The Skitovich-Darmois theorem for discrete and compact totally disconnected Abelian groups

TL;DR

This paper generalizes the Skitovich-Darmois theorem to a broad class of Abelian groups, showing that independence of certain linear forms of random variables implies their distributions are convolutions of Gaussian and idempotent distributions.

Contribution

It extends the classical Skitovich-Darmois theorem to discrete and compact totally disconnected Abelian groups, characterizing distributions via independence of linear forms.

Findings

Independence of linear forms implies distributions are convolutions of Gaussian and idempotent distributions.

Generalizes the Skitovich-Darmois theorem to a wider class of Abelian groups.

Provides a structural description of distributions on these groups.

Abstract

Let $X$ be an Abelian group of the form $X=\mathbb{R}^m\times K\times D$, where $m\geq 0$, $K$ is a compact totally disconnected group of the special form, $D$ is a discrete group. Let $\xi_i, i=1,2,...,n,n\geq 2,$ be independent random variables with values in $X$ and distributions $\mu_i$, and $\alpha_{ij},i,j=1,2,...,n,$ be topological automorphisms of $X$. We prove that the independence of the linear forms $L_j=\sum_{i=1}^{n}\alpha_{ij}\xi_i,j=1,2,...,n,$ implies that all $\mu_i$ are convolutions of Gaussian and idempotent distributions. This theorem can be considered as a generalization for the group X of the well-known Skitovich-Darmois theorem for $n$ linear forms.