Statistical Convergence Of Order Research Articles

Some Results of Generalized Weighted Norlund-Euler-<img src=image/13428701_01.gif> Statistical Convergence in Non-Archimedean Fields

Non-Archimedean analysis is the study of fields that satisfy the stronger triangular inequality, also known as ultrametric property. The theory of summability has many uses throughout analysis and applied mathematics. The origin of summability methods developed with the study of convergent and divergent series by Euler, Gauss, Cauchy and Abel. There is a good number of special methods of summability such as Abel, Borel, Euler, Taylor, Norlund, Hausdroff in classical Analysis. Norlund, Euler, Taylor and weighted mean methods in Non-Archimedan Analysis have been investigated in detail by Natarajan and Srinivasan. Schoenberg developed some basic properties of statistical convergence and also studied the concept as a summability method. The relationship between the summability theory and statistical convergence has been introduced by Schoenberg. The concept of weighted statistical convergence and its relations of statistical summability were developed by Karakaya and Chishti. Srinivasan introduced some summability methods namely y-method, Norlund method and Weighted mean method in p-adic Fields. The main objective of this work is to explore some important results on statistical convergence and its related concepts in Non-Archimedean fields using summability methods. In this article, Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence, generalized weighted summability using Norlund-Euler-<img src=image/13428701_02.gif> method in an Ultrametric field are defined. The relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and Statistical Norlund-Euler-<img src=image/13428701_02.gif> summability has been extended to non-Archidemean fields. The notion of Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence and inclusion results of Norlund-Euler statistical convergent sequence has been characterized. Further the relation between Norlund-Euler-<img src=image/13428701_02.gif> statistical convergence of order α & β has been established.

∆λ−statistical boundedness of order α on time scales

In this study, we introduce the notions ∆λ−statistical convergence of order α (for αϵ(0,1]) and λp−summable of order α (for αϵ(0,1]) on an arbitrary time scale. Moreover, some relations about these notions are obtained. We define ∆λ− statistically boundedness of order α (for αϵ(0,1]) on a time scale. Furthermore, We give connections between S (λ,α) T (b) , S (β,θ) T (b) and ST (b) for various sequences µ∆λ(t) and µ∆β(t) are determined in class Λ.

(∆mv , f)-lacunary statistical convergence of order α

In this paper, we define the space Sαθ (∆mv, f) of all (∆mv, f)-lacunary statistical convergent sequences of order α with the help of unbounded modulus function f, lacunary sequence (θ), generalized difference operator ∆ mv and real number α ∈ (0, 1]. We also introduce the space ωαθ (∆mv, f) of all strong (∆mv, f)-lacunary summable sequences of order α. Properties related to these spaces are studied. Inclusion relations between spaces Sαθ (∆mv, f) and ωα θ (∆mv, f) are established under certain conditions.

On (f, ρ)−Statistical convergence and strong (f, ρ)−summability of order α

Abstract The main object of this article is to introduce the concepts of (f, ρ)− statistical convergence of order α and strong (f, ρ)− summability of order α of sequences of real numbers and give some inclusion relations between these spaces.

Applications of deferred Cesàro statistical convergence of sequences of fuzzy numbers of order (ξ, ω)

The purpose of this article is to study deferred Cesrào statistical convergence of order (ξ, ω) associated with a modulus function involving the concept of difference sequences of fuzzy numbers. The study reveals that the statistical convergence of these newly formed sequence spaces behave well for ξ ≤ ω and convergence is not possible for ξ &gt; ω. We also define p-deferred Cesàro summability and establish several interesting results. In addition, we provide some examples which explain the validity of the theoretical results and the effectiveness of constructed sequence spaces. Finally, with the help of MATLAB software, we examine that if the sequence of fuzzy numbers is bounded and deferred Cesàro statistical convergent of order (ξ, ω) in (Δ, F, f), then it need not be strongly p-deferred Cesàro summable of order (ξ, ω) in general for 0 &lt; ξ ≤ ω ≤ 1.

