scholarly journals Limit theorems for weighted sums of random elements in separable Banach spaces

1979 ◽  
Vol 9 (3) ◽  
pp. 428-433 ◽  
Author(s):  
A Bozorgnia ◽  
M.Bhaskara Rao
Keyword(s):  
Banach Spaces ◽  
Limit Theorems ◽  
Weighted Sums ◽  
Random Elements

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

1979 ◽  
Vol 2 (2) ◽  
pp. 309-323
Author(s):  
W. J. Padgett ◽  
R. L. Taylor
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Real Numbers ◽  
Martingale Theory ◽  
Random Elements ◽  
Schauder Bases

Let{Xk}be independent random variables withEXk=0for allkand let{ank:n≥1, k≥1}be an array of real numbers. In this paper the almost sure convergence ofSn=∑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.

scholarly journals Convergence in mean of weighted sums of {an,k}-compactly uniformly integrable random elements in Banach spaces

1997 ◽  
Vol 20 (3) ◽  
pp. 443-450 ◽  
Author(s):  
M. Ordóñez Cabrera
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Weighted Sums ◽  
Uniform Integrability ◽  
Weighted Sum ◽  
Random Elements ◽  
New Hypothesis ◽  
Convergence In Mean

The convergence in mean of a weighted sum∑kank(Xk−EXk)of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the{ank}-compactly uniform integrability of{Xn}. This condition, which is implied by the tightness of{Xn}and the{ank}-uniform integrability of{‖Xn‖}, is weaker than the compactly miform integrability of{Xn}and leads to a result of convergence in mean which is strictly stronger than a recent result of Wang, Rao and Deli.

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