On the Skitovich-Darmois theorem for some locally compact Abelian groups

Abstract

Let $X$ be a locally compact Abelian group, ${\alpha }_j,{\beta }_j$ be topological automorphisms of $X$. Let ${\xi }_1,{\xi }_2$ be independent random variables with values in $X$ and distributions ${\mu }_j$ with non-vanishing characteristic functions. It is known that if $X$ contains no subgroup topologically isomorphic to the circle group $\mathbb{T}$, then the independence of the linear forms $L_1={\alpha }_1{\xi }_1+{\alpha }_2{\xi }_2$ and $L_2={\beta }_1{\xi }_1+{\beta }_2{\xi }_2$ implies that ${\mu }_j$ are Gaussian distributions. We prove that if $X$ contains no subgroup topologically isomorphic to ${\mathbb{T}}^2$, then the independence of $L_1$ and $L_2$ implies that ${\mu }_j$ are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of $X$ generated by an element of order 2. The proof is based on solving the Skitovich-Darmois functional equation on some locally compact Abelian groups.