ROUGH STATISTICAL CONVERGENCE IN 2-NORMED SPACES

In this paper, we have introduced the concept of the set of rough I3-lacunary limit points for triple sequences in 2-normed spaces. We have established statistical convergence requirements associated with this set. Furthermore, we have introduced the idea of rough I3-lacunary statistical convergence for triple sequences. Additionally, we have demonstrated that this set of rough I3-lacunary limit points is both convex and closed within the context of a 2-normed space. We have also explored the relationships between a sequence’s rough I3-lacunary statistical cluster points and its rough I3-lacunary statistical limit points in the same 2-normed space. Expanding upon the concept of triple sequence spaces, we have introduced the notion of Wijsman I3-Cesáro summability for triple sequences. In doing so, we have investigated the connections between Wijsman strongly I3-Cesáro summability and Wijsman statistical I3-Cesáro summability. Furthermore, we have introduced the concepts of Wijsman rough strongly p-lacunary summability of order α and Wijsman rough lacunary statistical convergence of order α for triple sequences. These new concepts have been subjected to a thorough examination to understand their characteristics, and we have explored potential connections between them. Additionally, we have investigated how these newly introduced concepts relate to existing notions in the literature.

The aim in our article is to introduce the notion of statistical convergence and statistically Cauchy sequences in intuitionistic fuzzy n-normed linear spaces. The paper shows that some properties of statistical convergence of real sequences also hold for sequences in this space. Characterization for statistically convergent and statistically Cauchy sequences is also given. Further, the concept of statistical limit points and statistical cluster points are introduced and their relation with limit points of sequences have been investigated.

In this work, using the concept of natural density, we introduce the notion of rough statistical convergence. We define the set of rough statistical limit points of a sequence and obtain two statistical convergence criteria associated with this set. Later, we prove that this set is closed and convex. Finally, we examine the relations between the set of statistical cluster points and the set of rough statistical limit points of a sequence.

In this study, we define a new type of statistical limit point using the notions of statistical convergence with respect to the $J_p$ power series method and then we present some examples to show the relations between these points and ordinary limit points. After that we also study statistical limit points of a sequence with the help of a modulus function in the sense of the $J_p$ power series method. Namely, we define $f-J_p$-statistical limit and cluster points of the real sequences and compare the set of these limit points with the set of ordinary points.

In this paper, using the concept of natural density, we introduce the notion of rough statistical convergence of triple sequences. We define the set of rough statistical limit points of a triple sequence and obtain rough statistical convergence criteria associated with this set. Later, we prove this set is closed and convex and also examine the relations between the set of rough statistical cluster points and the set of rough statistical limit points of a triple sequence.

In this paper we study the set of statistical cluster points of sequences in m-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m-dimensional spaces too. We also define a notion of Γ-statistical convergence. A sequence xis Γ-statistically convergent to a set Cif Cis a minimal closed set such that for every ∈ > 0 the set $$\{ k:\varrho (C,x_{{\text{ }}k} ) \geqslant \varepsilon \} $$ has density zero. It is shown that every statistically bounded sequence is Γ-statistically convergent. Moreover if a sequence is Γ-statistically convergent then the limit set is a set of statistical cluster points.

The aim of present work is to introduce and study lacunary statistical limit and lacunary statistical cluster points for generalized difference sequences of fuzzy numbers. Some inclusion relations among the sets of ordinary limit points, statistical limit points, statistical cluster points, lacunary statistical limit points, and lacunary statistical cluster points for these type of sequences are obtained.

The aim of this paper is to establish some relationship between the set of strong uniform statistical cluster points and the set of strong statistical cluster points of a given sequence in the probabilistic normed space. To this aim, let the uniform density be on the positive integers N for a sequence in the probabilistic normed space, that is, cases as equal of the lower and upper uniform density of a subset of N. We introduce the concept of strong uniform statistical cluster points and give a new type convergence in the probabilistic normed space. Note that the set of strong uniform statistical cluster points is a non-empty compact set. We also investigate some properties of the set all strong uniform cluster points of a sequence in the probabilistic normed space.

