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

Abstract

Let $X$ be an Abelian group of the form $X={\mathbb{R}}^m\times K\times D$, where $m\ge 0$, $K$ is a compact totally disconnected group of the special form, $D$ is a discrete group. Let ${\xi }_i,i=1,2,...,n,n\ge 2,$ 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.