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

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.