On rough statistical convergence of double sequences in normed linear spaces

In this paper, we introduce the concept of lacunary statistical convergence of double sequences in n-normed spaces. Some inclusion relations between the sets of statistically convergent and lacunary statistically convergent cases of double sequences are established. In addition, we also define lacunary statistical Cauchy double sequences and prove that this notion is equivalent to the lacunary statistical convergence of double sequences.

Let K be a complete, non-trivially valued non-archimedean field. In this paper, weighted statistical convergence and statistical summability of double sequences are studied. Also the notion of (, p, q) - summability and inclusion relations have been discussed in such fields.

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.

On the ideal convergence of double sequences in intuitionistic fuzzy normed 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.

In this paper, we introduce the concepts of ?deferred statistical convergence? and ?strong deferred convergence? for double sequences in the setting of gradual normed linear spaces. Our study uncovers the differences between these two concepts. Additionally, we perform a thorough analysis of the mathematical properties associated with these notions and establish several implication relationships.

ABSTRACTThe article explains and analyzes the conception of rough lacunary ideal statistical convergence on random ‐normed spaces for double sequences. The geometrical and topological results of rough lacunary ideal statistical limit points, rough lacunary ideal cluster points, and rough lacunary ideal boundedness are considered on these spaces. Through an exploration of properties associated with rough lacunary ideal convergence, equivalent conditions establish for set of lacunary ideal statistical limit points in the context of rough lacunary ideal statistically convergent double sequences.

Lacunary statistical convergence of multiple sequences

In this study, we introduced the notions of rough statistical convergence and defined the set of rough statistical limit points of a sequence and obtained statistical convergence criteria associated with this set in 2-normed space. Then, we proved that this set is closed and convex in 2-normed space. Also, we examined the relations between the set of statistical cluster points and the set of rough statistical limit points of a sequence in 2-normed space.

Recently, Patterson and Savas (Math. Commun. 10:55-61, 2005), defined the lacunary statistical analog for double sequences as follows: A real double sequences is said to be P-lacunary statistically convergent to L provided that for each , . In this case write or . In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems. MSC:42B15, 40C05.

The concept of statistical convergence for double sequences on probabilistic normed spaces was presented by Karakus and Demirci in 2007. The purpose of this paper is to introduce the concept of statistical e-convergence for double sequences and study some fundamental properties of statistical e-convergence for double sequences on probabilistic normed spaces.

The notion of statistical convergence of double sequences in topological groups is introduced by Çakalli and Savaş. The goal of this paper is to introduce a class of summability methods that can be applied to \(\lambda \)- double statistical convergence in topological groups and is to prove some theorems.Keywords\(\lambda \)- double statistical convergenceTopological groupsPringsheim limit

In the present paper, using the triangular $A-$statistical convergence for double sequences, which is an interesting convergence method, we prove a Korovkin-type approximation theorem for positive linear operators on the space of all real-valued continuous functions on $\left[ 0,\infty \right)\times \left[ 0,\infty \right) $ with the property that have a finite limit at the infinity. Moreover, we present the rate of convergence via modulus of continuity. Finally, we give some further developments.

In this paper, we have established some new theorems on double weighted mean statistical convergence of double sequences, which gives some new results and generalizes the some previous known results of Karakaya.

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.