The Skitovich-Darmois theorem for finite Abelian groups

TL;DR

This paper extends the Skitovich-Darmois theorem to finite Abelian groups, showing that independence of certain linear forms of independent variables implies their distributions are shifts of Haar measures on subgroups.

Contribution

It provides a new characterization of distributions of independent variables via linear forms in the setting of finite Abelian groups, generalizing classical results.

Findings

Independence of linear forms implies distributions are Haar shifts

The theorem characterizes distributions in finite Abelian groups

Extension of classical Skitovich-Darmois theorem

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_{1j}xi_1+alpha_{2j}xi_2+...+alpha_{nj}xi_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.