Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

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.     

Statistical convergence and measure convergence generated by a single statistical measure

The purpose of this paper is to discuss those kinds of statistical convergence,in terms of filter F,or ideal L-convergence,which are equivalent to measure convergence defined by a single statistical measure.We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l_∞/l_∞(I_μ).We also show that the usual sequential convergence is not equivalent to a single measure convergence.     

Statistical approximation properties of q-Bleimann,Butzer and Hahn operators

The main aim of this study is to introduce a new generalization of q-Bleimann, Butzer and Hahn operators and obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical BBH operators. The second aim of this study is to construct a bivariate generalization of the operator and also obtain the statistical approximation properties.     

On lacunary statistical convergence of order α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong N_θ^α(p)-summability. Furthermore, some relations between the spaces N_θ^αS_θ^α are examined.     

Moment convergence in regularized estimation under multiple and mixed-rates asymptotics

In M-estimation under standard asymptotics, the weak convergence combined with the polynomial type large deviation estimate of the associated statistical random field Yoshida (2011) provides us with not only the asymptotic distribution of the associated M-estimator but also the convergence of its moments, the latter playing an important role in theoretical statistics. In this paper, we study the above program for statistical random fields of multiple and also possibly mixedrates type in the sense of Radchenko (2008) where the associated statistical random fields may be nondifferentiable and may fail to be locally asymptotically quadratic. Consequently, a very strong mode of convergence of a wide range of regularized M-estimators is ensured.Our results are applied to regularized estimation of an ergodic diffusion observed at high frequency.     

Restricting statistical convergence

We first introduce a new notion called statistical convergence of order α and primarily show that it gives rise to a decreasing chain of closed linear subspaces of the space of all bounded real sequences with sup norm which never coincides with the class of convergent sequences and in fact their intersection properly contains the class of convergent sequences. We then show that the same method can be applied for double sequences also and introduce the notion of statistical convergence of order (α,β).     

Inclusion results on statistical cluster points

We study the concepts of statistical cluster points and statistical core of a sequence for A _λ methods defined by deleting some rows from a nonnegative regular matrix A. We also relate A _λ-statistical convergence to A _μ-statistical convergence. Finally we give a consistency theorem for A-statistical convergence and deduce a core equality result.     

On convergence rate of a rectangular partition based global optimization algorithm

The convergence rate of a rectangular partition based algorithm is considered. A hyper-rectangle for the subdivision is selected at each step according to a criterion rooted in the statistical models based theory of global optimization; only the objective function values are used to compute the criterion of selection. The convergence rate is analyzed assuming that the objective functions are twice- continuously differentiable and defined on the unit cube in d-dimensional Euclidean space. An asymptotic bound on the convergence rate is established. The results of numerical experiments are included.     

Korovkin-Type Approximation Theorem for Double Sequences of Positive Linear Operators via Statistical A-Summability

In this paper, using the concept of statistical A-summability which is stronger than the A-statistical convergence we provide a Korovkin-type approximation theorem on the space of all continuous real valued functions defined on any compact subset of the real two-dimensional space. We also study the rates of statistical A-summability of positive linear operators.     

A Baskakov type generalization of statistical Korovkin theory

In this paper using the notion of A-statistical convergence, where A is a nonnegative regular summability matrix, we obtain some statistical variants of Baskakov's results on the Korovkin type approximation theorems.     

Statistically boundedness and statistical core of double sequences

The concept of statistical convergence was presented by Steinhaus in 1951. This concept was extended to the double sequences by Mursaleen and Edely in 2003. Throughout this paper we will present multidimensional analogues of the results presented by Fridy and Orhan in 1997. To achieve this goal multidimensional analogues of the definition for bounded statistically sequences, statistical inferior and statistical superior will be presented. In addition to these results we will investigate statistical core for double sequences and study an inequality related to the statistical and P-cores of bounded double sequences.     

Statistical lacunary summability and a Korovkin type approximation theorem

In this paper, we introduce statistical lacunary summability and strongly ?? _q -convergence (0 < q < ??) and establish some relations between lacunary statistical convergence, statistical lacunary summability, and strongly ?? _q -convergence. We further apply our new notion of summability to prove a Korovkin type approximation theorem.     

Moment convergence of <Emphasis Type="Italic">Z</Emphasis>-estimators

The problem to establish the asymptotic distribution of statistical estimators as well as the moment convergence of such estimators has been recognized as an important issue in advanced theories of statistics. This problem has been deeply studied for M-estimators for a wide range of models by many authors. The purpose of this paper is to present an alternative and apparently simple theory to derive the moment convergence of Z-estimators. In the proposed approach the cases of parameters with different rate of convergence can be treated easily and smoothly and any large deviation type inequalities necessary for the same result for M-estimators do not appear in this approach. Applications to the model of i.i.d. observation, Cox’s regression model as well as some diffusion process are discussed.     

Some results related with statistical convergence and Berezin symbols

The concept of discrete statistical Abel convergence is introduced. In terms of Berezin symbols we present necessary and sufficient condition under which a series with bounded sequence {an}_n?0 of complex numbers is discrete statistically Abel convergent. By using concept of statistical convergence we also give slight strengthening of a result of Gokhberg and Krein on compact operators.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Superlinear Convergence of a General Algorithm for the Generalized Foley–Sammon Discriminant Analysis

Linear Discriminant Analysis (LDA) is one of the most efficient statistical approaches for feature extraction and dimension reduction. The generalized Foley–Sammon transform and the trace ratio model are very important in LDA and have received increasing interest. An efficient iterative method has been proposed for the resulting trace ratio optimization problem, which, under a mild assumption, is proved to enjoy both the local quadratic convergence and the global convergence to the global optimal solution (Zhang, L.-H., Liao, L.-Z., Ng, M.K.: SIAM J. Matrix Anal. Appl. 31:1584, 2010). The present paper further investigates the convergence behavior of this iterative method under no assumption. In particular, we prove that the iteration converges superlinearly when the mild assumption is removed. All possible limit points are characterized as a special subset of the global optimal solutions. An illustrative numerical example is also presented.     

Convergence of Metropolis-type algorithms for a large canonical ensemble

In this paper, we study the convergence of Metropolis-type algorithms used in modeling statistical systems with a fluctuating number of particles located in a finite volume. We justify the use of Metropolis algorithms for a particular class of such statistical systems. We prove a theorem on the geometric ergodicity of the Markov process modeling the behavior of an ensemble with a fluctuating number of particles in a finite volume whose interaction is described by a potential bounded below and decreasing according to the law r ^?3?α, α ≥ 0, as r → 0.     

A-statistical relative modular convergence of positive linear operators

In this paper, we investigate the problem of statistical approximation to a function f by means of positive linear operators defined on a modular space. Particularly, in order to get stronger results than the classical aspects we mainly use the concept of statistical convergence. Also, a non-trivial application is presented.     

Discussion of “human life is unlimited-but short” by H Rootzén and D Zholud

The statistical paper human life is short-but unlimited is briefly discussed. The possibilities and limitations of statistical inference concerning very long human life spans are considered. The restricted models of tail distributions that arise from assumption of renormalized convergence of max- or conditional peaks over thresholds-distributions are questioned in the application context of the reviewed paper. The restrictions of natural systems designs on possibilities of extreme life spans, and the potential to adopt modified extreme value models, allowing seasonal variation of death rates, are also pointed out.