I want to know if there is a necessary and sufficient condition on the sequence $(u_n)_n$ such that $\sum_n u_nX_n$ converges a.s. wehere $(X_n)_n$ are independent. I know that this series converges if and only if the three series of Kolmogorov converges, so can we reduce the conditions only on the sequence $(u_n)_n$ ? (for example if $\sum_n u_n$ is convergent...)
2026-03-25 11:16:29.1774437389
necessary and sufficient condition on $(u_n)$ for a.s convergence of $\sum_n u_nX_n$
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