ROUGH STATISTICAL CONVERGENCE IN 2-NORMED SPACES

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.

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.

We introduce the concepts statistical cluster and statistical limit points of a sequence of fuzzy numbers in a fuzzy valued metric space. Then we obtain some inclusion relations between the sets of limit points, statistical limit points and statistical cluster points for a sequence of fuzzy numbers.

In this paper, we introduce and study the concept of rough I–αβ–statistical convergence of order γ in neutrosophic normed spaces. This new mode of convergence combines the principles of rough convergence, statistical convergence with respect to an ideal, and the flexible structure of neutrosophic norms to handle indeterminacy and vagueness in sequence behavior. We establish fundamental properties of this convergence type and investigate the structure of its limit set. Specifically, we prove that the set of rough I–αβ–statistical limit points of order γ is convex and closed under certain conditions. We further analyze the relationship between cluster points and rough statistical limits in this context. The theoretical results are supported by illustrative examples to demonstrate the validity and applicability of the proposed notions. Our findings generalize several existing convergence concepts and contribute to the growing body of research in neutrosophic functional analysis.

Let $\mathcal{I}$ be an admissible (i.e., proper and containing all finite subsets of $\mathbb{N}$) ideal of subsets of the set $\mathbb{N}^+$ of positive integers. The concept of $\mathcal{I}$-convergence of sequences in metric spaces generalizes the concept of statistical convergence and also the usual concept of convergence of sequences. In this paper we investigate some problems concerning the sets of $\mathcal{I}$-cluster points and, in particular, the sets of statistical cluster points of sequences in metric spaces which are known to be closed sets. In the first part of the paper we give a sufficient condition on a sequence $x=(x_n)_{n=1}^\infty$ in a boundedly compact metric space which ensures the connectedness of the set of all statistical cluster points of $x$. If $\Gamma_x(\mathcal{I})$ denotes the set of all $\mathcal{I}$-cluster points of the sequence $x$ and $M$ is a set of sequences in a metric space $X$ such that, for each $x\in M$, $\Gamma_x(\mathcal{I})\ne\emptyset$, then the assignment $x\mapsto\Gamma_x(\mathcal{I})$ gives a map $M\to\mathcal{F}$ where $\mathcal{F}$ is the set of all non-empty closed subsets of the space $X$ or $M\to\mathcal{C}$ where $\mathcal{C}$ is a suitable subset of $\mathcal{F}$. In the second part of the paper we study the continuity of this map with respect to the sup-metric on $M$ and some standard hypertopologies on $\mathcal{C}$ (the Vietoris topology, the Fell topology, the proximal topology and the topology given by the Hausdorff metric). We obtain some positive results in the case of locally compact and, particularly, boundedly compact metric spaces.

In this paper, following the works of Phu (Numer Funct Anal Optimiz 22:201–224, 2001), Aytar (Numer Funct Anal Optimiz 29(34):291–303, 2008) and Ghosal (Appl Math 58(4):423–437, 2013) we make a new approach to extend the application area of rough statistical convergence usually used in sequence of real numbers to the theory of probability distributions. The introduction of these concepts in probability of rough statistical convergence, rough strong Cesaro summable, rough lacunary statistical convergence, rough \(N_\theta \)-convergence, rough \(\lambda \)-statistical convergence and rough strong \((V,\lambda )\)-summable generalize the convergence analysis to accommodate any form of distribution of random variables. Among these six concepts in probability only three convergences are distinct- rough statistical convergence, rough lacunary statistical convergence and rough \(\lambda \)-statistical convergence where rough strong Cesaro summable is equivalent to rough statistical convergence, rough \(N_\theta \)-convergence is equivalent to rough lacunary statistical convergence, rough strong \((V,\lambda )\)-summable is equivalent to rough \(\lambda \)-statistical convergence. Basic properties and interrelations of above mentioned three distinct convergences are investigated and make some observations about these classes and in this way we show that rough statistical convergence in probability is the more generalized concept than usual rough statistical convergence.

A sequence in a metric space is said to be slow if the distance between consecutive terms approaches zero. If a slow sequence lies in a compact metric space or in the real line, then the set of its cluster points is connected. The aim of the present note is to give a characterization of the set of all cluster points of a slow sequence in a normed space.

