%0 Journal Article
%T Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables
%A W. J. Padgett
%A R. L. Taylor
%J International Journal of Mathematics and Mathematical Sciences
%D 1979
%I Hindawi Publishing Corporation
%R 10.1155/s0161171279000272
%X Let {Xk} be independent random variables with EXk=0 for all k and let {ank:n ¡é ¡ë £¤1, ¡é ? ¡ëk ¡é ¡ë £¤1} be an array of real numbers. In this paper the almost sure convergence of Sn= ¡é   ¡®k=1nankXk, n=1,2, ¡é ? |, 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.
%K weighted sums
%K strong law of large numbers
%K almost sure convergence
%K generlized Gaussian random variables
%K random elements in Banach space
%K Schauder basis.
%U http://dx.doi.org/10.1155/S0161171279000272