On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements

2002 ◽  
Vol 47 (3) ◽  
pp. 533-547 ◽  
Author(s):  
Tien-Chung Hu ◽  
Tien-Chung Hu ◽  
Deli Li ◽  
Deli Li ◽  
Andrew Rosalsky ◽  
...  
Keyword(s):  
Banach Space ◽  
Complete Convergence ◽  
Weighted Sums ◽  
Random Elements

Complete convergence for weighted sums of arrays of banach-space-valued random elements*

2012 ◽  
Vol 52 (3) ◽  
pp. 316-325 ◽  
Author(s):  
De Hua Qiu ◽  
Tien-Chung Hub ◽  
Manuel Ordóñez Cabrera ◽  
Andrei Volodin
Keyword(s):  
Banach Space ◽  
Complete Convergence ◽  
Weighted Sums ◽  
Random Elements

scholarly journals Complete convergence for weighted sums of arrays of random elements

1983 ◽  
Vol 6 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Robert Lee Taylor
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Kernel Density ◽  
Weighted Sums ◽  
Real Numbers ◽  
Density Estimates ◽  
Kernel Density Estimates ◽  
Large Numbers ◽  
Random Elements

Let{Xnk:k,n=1,2,…}be an array of row-wise independent random elements in a separable Banach space. Let{ank:k,n=1,2,…}be an array of real numbers such that∑k=1∞|ank|≤1and∑n=1∞exp(−α/An)<∞for eachα ϵ R+whereAn=∑k=1∞ank2. The complete convergence of∑k=1∞ankXnkis obtained under varying moment and distribution conditions on the random elements. In particular, laws of large numbers follow for triangular arrays of random elements, and consistency of the kernel density estimates is obtained from these results.

Some mean convergence theorems for weighted sums of Banach space valued random elements

Stochastics ◽  
2021 ◽  
pp. 1-19
Author(s):  
Pingyan Chen ◽  
Manuel Ordóñez Cabrera ◽  
Andrew Rosalsky ◽  
Andrei Volodin
Keyword(s):  
Banach Space ◽  
Weighted Sums ◽  
Convergence Theorems ◽  
Mean Convergence ◽  
Random Elements

scholarly journals Strong laws of large numbers for arrays of rowwise independent random elements

1987 ◽  
Vol 10 (4) ◽  
pp. 805-814 ◽  
Author(s):  
Robert Lee Taylor ◽  
Tien-Chung Hu
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Random Elements ◽  
Rowwise Independent ◽  
Uniformly Bounded

Let{Xnk}be an array of rowwise independent random elements in a separable Banach space of typep+δwithEXnk=0for allk,n. The complete convergence (and hence almost sure convergence) ofn−1/p∑k=1nXnk to 0,1≤p<2, is obtained when{Xnk}are uniformly bounded by a random variableXwithE|X|2p<∞. When the array{Xnk}consists of i.i.d, random elements, then it is shown thatn−1/p∑k=1nXnkconverges completely to0if and only ifE‖X11‖2p<∞.

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.

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