Statistical limit superior and limit inferior

Abstract

Abstract. Following the concept of statistical convergence and statistical cluster points of a sequence x, we give a definition of statistical limit superior and inferior which yields natural relationships among these ideas: e.g., x is statistically convergent if and only if st-liminfx = st-limsupx. The statistical core of x is also introduced, for which an analogue of Knopp's Core Theorem is proved. Also, it is proved that a bounded sequence that is C 1 -summable to its statistical limit superior is statistically convergent.

Some results on uniform statistical cluster points

“…In [21], we studied some relationships between convergence and uniform statistical convergence of a given sequence and its subsequences. The related notions of statistical limit superior and inferior and statistical cluster points have been studied in recent papers including [12,13] In the present paper, we are concerned with the relationships between the uniform statistical limit superior and inferior, almost convergence and uniform statistical convergence of a sequence. We also show some results about the set of uniform statistical cluster points of a given sequence and its subsequences and stretchings, including the discussion of the Lebesgue measure of the set of subsequences that retain the same set of uniform statistical cluster points.…”

“…Orhan and Miller [18] have shown that if x almost converges to L , then the set of In [12], Fridy and Orhan introduced the definitions of the statistical limit superior and statistical limit inferior of a sequence and proved some results concerning these notions. Here we proceed analogously for the case of uniform statistical convergence.…”

Some new paranormed sequence spaces and core theorems

“…(ii) C = (c nk ) ∈ (c : c λ ) reg if and only if (20) holds and (21), (22) also hold with α k = 0 for all k ∈ N and α = 1, respectively.…”

“…Let K be a subset of N. The natural density δ (K) of K ⊆ N is lim n→∞ n −1 {|{k ≤ n : k ∈ K}|} provided it exists, where |E| denotes the cardinality of the set E. A sequence x = (x k ) is called statistically convergent (st-convergent) to the number l if every ε > 0, δ ({k : |x k − l| ≥ ε}) = 0, [20] and is written as st − lim x = l. We write st and st 0 to denote the sets of all statistically convergent and statistically null sequences. The notion of the statistical core (or st-core) of a complex valued sequence has been introduced by Fridy and Orhan [21] and it is shown for a statistically bounded sequence x that st − core(x) = z∈C C x (z) for any x ∈ ∞ , where C x (z) = {w ∈ C : |w − z| ≤ st − lim sup k |x k − z|}.…”

Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function

“…Later on it was further investigated by Fridy and Orhan [4]. The idea depends on the notion of density of subset of  .…”