Stability of sums of weighted nonnegative random variables (Q788387)

An almost sure stability theorem for weighted sums \(S_ n = \sum^{n}_{j=1} w_ j X_ j\) of nonnegative random variables \(X_ j\) with finite second moments and \(\lim_{n\to\infty} w_ n \cdot \left(\sum^{n}_{j=1} w_ j\right)^{-1} = 0\) as \(n\to\infty\), is proved. Some corollaries are considered, among others concerning the similar statements for pairwise independent random variables and some generalization of the divergence part of the Borel-Cantelli lemma.