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Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 49561, 15 pages

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

Department of Mathematics, Vinh University, Nghe An 42118, Vietnam

Received 26 June 2005; Revised 11 September 2005; Accepted 16 September 2005

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.


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=1kmj=1lnamnij(XijEXij)Lr0(0<r2) where {amnij;m,n,i,j1} are constants, and {kn,n1} and {ln,n1} are sequences of positive integers. The weak law results provide conditions for i=1Tmj=1τnamnij(XijEXij)p0 to hold where {Tm,m1} and {τn,n1} are sequences of positive integer-valued random variables. The sharpness of the results is illustrated by examples.