Statistical Convergence of Order α for Complex Uncertain Sequences

In this paper, we introduce convergence concepts namely, statistical convergence of order [Formula: see text], statistical convergence of order [Formula: see text] almost surely, statistical convergence of order [Formula: see text] in measure, statistical convergence of order [Formula: see text] in mean, statistical convergence of order [Formula: see text] in distribution in complex uncertain theory. We also investigate some relationships among them.

Inscription on statistical convergence of order α

In this article we recall a remarkable result stated as "For a fixed ?, 0 &lt; ? ? 1, the set of all bounded statistically convergent sequences of order ? is a closed linear subspace of m (m is the set of all bounded real sequences endowed with the sup norm)" by Bhunia et al. (Acta Math. Hungar. 130 (1-2) (2012), 153-161) and to develop the objective of this perception we demonstrate that the set of all bounded statistically convergent sequences of order ? may not form a closed subspace in other sequence spaces. Also we determine two different sequence spaces in which the set of all statistically convergent sequences of order ? (irrespective of boundedness) forms a closed set.

Applications of statistical convergence of order (η, δ + γ) in difference sequence spaces of fuzzy numbers

In this article, we introduce and study some difference sequence spaces of fuzzy numbers by making use of λ-statistical convergence of order (η, δ + γ) . With the aid of MATLAB software, it appears that the statistical convergence of order (η, δ + γ) is well defined every time when (δ + γ) &gt; η and this convergence fails when (δ + γ) &lt; η. Moreover, we try to set up relations between (Δv, λ)-statistical convergence of order (η, δ + γ) and strongly (Δv, p, λ)-Cesàro summability of order (η, δ + γ) and give some compelling instances to show that the converse of these relations is not valid. In addition to the above results, we also graphically exhibits that if a sequence of fuzzy numbers is bounded and statistically convergent of order (η, δ + γ) in (Δv, λ), then it need not be strongly (Δv, p, λ)-Cesàro summable of order (η, δ + γ).

Statistical Convergence of order β for (λ, μ) double sequences of fuzzy numbers

In this study, we introduce the concepts of φλ,μ-double statistically convergence of order β in fuzzy sequences and strongly λ- double Cesaro summable of order β for sequences of fuzzy numbers. Also we give some inclusion theorems.

(A,φ)-lacunary statistical convergence of order α

In the present paper, we introduce and study (A,?) -statistical convergence of order ?, using the ?-function, infinite matrix and we establish some inclusion theorems.

Lacunary statistical convergence of order $$\alpha, \beta$$ for generalized vector-valued difference sequence spaces

In this paper, we defined space $$S_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q)$$ of all vector-valued lacunary $$\Delta _v^m$$ -statistical convergent sequences of order $$(\alpha , \beta )$$ and space $$N_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q,p)$$ of all vector-valued strongly $$\Delta _v^m$$ -lacunary summable sequences of order $$(\alpha ,\beta )$$ by taking sequence $$(E_k, q_k)$$ of normed spaces, where p is a positive real number and $$\alpha ,\beta$$ are real numbers with $$0<\alpha \leqslant \beta \leqslant 1$$ . Some inclusion relations between these spaces are obtained. We also studied space $$\omega _\theta ^{\alpha ,\beta } (\Delta _v^m,f,E,q,p)$$ of all vector-valued strongly $$\Delta _v^m$$ -lacunary summable sequences of order $$(\alpha , \beta )$$ with respect to modulus function f by taking a bounded sequence $$(p_k)$$ of strictly positive real numbers with $$\displaystyle \inf _k p_k>0$$ . The inclusion relations between spaces $$\omega _\theta ^{\alpha ,\beta } (\Delta _v^m,f,E,q,p)$$ and $$S_\theta ^{\alpha ,\beta } (\Delta _v^m,E,q)$$ are determined.

On S(λ,A)-convergence and strong (V,λ,A)-convergence of order α

In the paper [B. de Malafosse and V. Rakocevic, Linear Algebra Appl. 420, no. 2-3, (2007), 377--387], authors defined the concept of (λ,A)-statistical convergence. In this paper, the concept of (λ,A)-statistical convergence is generalized to (λ,A)-statistical convergence of order α. Also, we introduce the concept of strong (V,λ,A)-convergence of order α and give some inclusion relations between the concepts of (λ,A)-statistical convergence of order α and strong (V,λ,A)-convergence of order α.

