Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables - W. J. Padgett; R. L. Taylor (International Journal of Mathematics and Mathematical Sciences)
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.