The principle of convergence ``almost everywhere'' in Lie groups

V. M. Maksimov

Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 543-555

Résumé

Let $U$ be a neighborhood of the identity in an arbitrary Lie group with a fixed system of local coordinates $(x)$ and let $\xi_n$ be independent random variables taking values in the neighborhood $U$ and $\widetilde\xi_n$ be real variables naturally induced by the variables $\xi_n$ in the system of local coordinates $(x)$. If the $\widetilde\xi_n$ have zero means, then the product $\xi_1\cdots\xi_n$, $n\to\infty$, converges or diverges a.e. with $$ \widetilde\xi_1+\widetilde\xi_2+\dots+\widetilde\xi_n+\cdots. $$ Bibliography: 6 titles.