On statistical limit points and the consistency of statistical convergence

This article extends the concept of a statistical limit (cluster) point of a sequence x (as introduced by Fridy) to a T-statistical limit (cluster) point, where T is a nonnegative regular matrix summability method. These definitions are reformulated in the setting of beta N \ N. It is shown that for a bounded sequence x, the set of T-statistical cluster points of x forms a compact subset of R. It is also shown that, if T and R are two nonnegative regular summability matrices, then T-statistical convergence and R-statistical convergence are consistent if and only if the support sets of T and R have nonempty intersection. (C) 1996 Academic Press, Inc.