Statistical limit points

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 $\{ k \in \mathbb {N}:|{x_k} - \gamma | < \varepsilon \}$ 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.