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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Statistical limit points


Author: J. A. Fridy
Journal: Proc. Amer. Math. Soc. 118 (1993), 1187-1192
MSC: Primary 40C99
MathSciNet review: 1181163
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Abstract: 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 \mathbb{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}:\vert{x_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 \mathbb{N}:{x_k} > {x_{k + 1}}\} $ has density one and $ x$ is bounded on $ M$, then $ x$ is statistically convergent.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1181163-6
PII: S 0002-9939(1993)1181163-6
Keywords: Natural density, statistically convergent sequence
Article copyright: © Copyright 1993 American Mathematical Society