Notion Of Statistical Convergence Research Articles

On lacunary statistical convergence of double sequences in credibility theory

In 1951, the notion of statistical convergence was a more general concept than the idea of convergence as by presented by Fast. Since then statistical convergence has been was investigated by more and more researchers. On the other hand, in 2006 Li has began the survey of credibility theory and presented some convergence concepts in credibility theory. Because of the esoteric of both credibility and summability theory, the body of literature is limited. As such the only accessible notion to both theories is Pringsheim limits. The goal of this paper is to present a natural multidimensional extension of credibility theory via summability methods. To accomplish this, in this paper we present useful tools to generalize the well-known convergence definition to the concept of lacunary statistical convergence for double sequences in credibility theory. Additionally, some important results and examples will be examined.

Statistically Convergent Sequences in Neutrosophic Metric Spaces

In this study, the introduction of statistical convergence and statistical Cauchy sequences with respect to neutrosophic metric spaces is motivated by the notion of statistical convergence in fuzzy metric spaces. We offer useful characterizations for statistically convergent and statistically Cauchy sequences.

On statistical mathfrak{A}-Cauchy and statistical mathfrak{A}-summability via ideal

The notion of statistical convergence was extended to mathfrak{I}-convergence by (Kostyrko et al. in Real Anal. Exch. 26(2):669–686, 2000). In this paper we use such technique and introduce the notion of statistically mathfrak{A}^{mathfrak{I}}-Cauchy and statistically mathfrak{A}^{mathfrak{I}^{ast }}-Cauchy summability via the notion of ideal. We obtain some relations between them and prove that under certain conditions statistical mathfrak{A}^{mathfrak{I}}-Cauchy and statistical mathfrak{A}^{mathfrak{I}^{ast }}-Cauchy summability are equivalent. Moreover, we give some Tauberian theorems for statistical mathfrak{A}^{mathfrak{I}}-summability.

Statistical Tauberian theorems for Ces\xe0ro integrability mean based on post-quantum calculus

The notion of statistical convergence is more general than the classical convergence. Tauberian theorems via different ordinary summability means have been established by many researchers. In the present work, we have established some new Tauberian theorems based on post-quantum calculus via statistical Cesàro summability mean of real-valued continuous function of one variable under oscillating behavior and De la vallée Poussin mean of a single integral. Moreover, some remarks and corollaries are provided here to support our theorems.

$$I_\Delta $$ and $$I^*_\Delta $$-convergence on time scales

In this paper, we introduce the concept of $$\Delta $$ -weak ideal on a time scale T and using it we introduce and study the notions of $$I_{\Delta }$$ -convergence and $$I^{*}_{\Delta }$$ -convergence of a $$\Delta $$ -measurable real-valued function f defined on T. The notion of $$I_{\Delta }$$ -convergence generalizes the notion of statistical convergence of a $$\Delta $$ -measurable real-valued function f defined on a time scale T.

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ℝ. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δα and interpret them to those of previous works. Also, by using the convergence of Δα-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δα-statistical convergence of positive linear operators.

Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$ such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

Density by moduli and statistical convergence

AbstractBy using modulus functions we introduce a new concept of density for sets of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence which is studied and characterized. As an application, we prove that the ordinary convergence is equivalent to the module statistical conver- gence for every unbounded modulus function.

Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces

The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and -convergence through de la Vallée-Poussin mean in a locally solid Riesz space. MSC:40A35, 40G15, 46A40.

Statistical convergence in a paranormed space

In this article, we define the notion of statistical convergence, statistical Cauchy and strongly p-Cesaro summability in a paranormed space. We establish some relations between them. AMS subject classification (2000): 41A10; 41A25; 41A36; 40A05; 40A30.

Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statistical τ‐convergence, statistical τ‐Cauchy and S∗(τ)‐convergence of double sequences in a locally solid Riesz space.

Statistical Convergence in Intuitionistic Fuzzy n-normed Linear Spaces

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.

A CHARACTERIZATION OF u-UNIFORMLY COMPLETENESS OF RIESZ SPACES IN TERMS OF STATISTICAL u-UNIFORMLY PRE-COMPLETENESS

In this paper we introduce statistically u-uniformly convergent sequences in Riesz spaces (vector lattices) and then we give a characterization of u-uniformly completeness of Riesz spaces. The notion of statistical convergence of sequences was introduced by Steinhauss [6] at a conference held at Wroclaw University, Poland, in 1949 (see also [1]). A sequence (xn) of real numbers is said to be statistically convergent to a real number x if lim n→∞ 1 n |{k : k ≤ n, | xk − x |≥ }| = 0 for each 0 < , where the vertical bars denote the cardinality of the set which they enclose. Maddox [4] has generalized the notion statistical convergent sequence for locally convex spaces as follows: A sequence (xn) in a locally convex space X which determined by the seminorms (qi)i∈I , is said to be statistical convergent to x ∈ X if: lim n→∞ 1 n |{k : k ≤ n, qi(xn − x) ≥ }| = 0 for each 0 < and i ∈ I. A vector space X with a partial order ≤ is called an ordered vector space if αx + z ≤ αy + z for each z ∈ X whenever x ≤ y, 0 ≤ α ∈ R. An ordered vector space X is called a Riesz space (or vector lattice) if supremum of ∗2000 Mathematics Subject Classification. Primary:40A05,46A40.

ON STATISTICALLY CONVERGENT IN FINITE DIMENSIONAL SPACES

Abstract: In this paper, the notion of statistical convergence, which was introduced by Steinhaus (1951), was studied in Rm ; and some concepts and theorems, whose statistical correspondence for the real number sequences were given, were carried to Rm . In addition, the concepts of the statistical limit point and the statistical cluster point were given and it was mentioned that these two concepts were'nt equal in Fridy's study in 1993. These concepts were given in Rm and the inclusion correlation between these concepts was studied. Key words: Statistical convergence, statistical boundedness, statistically Cauchy sequence, statistical limit point, statistical cluster point. Mathematics Subject Classifications (2000): 40A05 SONLU BOYUTLU UZAYLARDA ISTATISTIKSEL YAKINSAKLIK UZERINE Ozet: Bu makalede Steinhaus (1951) tarafindan verilen istatistiksel yakinsaklik kavrami Rm de incelenip, reel sayi dizileri icin istatistiksel benzerleri verilen bazi kavram ve teoremler R m e tasinmistir. 1993'de Fridy tarafindan yapilan calismada istatistiksel limit noktasi ve istatistiksel yigilma noktasi kavramlari verilip bu iki kavramin birbirine denk olmadigindan bahsedilmistir. Bu kavramlar R m de verilip aralarindaki kapsama bagintisi incelenmistir. Anahtar kelimeler: Istatistiksel Yakinsaklik, Istatistiksel Sinirlilik, Istatistiksel Cauchy Dizisi, Istatistiksel Limit Noktasi, Istatistiksel Yigilma Noktasi.

Generalized vector valued double sequence space using modulus function

In this paper, we introduce a generalized vector valued paranormed double sequence space $ F^{2}(E,p,f,s) $, using modulus function $ f $, where $ p=(p_{nk}) $ is a sequence of non-negative real numbers, $ s\geq 0 $ and the elements are chosen from a seminormed space $ (E, q_{E}) $. Results regarding completeness, normality, $ K_{2} $-space, co-ordinatewise convergence etc. are derived. Further, a study of multiplier sets, ideals, notion of statistical convergence and ($ p_{nk} $ )-Ces\'aro summability in the space $ F^{2}(E,p,f,s) $ is also made.

Statistical convergence in a locally convex space

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.