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

1993 ◽  
Vol 16 (3) ◽  
pp. 587-591 ◽  
Author(s):  
Abolghassem Bozorgnia ◽  
Ronald Frank Patterson ◽  
Robert Lee Taylor
Keyword(s):  
Banach Space ◽  
Density Estimation ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Elements ◽  
Rowwise Independent

Let{Xnk}be an array of rowwise independent random elements in a separable Banach space of typer,1≤r≤2. Complete convergence ofn1/p∑k=1nXnkto0,0<p<r≤2is obtained whensup1≤k≤nE ‖Xnk‖v=O(nα),α≥0withv(1p−1r)>α+1. An application to density estimation is also given.

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 Strong laws of large numbers for arrays of rowwise conditionally independent random elements

1993 ◽  
Vol 6 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ronald Frank Patterson ◽  
Abolghassem Bozorgnia ◽  
Robert Lee Taylor
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Moment Conditions ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Random Elements ◽  
Conditional Means

Let {Xnk} be an array of rowwise conditionally independent random elements in a separable Banach space of type p, 1≤p≤2. Complete convergence of n−1r∑k=1nXnk to 0, 0<r<p≤2 is obtained by using various conditions on the moments and conditional means. A Chung type strong law of large numbers is also obtained under suitable moment conditions on the conditional means.

scholarly journals On Chung-Teicher type strong law for arrays of vector-valued random variables

2004 ◽  
Vol 2004 (9) ◽  
pp. 443-458
Author(s):  
Anna Kuczmaszewska
Keyword(s):  
Banach Space ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Elements ◽  
Vector Valued

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach spaceℬ. The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of specific series ando(1)requirements on specific weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space.

scholarly journals Strong laws of large numbers for arrays of row-wise exchangeable random elements

1985 ◽  
Vol 8 (1) ◽  
pp. 135-144 ◽  
Author(s):  
Robert Lee Taylor ◽  
Ronald Frank Patterson
Keyword(s):  
Banach Space ◽  
Almost Sure Convergence ◽  
Kernel Density ◽  
Triangular Array ◽  
Density Estimates ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Kernel Density Estimates ◽  
Large Numbers ◽  
Random Elements

Let{Xnk,1≤k≤n,n≤1}be a triangular array of row-wise exchangeable random elements in a separable Banach space. The almost sure convergence ofn−1/p∑k=1nXnk,1≤p<2, is obtained under varying moment and distribution conditions on the random elements. In particular, strong laws of large numbers follow for triangular arrays of random elements in(Rademacher) typepseparable Banach spaces. Consistency of the kernel density estimates can be obtained in this setting.

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.

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).

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