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

W J Padgett,R L Taylor

doi:10.1155/s0161171279000272

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 , 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.

Highlights

Special cases of generalizedGaussian random variables are normal with zero means or bounded, symmetric random variables
{ank: and let n > i, k > i} be an array of real numbers
The results are extended to weighted sums of random elements in Banach spaces which have Schauder bases

Summary

Special cases of generalized

Gaussian random variables are normal with zero means or bounded, symmetric random variables. This extension will provide a convergence theory for weighted sums of stochastic processes which mav be considered as random elements in function spaces, such as the Wiener process on a closed interval [0, T] for some T > 0 [see Billingsley (1968)].

For each n

EXI Let n

Let p be a fixed positive integer and consider n