A Complete Convergence Theorem for Row Sums from Arrays of Rowwise Independent Random Elements in Rademacher TypepBanach Spaces

2012 ◽  
Vol 30 (2) ◽  
pp. 343-353 ◽  
Author(s):  
Tien-Chung Hu ◽  
Andrew Rosalsky ◽  
Andrei Volodin
Keyword(s):  
Convergence Theorem ◽  
Complete Convergence ◽  
Random Elements ◽  
Rowwise Independent ◽  
Complete Convergence Theorem

scholarly journals Complete convergence for sums of arrays of random elements

2000 ◽  
Vol 23 (11) ◽  
pp. 789-794 ◽  
Author(s):  
Soo Hak Sung
Keyword(s):  
Complete Convergence ◽  
Moment Conditions ◽  
Random Elements ◽  
Rowwise Independent

Let{Xni}be an array of rowwise independentB-valued random elements and{an}constants such that0<an↑∞. Under some moment conditions for the array, it is shown that∑i=1nXni/anconverges to0completely if and only if∑i=1nXni/anconverges to0in probability.

scholarly journals On some threshold-one attractive interacting particle systems on homogeneous trees

2020 ◽  
Vol 57 (3) ◽  
pp. 866-898
Author(s):  
Y. X. Mu ◽  
Y. Zhang
Keyword(s):  
Convergence Theorem ◽  
Complete Convergence ◽  
Interacting Particle Systems ◽  
Initial Conditions ◽  
Contact Process ◽  
Particle Systems ◽  
Voter Model ◽  
Interacting Particle ◽  
Homogeneous Trees ◽  
Complete Convergence Theorem

AbstractWe consider the threshold-one contact process, the threshold-one voter model and the threshold-one voter model with positive spontaneous death on homogeneous trees $\mathbb{T}_d$ , $d\ge 2$ . Mainly inspired by the corresponding arguments for the contact process, we prove that the complete convergence theorem holds for these three systems under strong survival. When the system survives weakly, complete convergence may also hold under certain transition and/or initial conditions.

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 ON COMPLETE CONVERGENCE FOR ARRAYS OF ROWWISE INDEPENDENT RANDOM ELEMENTS

2007 ◽  
Vol 44 (2) ◽  
pp. 467-476 ◽  
Author(s):  
Soo-Hak Sung ◽  
Manuel Ordonez Cabrera ◽  
Tien-Chung Hu
Keyword(s):  
Complete Convergence ◽  
Random Elements ◽  
Rowwise Independent

A Complete Convergence Theorem for Attractive Reversible Nearest Particle Systems

1997 ◽  
Vol 49 (2) ◽  
pp. 321-337 ◽  
Author(s):  
T. S. Mountford
Keyword(s):  
Convergence Theorem ◽  
Invariant Measures ◽  
Complete Convergence ◽  
Regularity Condition ◽  
Particle Systems ◽  
Complete Convergence Theorem

AbstractIn this paper we prove a complete convergence theorem for attractive, reversible, super-critical nearest particle systems satisfying a natural regularity condition. In particular this implies that under these conditions there exist precisely two extremal invariant measures. The result we prove is relevant to question seven of Liggett (1985), Chapter VII.

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.

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