On the Convergence of Some Random Series

Abstract

Let $\{X_n\}$ be a sequence of independent random elements with values in a separable Banach space, $S_n=X_1+\cdots +X_n$ and $\{a_n\}$ a sequence of nonnegative numbers. There is proved a criterion for the almost sure convergence of the series $\sum a_n{S}_n$. The special case $a_n=n^{–\alpha },\alpha >0$ is also studied.