Mean convergence theorems and weak laws of large numbers for
double arrays of random variables

Van Thanh, Le

doi:https://doi.org/10.1155/JAMSA/2006/49561

International Journal of Stochastic Analysis

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Volume 2006 | Article ID 049561 | https://doi.org/10.1155/JAMSA/2006/49561

Mean convergence theorems and weak laws of large numbers for double arrays of random variables

Le Van Thanh¹

Received26 Jun 2005

Revised11 Sept 2005

Accepted16 Sept 2005

Published26 Apr 2006

Abstract

For a double array of random variables {Xmn, m ≥ 1, n ≥ 1}, mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which ∑i=1km∑j=1lnamnij(Xij−EXij)→Lr0(0<r≤2) where {amnij;m,n,i,j≥1} are constants, and {kn,n≥1} and {ln,n≥1} are sequences of positive integers. The weak law results provide conditions for ∑i=1Tm∑j=1τnamnij(Xij−EXij)→p0 to hold where {Tm,m≥1} and {τn,n≥1} are sequences of positive integer-valued random variables. The sharpness of the results is illustrated by examples.

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Copyright

Copyright © 2006 Le Van Thanh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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