International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 2, Pages 309-323
doi:10.1155/S0161171279000272
Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables
Department of Mathematics, Computer Science, and Statistics, University of South Carolina, Columbia, Columbia 29208, South Carolina, USA
Received 23 June 1978
Copyright © 1979 W. J. Padgett and R. L. Taylor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.