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On statistical limit points and the consistency of statistical convergence. (English) Zbl 0867.40001
Summary: This article extends the concept of a statistical limit (cluster) point of a sequence $x$ (as introduced by {\it J. A. Fridy} [Proc. Am. Math. Soc. 118, No. 4, 1187-1192 (1993; Zbl 0776.40001)]) to a $T$-statistical limit (cluster) point, where $T$ is a nonnegative regular matrix summability method. These definitions are reformulated in the setting of $\beta\bbfN\setminus \bbfN$. It is shown that for a bounded sequence $x$, the set of $T$-statistical cluster points of $x$ forms a compact subset of $\bbfR$. 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.

40A05Convergence and divergence of series and sequences
40C05Matrix methods in summability
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