Independent Linear Statistics on the Cylinders

Let either $X={\bf R}\times{\bf T}$ or $X=\Sigma_{\boldsymbol a}\times{\bf T}$, where ${\bf R}$ is an additive group of real number, ${\bf T}$ is a cycle group, and $\Sigma_{\boldsymbol a}$ is an ${\boldsymbol a}$-adic solenoid. Let $\alpha_{ij}$, where $i, j=1,2,3,$ be a topological automorphism 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.

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