Strong laws of large numbers and mean convergence theorems for randomly weighted sums of arrays under a condition of integrability

摘要

Let (X_(ni), u_n ≤ i ≤ v_n, n ≥ 1) and (A_(ni), u_n ≤io_n,n≥1) be two arrays of random variables. A new concept of integrability for an array of random variables (X_(ni)) with respect to an array of random variables (A_(ni)) is introduced. Under these notions of integrability and appropriate conditions on the array of random weights, strong laws of large numbers and mean convergence theorems for the randomly weighted sums ∑_i=u_n~u_n (A_(ni)X_(ni) - EA_(ni)X_(ni)) are obtained. Our results extend and sharpen the known results in the literature.