scholarly journals On the Almost Sure Convergence of Randomly Weighted Sums of Random Elements

1983 ◽  
Vol 11 (3) ◽  
pp. 795-797 ◽  
Author(s):  
R. L. Taylor ◽  
C. A. Calhoun
Keyword(s):  
Almost Sure Convergence ◽  
Weighted Sums ◽  
Randomly Weighted Sums ◽  
Random Elements

scholarly journals On the almost sure convergence of weighted sums of random elements inD[0,1]

1981 ◽  
Vol 4 (4) ◽  
pp. 745-752
Author(s):  
R. L. Taylor ◽  
C. A. Calhoun
Keyword(s):  
Almost Sure Convergence ◽  
Growth Conditions ◽  
Weighted Sums ◽  
Integral Conditions ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Convergence Results ◽  
Large Numbers ◽  
Random Elements

Let{wn}be a sequence of positive constants andWn=w1+…+wnwhereWn→∞andwn/Wn→∞. Let{Wn}be a sequence of independent random elements inD[0,1]. The almost sure convergence ofWn−1∑k=1nwkXkis established under certain integral conditions and growth conditions on the weights{wn}. The results are shown to be substantially stronger than the weighted sums convergence results of Taylor and Daffer (1980) and the strong laws of large numbers of Ranga Rao (1963) and Daffer and Taylor (1979).

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.

Stochastic convergence of randomly weighted sums of random elements

1984 ◽  
Vol 2 (3) ◽  
pp. 299-321 ◽  
Author(s):  
Robert Lee Taylor ◽  
Carol Calhoun Raina ◽  
Peter Z. Daffer
Keyword(s):  
Weighted Sums ◽  
Stochastic Convergence ◽  
Randomly Weighted Sums ◽  
Random Elements

scholarly journals Stability for randomly weighted sums of random elements

1999 ◽  
Vol 22 (3) ◽  
pp. 559-568 ◽  
Author(s):  
Tien-Chung Hu ◽  
Hen-Chao Chang
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Weighted Sums ◽  
Randomly Weighted Sums ◽  
Random Elements ◽  
Necessary And Sufficient

Let{Xn:n=1,2,3,…}be a sequence of i.i.d. random elements taking values in a separable Banach space of typepand let{An,i:i=1,2,3,…;n=1,2,3,…}be an array of random variables. In this paper, under various assumptions of{An,i}, the necessary and sufficient conditions for∑i=1∞An,iXi→0a.s. are obtained. Also, the necessity of the assumptions of{An,i}is discussed.

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