The primary objective of this study is to introduce the concepts of $I_2$-deferred Cesàro summability and $I_2-$ deferred statistical convergence for double sequences in fuzzy normed spaces (FNS). Furthermore, the aim is to explore the connections between these concepts and subsequently establish several theorems pertaining to the notion of $I_2$-deferred statistical convergence in FNS for double sequences. We further define $I_2$-deferred statistical limit points and $I_2$-deferred statistical cluster points of a sequence within FNS and explore the relationships among these concepts.

In this paper, we define the concepts of rough statistical cluster point and rough statistical limit point of a sequence in a finite dimensional normed space. Then we obtain an ordinary statistical convergence criteria associated with rough statistical cluster point of a sequence. Applying these definitions to the sequences of functions, we come across a new concept called statistical condensation point. Finally, we observe the relations between the sets of statistical condensation points, rough statistical cluster points and rough statistical limit points of a sequence of functions.

In this paper, we introduced the space $$DS^\alpha (\varDelta _v^m,f,X)$$ of all deferred $$(\varDelta _v^m,f)$$ -statistically convergent sequences of order $$\alpha $$ in the normed space (X, q) with the help of generalized difference operator $$\varDelta _v^m$$ , unbounded modulus function f, $$\alpha \in (0, 1]$$ and deferred sequences $$(p_n),~(q_n)$$ of non-negative integers with $$p_n<q_n$$ for any $$n\in \mathbb {N}$$ , $$\displaystyle \lim _{n \rightarrow \infty }(q_n-p_n)=\infty $$ . We also introduced the space $$D\omega ^\alpha (\varDelta _v^m,f,X)$$ of all strongly deferred $$(\varDelta _v^m,f)$$ -summable sequences of order $$\alpha $$ in the normed space (X, q). Inclusion relations between spaces $$DS^\alpha (\varDelta _v^m,f,X)$$ and $$D\omega ^\alpha (\varDelta _v^m,f,X)$$ are established under certain conditions.

The notion of deferred weighted statistical probability convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the deferred weighted statistical convergence. Such concepts were introduced and studied by Srivastava et al. (Appl Anal Discrete Math, 2020). In the present work, we introduced and studied the notion of statistical probability convergence as well as statistical convergence for sequences of random variables and sequences of real numbers respectively defined over a Banach space via deferred Norlund summability mean. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrated that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). Finally, an illustrative example is presented here by the generalized Meyer-Konig and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

In this paper, we introduce the concepts of statistical monotone convergence and statistical order convergence in a Riesz space, and establish some basic facts. We show that the statistical order convergence and the statistical convergence in norm need not be equivalent in a normed Riesz space. Finally, we introduce the statistical order boundedness of a sequence in a Riesz space.

This article studies the strong conical hull intersection property (strong CHIP) for a general (possibly infinite) system of convex sets in normed linear spaces. The first part of the article presents necessary and sufficient conditions for strong CHIP in terms of tangent cones and feasible direction cones in the primal space, and reveals some differences between strong CHIP for finite and infinite systems. The second part of the paper proves that strong CHIP has the segment extension property, i.e. if strong CHIP is satisfied at some ‘base points’ then it can be extended from the base points along line segments to the entire set. Thus, the verification effort of strong CHIP can be greatly reduced in many cases. For example, if the intersection set is a translated cone, then strong CHIP at the vertex ensures strong CHIP for the entire cone. In the special case of Euclidean space, strong CHIP at the extreme points of the intersection set ensures strong CHIP for the entire intersection.

Neutrosophic normed spaces, one of the recent hot issues in mathematics, are covered in this paper. The Neutrosophic approach is based on the idea that the degree of uncertainty should be taken into consideration and that it is insufficient to categorise problems in daily life as either right or wrong. This paper proposes double sequences’ rough statistical convergence in Neutrosophic Normed Spaces. It then specifies the rough statistical limit points and cluster points of a double sequence in these spaces. This paper then looks at some of their fundamental characteristics. The necessity for more research is finally covered.

In this paper, we investigate the concept of Abel statistical ward continuity in 2-normed spaces. A function f defined on a 2-normed space X into X is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (xk) of points in X is called Abel statistically quasi Cauchy if limx→1−(1−x)∑k:||Δxk,z||≥εxk=0 for every ε > 0 and z ∈ X, where Δxk = xk+1 − xk for every k ∈ N. Some other types of compactness and continuities are also studied and interesting results are obtained.