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.