RH-conservative matrix characterization of P-convergence in probability

The goal of this paper is to characterize P-convergence in probability of four-dimensional weighted means using RH-conservative matrices. We begin with the presentation of the following theorem. Let (X-k.l) = (XkXl) be a double sequence of non-degenerate independently identically distributed random variables such that E(X-k,X-l) = mu and E(X-k,X-l) < infinity for each (k, l). Suppose that A = (a(m,n.k,l)) is an RH-conservative matrix; then the necessary and sufficient condition for Y-m,Y-n to P-converge to mu (a - Sigma(k,l) C-k,C-l) + Sigma(k,l) C X-k,l(k,l) in probability is that