Convergence of randomly weighted sums for arrays under a condition of integrability

Let {X_ni, u_n ≤ i ≤i≤v_n,n1} and {A ., w ≤ i ≤ v , n ≥ 1} be two arrays of random variables. A new notion of integrability for an array of random variables {X_ni} with respect to an array of random variables {A_ni} is introduced, weak laws of large numbers for the randomly weighted sums Σ^vn_i=un A_niX_niare obtained. Our results extend and sharpen the known results in the literature.