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.