ivan · 1 citation
#20D99 #43A05 #60B15 #62E10 #FOS: Mathematics #I.7.2 #Probability (math.PR)

paper · doi:10.48550/arxiv.1106.4134

Abstract

Let X be a finite Abelian group, xi_i, i=1,2,...,n,n>1, be independent random variables with values in X and distributions mu_i. Let alpha_ij,i,j=1,2,...,n, be automorphisms of X. We prove that the independence of n linear forms L_j=alpha_1jxi₁+alpha_2jxi₂+...+alpha_njxi_n implies that all mu_i are shifts of the Haar distributions on some subgroups of the group X. This theorem is an analogue of the Skitovich-Darmois theorem for finite Abelian groups.

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