Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables

Let {Xk} be independent random variables with EXk=0 for all k and let {ank:n ￠ ‰ ￥1, ￠ € ‰k ￠ ‰ ￥1} be an array of real numbers. In this paper the almost sure convergence of Sn= ￠ ‘k=1nankXk, n=1,2, ￠ € |, to a constant is studied under various conditions on the weights {ank} and on the random variables {Xk} using martingale theory. In addition, the results are extended to weighted sums of random elements in Banach spaces which have Schauder bases. This extension provides a convergence theorem that applies to stochastic processes which may be considered as random elements in function spaces.