Statistical Convergence Of Sequences Research Articles

Some further studies on strong ℐλ-statistical convergence in probabilistic metric spaces

Abstract We prove some basic properties of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence of sequences in probabilistic metric spaces and introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical cluster point. We also introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences in probabilistic metric spaces. Further, we establish a connection between strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence and strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences.

Lacunary ℐ -Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

The concepts of invariant convergence, invariant statistical convergence, lacunary invariant convergence, and lacunary invariant statistical convergence for set sequences were introduced by Pancaroğlu and Nuray (2013). We know that ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the lacunary ℐ -invariant convergence of sequence of sets in intuitionistic fuzzy metric spaces (briefly, IFMS). In this study, we examine the notions of lacunary ℐ -invariant convergence W ℐ σ θ η , ν (Wijsman sense), lacunary ℐ ∗ -invariant convergence W ℐ σ θ ∗ η , ν (Wijsman sense), and q -strongly lacunary invariant convergence W N σ θ η , ν q (Wijsman sense) of sequences of sets in IFMS. Also, we give the relationships among Wijsman lacunary invariant convergence, W N σ θ η , ν q , W ℐ σ θ η , ν , and W ℐ σ θ ∗ η , ν in IFMS. Furthermore, we define the concepts of W ℐ σ θ η , ν -Cauchy sequence and W ℐ σ θ ∗ η , ν -Cauchy sequence of sets in IFMS. Furthermore, we obtain some features of the new type of convergences in IFMS.

Ideal convergence of sequences in neutrosophic normed spaces

Statistical convergence of sequences has been studied in neutrosophic normed spaces (NNS) by Kirişci and Şimşek [39]. Ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the ideal convergence in NNS. In this paper, we study the concept of ideal convergence and ideal Cauchy for sequences in NNS.

Strongly Statistical Convergence

We introduce a concept of A-strongly statistical convergence for sequences of complex numbers, where A = (ank)n,k∈ℕ is an infinite matrix with nonnegative entries. A sequence (xn) is called strongly convergent to L if $$ \underset{n\to \infty }{\lim }{\sum}_{k=1}^{\infty }{a}_{nk}\left|{x}_k\right.-L\left|=0\right. $$ in the ordinary sense. In the definition of A-strongly statistical limit, we use the statistical limit instead of the ordinary limit with a convenient density. We study some densities and show that the (ank)-strongly statistical limit is an ( $$ {a}_{m_nk} $$ )-strong limit, where the density of the set {mn ∈ ℕ : n ∈ ℕ} is equal to 1. We introduce the notion of dense positivity for nonnegative sequences. A nonnegative sequence (rn) is dense positive provided the limit superior of a subsequence ( $$ {r}_{m_n} $$ ) is positive for all (mn) with density equal to 1. We show that the dense positivity of (rn) is a necessary and sufficient condition for the uniqueness of A-strongly statistical limit, where A = (ank) and rn = $$ {\sum}_{k=1}^{\infty }{a}_{nk} $$ . Furthermore, necessary conditions for the regularity, linearity and multiplicativity of the A-strongly statistical limit are established.

(Δm, f)-Statistical convergence for sequences of fuzzy numbers

(Δm, f)-Statistical convergence for sequences of fuzzy numbers

Quasi-Almost Lacunary Statistical Convergence of Sequences of Sets

In this study, we defined concepts of Wijsman quasi-almost lacunary convergence, Wijsman quasi-strongly almost lacunary convergence and Wijsman quasi q-strongly almost lacunary convergence. Also we give the concept of Wijsman quasi-almost lacunary statistically convergence. Then, we study relationships among these concepts. Furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too.

Triple rough statistical convergence of sequence of Bernstein operators

Triple rough statistical convergence of sequence of Bernstein operators

Density by moduli and Wijsman lacunary statistical convergence of sequences of sets

The main object of this paper is to introduce and study a new concept of f-Wijsman lacunary statistical convergence of sequences of sets, where f is an unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence with respect to a modulus for sequences of sets and it is shown that, under certain conditions on a modulus f, the concepts of Wijsman lacunary strong convergence with respect to a modulus f and f-Wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which mathit{WS}_{theta}^{f} = mathit{WS}^{f}, where mathit{WS}_{theta}^{f} and mathit{WS}^{f} denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively.

Lacunary statistical convergence of sequences of functions in intuitionistic fuzzy normed space

In this paper we study lacunary statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces. We define concept of lacunary statistical pointwise convergence and lacunary...

Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejer on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.

Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

The purpose of this work is to investigate types of convergence of sequences of functions in intuitionistic fuzzy normed spaces and some properties related with these concepts.

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.

Statistical convergence of sequences of fuzzy numbers and sequences of α−cuts

In this short paper, we show that the statistical convergence of a sequence of fuzzy numbers with respect to the supremum metric is equivalent to the uniform statistical convergence of the sequences of functions which are defined via the endpoints of α-cuts of the same fuzzy numbers sequence. We introduce the concept of levelwise statistical convergence and present the analogous relation between levelwise statistical convergence of a sequence of fuzzy numbers and pointwise statistical convergence of the sequences of endpoint functions.