Statistical convergence of order β in fuzzy normed linear spaces

Statistical convergence of order β in fuzzy normed linear spaces

On the Statistical Convergence of Order α in Paranormed Space

The aim of the present work is to introduce notions of statistical convergence, strongly p-Cesàro summability and the statistically Cauchy sequence of order α in paranormed spaces. Some certain topological properties of these new concepts are examined. Furthermore, we introduce the some inclusion relations among them.

Korovkin type approximation theorems proved viaweighted αβ-equistatistical convergence for bivariate functions

Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted ??-statistical convergence of order ?, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted ??-statistical convergence of order ?, which is an extension of ??-statistical convergence of order ?. Our definition, with sk = 1, for all k, reduces to ??-statistical convergence of order ?. Moreover, we use this definition of weighted ??-statistical convergence of order ?, to prove Korovkin type approximation theorems via, weighted ??-equistatistical convergence of order ? and weighted ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Also we prove Korovkin type approximation theorems via ??-equistatistical convergence of order ? and ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted ??-equistatistical convergence of order ? is introduced and discussed.

A notion of αβ-statistical convergence of order γ in probability

A sequence of real numbers {xn}n_N is said to be ɑβ-statistically convergent of order (where 0 < ᵞ ≤ 1) to a real number x [1] if for every b > 0, lim n͢1 1 (βn - ɑn + 1) |{k ∈ [ɑn, βn] : |xk - x| ≥ b}| = 0, where {ɑn}n2N and {βn}n2N are two sequences of positive real numbers such that {ɑn}n2N and {βn}n2N are both non-decreasing, βn ≥ ɑn for all n ∈ N, (βn -ɑn) → ∞ as n → ∞. In this paper we study a related concept of convergences in which the value xk is replaced by P(|Xk - X| ≥ e) and E(|Xk - X|r) respectively (where X,Xk are random variables for each k ∈ N, e > 0, P denotes the probability, and E denotes the expectation) and we call them ɑβ -statistical convergence of order in probability and ɑβ-statistical convergence of order in rth expectation respectively. The results are applied to build the probability distribution for ɑβ -strong p-Cesàro summability of order in probability and ɑβ -statistical convergence of order in distribution. So our main objective is to interpret a relational behaviour of above mentioned four convergences. We give a condition under which a sequence of random variables will converge to a unique limit under two different (ɑ, β) sequences and this is also use to prove that if this condition violates then the limit value of ɑβ - statistical convergence of order in probability of a sequence of random variables for two different (ɑ, β) sequences may not be equal.

Δmp - statistical convergence of order α

In this work, we generalize the concepts of statistically convergent sequence of order ? and statistical Cauchy sequence of order ? by using the generalized difference operator ?m. We prove that a sequence is ?mp-statistically convergent of order ? if and only if it is ?mp-statistically Cauchy of order ?.

(λ,μ) Statistical Convergence of Order γ for Double Sequences of Functions

In this present work, we investigate the concepts of Δr - statistical convergence of order [see formula in PDF] for double sequences of real valued functions. Also some definitions we given for Δr - statistical Cauchy sequence for double sequences of real valued functions and some relations between [see formula in PDF] statistical convergence and strong [see formula in PDF]-summability are given.

New approach to the concept of aᶥ -statistical convergence with the number of alpha

Following a very recent approach, we generalize recently introduced summability method, namely, I-statisticalconvergence. We use an infinite matrix of complex numbers and an a number to do this generalization. We call this new convergence as A^I statistical convergence of order a with respect to a sequence of modulus functions. We also define two new spaces by using strong convergence and Ces·ro summability and after giving these descriptions, we investigate their relationship and we obtain some result

Some Seminormed Difference Sequence Spaces over n-Normed Spaces Defined by a Musielak-Orlicz Function of Order (α,β)

We first define the notion of lacunary statistical convergence of order (α,β), and taking this notion into consideration, we introduce some seminormed difference sequence spaces over n-normed spaces with the help of Musielak-Orlicz function M=(Mk) of order (α,β). We also examine some topological properties and prove inclusion relations between the resulting sequence spaces.