In this article, we narrated the new notion of rough f-statistical convergence and rough f-statistical Cauchy sequences, successively becoming a more generalized version of rough statistical convergence and rough statistical Cauchy sequences. Consecutively, compare the following important theorems with those of Listán-García [Quaest. Math. 37(4) (2014), 525–530], Phu [Numer. Funct. Anal. Optim. 22(1–2) (2001), 199–222 and Numer. Funct. Anal. Optim. 24(3–4) (2003), 285–301], and Aytar [Numer. Funct. Anal. Optim. 29(3–4) (2008), 291–303]. Suppose x = {xn } n ∈ℕ is a bounded sequence in some finite dimensional normed space X. Let C denote the cluster point set of this sequence. Then, DX (C) is the minimal Cauchy degree and rX (C) is the minimal convergence degree of {xn } n ∈ℕ, where That means is a ρ-Cauchy sequence}, and Suppose ρ ≥ 0 and {xn } n ∈ℕ is a ρ-Cauchy sequence in some normed space X. Then, {xn } n ∈ℕ is r-convergent for every r > 2−1 J(X)ρ. If X is finite dimensional then {xn } n ∈ℕ is r-convergent for every r ≥ 2−1 J(X)ρ. If x = {xn } n ∈ℕ is a sequence in a finite dimensional normed space ℝ n , then If we reformulate the above three results regarding rough f -statistical convergence and rough f -statistical Cauchy sequences in an infinite dimensional normed space, subsequent assertions will be false. Apart from these, we find the minimal f -statistical convergence degree and the minimal f -statistical Cauchy degree of a sequence in any dimensional normed spaces.

In this paper, we address some imprecisions in the definition of neutrosophic normed space proposed by Kirişçi and Şimşek [16], Definition 4. We propose a modification to the definition and introduce the notion of rough lacunary statistical convergence in the neutrosophic normed space. Furthermore, we present the idea of rough lacunary statistical cluster points in neutrosophic normed spaces and investigate the relationship between the set of these cluster points and the set of rough lacunary statistical limit points of the aforementioned convergence.

The idea of rough statistical convergence for single sequences was introduced by Salih Aytar as a generalization of rough convergence. In this paper we define and study rough statistical convergence of double sequences, the set of rough statistical limits of a double sequence. We also study the relation between the set of statistical cluster points and the set of rough limit points of a double sequence.

In the current work we introduce the notions of levelwise statistical limit point and levelwise statistical cluster point of a sequence of fuzzy numbers and discuss the relations between the sets of ordinary limit points, levelwise limit points, levelwise statistical limit points, statistical cluster points and levelwise statistical cluster points of a sequence of fuzzy numbers. Finally, we show that the Bolzano–Weierstrass Theorem is not valid for levelwise statistical convergence.

In this paper, we introduce the concept of (lambda, μ)-statistical convergence in n- normed spaces, where = (r) and μ = (μs) be two non-decreasing sequences of positive real numbers, each tending to ∞ and such that r+1 ≤ r + 1, 1 = 1; μs+1 ≤ μs + 1, μ1 = 1. Some inclusion relations between the sets of statistically convergent and (, μ)-statistically convergent double sequences are established. We find its relation to statistical convergence, (C, 1, 1)-summability and strong (V, lambda, μ)-summability in n-normed spaces.

Abstract: In this paper, the notion of statistical convergence, which was introduced bySteinhaus (1951), was studied in Rm ; and some concepts and theorems, whosestatistical correspondence for the real number sequences were given, were carried toRm . In addition, the concepts of the statistical limit point and the statistical cluster pointwere given and it was mentioned that these two concepts were'nt equal in Fridy's studyin 1993. These concepts were given in Rm and the inclusion correlation between theseconcepts was studied.Key words: Statistical convergence, statistical boundedness, statistically Cauchysequence, statistical limit point, statistical cluster point.Mathematics Subject Classifications (2000): 40A05SONLU BOYUTLU UZAYLARDA İSTATİSTİKSEL YAKINSAKLIKÜZERİNEÖzet: Bu makalede Steinhaus (1951) tarafından verilen istatistiksel yakınsaklık kavramıRm de incelenip, reel sayı dizileri için istatistiksel benzerleri verilen bazı kavram veteoremler R m e taşınmıştır. 1993'de Fridy tarafından yapılan çalışmada istatistiksellimit noktası ve istatistiksel yığılma noktası kavramları verilip bu iki kavramın birbirinedenk olmadığından bahsedilmiştir. Bu kavramlar R m de verilip aralarındaki kapsamabağıntısı incelenmiştir.Anahtar kelimeler: İstatistiksel Yakınsaklık, İstatistiksel Sınırlılık, İstatistiksel CauchyDizisi, İstatistiksel Limit Noktası, İstatistiksel Yığılma Noktası.

The concept of statistical order convergence of sequences in Riesz spaces was introduced and studied. In the present paper, we define the statistical order limit points of a sequence (xn) as a vector x that is the order limit of a subsequence (xk)k?K of (xn) such that the set K does not have density zero. Moreover, we introduce the statistical order cluster points of sequences in Riesz spaces, and also, we give some relations between them.