Topology in a Group and Convergence of Distributions

The purpose of this paper is to prove the following result. Let $\xi_1,\xi_2,\dots,\xi_n,\dots$ be an arbitrary sequence of independent random variables on a locally compact group $G$. We construct the compositions
$$ \xi_n=\xi_1\xi_2\dots\xi_n. $$
If elements $a_n\in G$ can be found so that the sequence of normalized compositions
$$ \eta_n=\zeta_n a_n $$
as a limiting distribution, then the group $G$ is compact.