Independent linear statistics on the cylinders

Let either $X=\mathbf{R}\times \mathbf{T}$ or X=\Sigma_\text{\boldmatha}\times\mathbf{T}, where $\mathbf{R}$ is the additive group of real number, $\mathbf{T}$ is the cycle group and \Sigma_\text{\boldmatha} is an \text{\boldmatha}-adic solenoid . Let ${\alpha }_{ij}$, 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 ${\xi }_j$, where $j=1,2,3$, be independent random variables with values in the group $X$ and distributions ${\mu }_j$ such that their characteristic functions do not vanish. If the linear statistics $L_1={\alpha }_{11}{\xi }_1+{\alpha }_{12}{\xi }_2+{\alpha }_{13}{\xi }_3$, $L_2={\alpha }_{21}{\xi }_1+{\alpha }_{22}{\xi }_2+{\alpha }_{23}{\xi }_3$, and $L_3={\alpha }_{31}{\xi }_1+{\alpha }_{32}{\xi }_2+{\alpha }_{33}{\xi }_3$ are independent, then all ${\mu }_j$ are Gaussian distributions.