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

TL;DR

This paper extends the Skitovich-Darmois theorem to certain locally compact Abelian groups, characterizing distributions of independent random variables based on the independence of specific linear forms.

Contribution

It generalizes previous results by identifying conditions under which distributions are Gaussian or convolutions involving Gaussian distributions on broader classes of groups.

Findings

If $X$ contains no subgroup isomorphic to $ op$, independence implies Gaussianity.

If $X$ contains no subgroup isomorphic to $ op^2$, independence implies Gaussian or Gaussian convolved with signed measures.

The proof involves solving the Skitovich-Darmois functional equation on these 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.