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Zbl 0776.40001
Fridy, J.A.
Statistical limit points.
(English)
[J] Proc. Am. Math. Soc. 118, No.4, 1187-1192 (1993). ISSN 0002-9939; ISSN 1088-6826/e

Summary: Following the concept of a statistically convergent sequence $x$, we define a statistical limit point of $x$ as a number $\lambda$ that is the limit of a subsequence $\{x\sb{k(j)}\}$ of $x$ such that the set $\{k(j):j\in{\bold N}\}$ does not have density zero. Similarly, a statistical cluster point of $x$ is a number $\gamma$ such that for every $\varepsilon>0$ the set $\{k\in\bbfN:\vert x\sb k- \gamma\vert<\varepsilon\}$ does not have density zero. These concepts, which are not equivalent, are compared to the usual concept of limit point of a sequence. Statistical analogues of limit point results are obtained. For example, if $x$ is a bounded sequence then $x$ has a statistical cluster point but not necessarily a statistical limit point. Also, if the set $M:=\{k\in\bbfN:x\sb k>x\sb{k+1}\}$ has density one and $x$ is bounded on $M$, then $x$ is statistically convergent.
MSC 2000:
*40A05 Convergence of series and sequences
26A03 Elementary topology of the real line
11B05 Topology etc. of sets of numbers

Keywords: statistically convergent sequence; statistical limit point

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