Weighted Sums of Independent Identically Distributed Random Variables

Abstract

We characterize the sequences $(\alpha_i)$ of real numbers such that $\sum^\infty_{i = 1} \alpha_i f_i$ exists a.e. or in $L_p$ for all sequences of independent identically distributed symmetric random variables with $p$th moment. Moreover, we also treat the case $\sup|\alpha_i f_i| < 0\infty$ a.e.