A convergence property of products of independent random variables on compact Lie groups

We consider products of independent random variables $\xi_1\xi_2\cdots\xi_n$, $n=\overline{1,\infty}$, taking values in an arbitrary compact Lie group. In some neighborhood of the identity let the coordinates of the group be given by a mapping $\psi$ of $G$ into a neighborhood of the zero of $R_s$, where $s$ is the dimension of the group. It is shown that for no mappings $\psi$ is it necessarily true that the sum $\psi(\xi_1)+\psi(\xi_2)+\cdots$ converges almost everywhere if the product $\xi_1\xi_2\cdots\xi_n$ converges almost everywhere. Nevertheless it is established that there exist elements $\alpha_n$ of $G$ such that for $\xi'_n=\alpha_n^{-1}\xi_n\alpha_{n+1}$ the sum $\psi(\xi'_1)+\dots+\psi(\xi'_n)+\nobreak\cdots$ and the product $\xi_1\xi_2\cdots\xi_n$ are both convergent almost everywhere or else neither of them has this property.
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