Asymptotic equivalence of double sequences

Abstract

The goal of this paper is to present a four-dimensional matrix char- acterization of asymptotic equivalence of double sequences. This will be accomplished with the following notion of asymptotic equivalence of double sequences. Two double sequences are asymptotic equivalent if and only if P − $lim_{k,l}frac{x_k,l}{y_k,l}$ = 1, where x and y are selected judicially. Using this notion necessary and sufficient conditions on the entries of a four-dimensional matrix are given to ensure that the transformation will preserve asymptotic equivalence.