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Let λ = (λn)n≥1be a nondecreasing sequence of positive numbers tending to infinity such that λ1 = 1 and λn+1 ≥ λn + 1 for all n, and let In = [n -- λn + 1; n] for n = 1; 2;,.... Then for any given nonzero sequence µ, we define by Δ+(µ) the operator that generalizes the operator of the First difference and is defined by Δ+(µ)xk = µk(xk -- xk+1). In this article, for any given integer r ≤ 1, we deal with the λ+r(µ)-statistical convergence that generalizes in a certain sense the well-known λE-rstatistical convergence, main results consist in determining sets of sequences χ and χ' of the form sξ0 satisfying χ ⊂ [V; λ]0(Δ+r(µ)) ⊂ χ' and sets κ and κ' of the form sξ satisfying χ ⊂ [V, λ]∞(Δ+r(µ)) ≥ κ'. This study is justified since the infinite matrix associated with the operator Δ+r(µ) cannot be explicitly calculated for all r. [ABSTRACT FROM AUTHOR]

We study the rate of convergence in the strong law of large numbers expressed in terms of complete convergence of Baum-Katz type for sequences of random variables satisfying Petrov's condition. [ABSTRACT FROM AUTHOR]

We study the growth at the golden rotation number ω = ( 5 − 1 ) / 2 of the function sequence P n ( ω ) = ∏ r = 1 n | 2 sin ⁡ π r ω | . This sequence has been variously studied elsewhere as a skew product of sines, Birkhoff sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence P n , namely the sub-sequence Q n = | ∏ r = 1 F n 2 sin ⁡ π r ω | for Fibonacci numbers F n , and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of P n ( ω ) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (2011) [10] . [ABSTRACT FROM AUTHOR]

We investigate a multi-index analogue of unconditional convergence theorem of Tandori. Namely, we formulate a condition regarding a multiindex sequence of numbers ( a) which ensures that for all orthonormal fields ( $${\Phi_n}$$ ) the series $${\sum_{n} a_n\Phi_n}$$ is unconditionally almost everywhere convergent. We consider convergence in Pringsheim's and in regular sense. Our condition is in a way optimal. [ABSTRACT FROM AUTHOR]

Letbe a sequence of independent and identically distributedB-valued random variables, and set. Let,andq>0, and putWe strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that, if and only ifand [ABSTRACT FROM PUBLISHER]

Let { X, n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L-convergence theorem and Marcinkiewicz-Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables. [ABSTRACT FROM AUTHOR]

In the paper, we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As an application, a strong law of large numbers for martingale difference sequence is obtained. [ABSTRACT FROM AUTHOR]

A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem. Our result generalizes and improves on the corresponding one in Li and Yang (2008). We also obtain a complete convergence result for an array of rowwise pairwise NQD random variables. [ABSTRACT FROM PUBLISHER]

Let $${\mathcal{F}}$$ be a separable uniformly bounded family of measurable functions on a standard measurable space $${(X, \mathcal{X})}$$, and let $${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$$ be the smallest number of $${\varepsilon}$$ -brackets in L( μ) needed to cover $${\mathcal{F}}$$. The following are equivalent: It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures. [ABSTRACT FROM AUTHOR]

We give a new characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space endowed with the level convergence topology. Our results imply that some previous compactness criteria are wrong. A counterexample also is given to validate this judgment. [ABSTRACT FROM AUTHOR]

Abstract: The notation means the finite sum of all numbers as ranges over the integers that divide including and . The purpose of this paper is to define a theory of summability suggested by this type of sum and an associated theory of convergence. This paper studies five sequence spaces that are related to the arithmetical summability thus defined. [Copyright &y& Elsevier]

