Characterization of the Gaussian distribution on groups by the independence of linear statistics (Q1813806)

\textit{V. P. Skitovich} [Dokl. Akad. Nauk SSSR 89, 217--219 (1953; Zbl 0053.27401)] and \textit{G. Darmois} [Rev. Inst. Int. Stat. 21, 2--8 (1953; Zbl 0051.36003)] proved the theorem: Let \(\xi_1,\ldots,\xi_s\) be independent random variables; if \(\alpha_1\xi_1+\ldots+\alpha_s\xi_s\) and \(\beta_1\xi_1+\ldots+\beta_s\xi_s\) are independent (\(\alpha_i\neq 0\), \(\beta_i\neq 0\)), then the distributions of \(\xi_i\) are normal. The author completely describes those locally compact Abelian groups to which it is possible to transfer this theorem.