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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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