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.