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Statistical limit points. (English) Zbl 0776.40001
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 $$\{x_{k(j)}\}$$ of $$x$$ such that the set $$\{k(j):j\in{\mathbf N}\}$$ does not have density zero. Similarly, a statistical cluster point of $$x$$ is a number $$\gamma$$ such that for every $$\varepsilon>0$$ the set $$\{k\in\mathbb{N}:| x_ k- \gamma|<\varepsilon\}$$ 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 $$M:=\{k\in\mathbb{N}:x_ k>x_{k+1}\}$$ has density one and $$x$$ is bounded on $$M$$, then $$x$$ is statistically convergent.

##### MSC:
 40A05 Convergence and divergence of series and sequences 26A03 Foundations: limits and generalizations, elementary topology of the line 11B05 Density, gaps, topology
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