On Statistical Limit Points and the Consistency of Statistical Convergence

Abstract

This article extends the concept of a statistical limit (cluster) point of a sequencex(as introduced by Fridy) to aT-statistical limit (cluster) point, whereTis a nonnegative regular matrix summability method. These definitions are reformulated in the setting of βN\\N. It is shown that for a bounded sequencex, the set ofT-statistical cluster points ofxforms a compact subset of R. It is also shown that, ifTandRare two nonnegative regular summability matrices, thenT-statistical convergence andR-statistical convergence are consistent if and only if the support sets ofTandRhave nonempty intersection.