Independent linear statistics on the cylinders

Abstract

Let either $X=\mathbf{R}\times\mathbf{T}$ or $X=Σ_\text{\boldmath $a$}\times\mathbf{T}$, where $\mathbf{R}$ is the additive group of real number, $\mathbf{T}$ is the cycle group and $Σ_\text{\boldmath $a$}$ is an $\text{\boldmath $a$}$-adic solenoid . Let $α_{ij}$, where $i, j=1,2,3,$ be topological automorphisms of the group $X$. We prove the following analogue of the well-known Skitovich--Darmois theorem for the group $X$. Let $ξ_j$, where $j=1, 2, 3$, be independent random variables with values in the group $X$ and distributions $μ_j$ such that their characteristic functions do not vanish. If the linear statistics $L_1=α_{11}ξ_1+α_{12}ξ_2+α_{13}ξ_3$, $L_2=α_{21}ξ_1+α_{22}ξ_2+α_{23}ξ_3$, and $L_3=α_{31}ξ_1+α_{32}ξ_2+α_{33}ξ_3$ are independent, then all $μ_j$ are Gaussian distributions.