Statistical limit points

Abstract

Following the concept of a statistically convergent sequence x x , we define a statistical limit point of x x as a number λ \lambda that is the limit of a subsequence { x k ( j ) } \{ {x_{k(j)}}\} of x x such that the set { k ( j ) : j ∈ N } \{ k(j):j \in \mathbb {N}\} does not have density zero. Similarly, a statistical cluster point of x x is a number γ \gamma such that for every ε > 0 \varepsilon > 0 the set { k ∈ N : | x k − γ | > ε } \{ 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 x is a bounded sequence then x x has a statistical cluster point but not necessarily a statistical limit point. Also, if the set M := { k ∈ N : x k > x k + 1 } M: = \{ k \in \mathbb {N}:{x_k} > {x_{k + 1}}\} has density one and x x is bounded on M M , then x x is statistically convergent.