On P-convergence of four dimensional weighted sums of double random variables

Abstract

The goal of this paper was to present a series of limit theorems that characterizes independent double random variables via four dimensional summability transformation. In order to accomplish this goal we began with the presentation of the following theorem that characterize pairwise independent random variables: let [xk,l] be a double sequence of pairwise independent random variables such that [xk, l] was uniformly integrable. Let [am, n, k, l] be a four dimensional matrix such that ≤ C for all ordered pair (m, n) and for some C and converges to 0 in probability Then (xk,l – E(xk,l) converges in mean to 0. Other extensions and variations via multidimensional transformation shall also be presented.