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

W. J. Padgett and R. L. Taylor

International Journal of Mathematics and Mathematical Sciences, 1979, vol. 2, 1-15

Abstract:

Let { X k } be independent random variables with E X k = 0 for all k and let { a n k : n ≥ 1 , k ≥ 1 } be an array of real numbers. In this paper the almost sure convergence of S n = ∑ k = 1 n a n k X k , n = 1 , 2 , … , to a constant is studied under various conditions on the weights { a n k } and on the random variables { X k } 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.

Date: 1979
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:871020

DOI: 10.1155/S0161171279000272

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