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 (Xk,l)=(XkXl)(Xk,l)=(XkXl) be a double sequence of non-degenerate independently identically distributed random variables such that E(Xk,l)=μ(Xk,l)=μ and E(Xk,l)<∞ for each (k,l). Suppose that A=(am,n,k,l)A=(am,n,k,l) is an RH-conservative matrix; then the necessary and sufficient condition for Ym,nYm,n to P-converge to μ(a−∑k,lck,l)+∑k,lck,lXk,lμ(a−∑k,lck,l)+∑k,lck,lXk,l in probability is that P-limm,nsupk,l|am,n,k,l−ck,l|=0. Other variations and implications will also be presented.