Subsequential Results on Uniform Statistical Convergence

Abstract

Abstract. In this paper, we present some relationships between convergence and uniform statistical convergence of a given sequence and its subsequences. The results concerning uniform statistical convergence presented here are also closely related to earlier results regarding statistical convergence and almost convergence of sequences, and are dealing with measure and in a minor case with category. Finally, we present a Cauchy type characterization of uniform statistical convergence and a result concerning uniform statistical convergence of subseries of a series.

Some results on uniform statistical cluster points

“…as in part b) of the proof of Theorem 2.3. in [18]. Then x is almost convergent and hence from Lemma 1 of [21] uniformly statistically convergent to 0 and from the proof of Theorem 1 of [21], the set {t ∈ (0, 1] : (xt) converges uniformly statistically to 0} has measure 0. However, that set is the same as the set T, and so for this sequence m(T ) = 0.…”

“…Moreover, Dawson [8] and Fridy [11] have studied analogous results by replacing subsequences with stretching and rearrangements, respectively. 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.…”

Some new insights into ideal convergence and subsequences

“…It is well known that every x ∈ (0, 1] has a unique binary expansion x = ∑ ∞ n=1 2 −n d n (x) such that d n (x) = 1 for infinitely many positive integers n, and for every x ∈ (0, 1] and any sequence s = (s n ) we can generate a subsequence (sx) of s in such a way that: if d n (x) = 1, then (sx) n = s n . In the existing literature the relationships between a given sequence and its subsequences have been studied in two directions: the first direction is changing the concept of convergence by statistical convergence, A-statistical convergence, uniform statistical convergence, ideal convergence, and the other direction is using measure or category to study the measure and topological largeness of the sets of subsequences (see [2,4,15,[18][19][20][21][22][23][24]). There are still gaps to examine in this area.…”

“…, then I d -convergence coincides with statistical convergence where d(A) denotes the natural density of A [10], and if I = I u = {A ⊂ N : u(A) = 0}, then I u -convergence reduces to uniform statistical convergence where u(A) denotes the uniform density of A [22,23]. Here, the term meager will refer to sets of first Baire category, while the term comeager will refer to sets whose complement is of first category.…”

Category theoretical view of I-cluster and I-limit points of subsequences

“…This has been extended to a result of great generality in [5,6,7] with the use of summability. Agnew [1], Miller and Orhan [16], Yurdakadim and Miller-Van Wieren [19], and Zeager [21] have studied the relationship between sequences and their subsequences.…”