Abstract: We give a constructive analysis of learning as it arises in various computational interpretations of classical Peano Arithmetic, such as Aschieri and Berardi learning based realizability, Avigad’s update procedures and epsilon substitution method. In particular, we show how to compute in Gödel’s system upper bounds on the length of learning processes, which are themselves represented in through learning based realizability. The result is achieved by the introduction of a new non standard model of Gödel’s , whose new basic objects are pairs of non standard natural numbers (convergent sequences of natural numbers) and moduli of convergence, where the latter are objects giving constructive information about the former. As a foundational corollary, we obtain that that learning based realizability is a constructive interpretation of Heyting Arithmetic plus excluded middle over formulas (for which it was designed) and of all Peano Arithmetic when combined with Gödel’s double negation translation. As a byproduct of our approach, we also obtain a new proof of Avigad’s theorem for update procedures and thus of the termination of the epsilon substitution method for . [Copyright &y& Elsevier]

In this paper we introduce the concepts of double lacunary strongly convergence and double lacunary statistical convergence of double interval numbers. We prove some inclusion relations and study some of their properties. [ABSTRACT FROM AUTHOR]

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years. For an admissible ideal ∮ ⊂ ℔ × ℔, the aim of the present paper is to introduce the concepts of ∮-convergence and ∮-convergence for double sequences on probabilistic normed spaces (PN spaces for short). We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide. We also define ∮ -Cauchy and ∮-Cauchy double sequences on PN spaces and show that ∮ -convergent double sequences are ∮-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general. [ABSTRACT FROM AUTHOR]

By the inequalities established in this article, we obtain the convergence rate of strong law of large numbers for positively associated sequences. The results derived extend and improve the corresponding ones in Vronskii (1999). [ABSTRACT FROM AUTHOR]

We obtain the following very fast sequences convergent to Euler-Mascheroni constant: θn = Hn-log (n +1/2+1/24n-1/48n² + 23/5760n³ + 17/3840n4 - 10099/2903040n5) and φn = Hn-log (n +1/2+1/24(n + 1/2 ) -- 37/5760(n + 1/2 )³ + 10313/2903040(n+1/2)5), where Hn are the harmonic numbers defined by Hn = 1+1/2+1/3+ ... + 1/n. subjclass [2000] : Primary 11Y60; Secondary 40A05. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

We study the complete convergence and complete moment convergence for martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for martingale difference sequence is obtained. Our results generalize the corresponding ones of Stoica (2007, 2011). [ABSTRACT FROM AUTHOR]

Let {Xni,i ≥ 1, n ≥ 1} be an array of rowwise asymptotically almost negatively associated (AANA, in short) random variables. The complete convergence for weighted sums of arrays of rowwise AANA random variables is studied, which complements and improves the corresponding result of Baek et al. (2008). As applications, the Baum and Katz type result for arrays of rowwise AANA random variables and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of AANA random variables are obtained. [ABSTRACT FROM AUTHOR]

Abstract: Let be a mapping from to , satisfying that for and , is the th largest value (median value) of the numbers . In [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775–2781], we conjectured that for , if there exists such that is locally finitely convergent with respect to on , then is finitely convergent with respect to . In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true. [Copyright &y& Elsevier]

Abstract: We prove a Baum–Katz–Nagaev type rate of convergence in the Marcinkiewicz–Zygmund and Kolmogorov strong laws of large numbers for norm bounded martingale difference sequences. [Copyright &y& Elsevier]

In this article we introduce the fuzzy real valued double sequence spaces 2ℓF (p) where p = (Pnk) is a double sequence of bounded strictly positive numbers. We study their different properties like completeness , solidness , symmetricity, convergence free etc. We prove some inclusion results also. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for . This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity. [Copyright &y& Elsevier]

We consider the classical Kolmogorov condition for strong law of large numbers for sequences of dependent random variables; the so-called φ-mixing and Rademacher-Menchoff condition for ρ-mixing sequences. [ABSTRACT FROM AUTHOR]

Abstract: We consider the problem of approximating the global minimum of an -times continuously differentiable function on the unit interval, based on sequentially chosen function and derivative evaluations. Using a probability model based on the -fold integrated Wiener measure, we establish a lower bound on the expected number of function evaluations required to approximate the minimum to within on average. [Copyright &y& Elsevier]

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean. [ABSTRACT FROM AUTHOR]

Abstract: Analyzing several classical tests for convergence/divergence of number series, we relax the monotonicity assumption for the sequence of terms of the series. We verify the sharpness of the obtained results on corresponding classes of sequences and functions. [ABSTRACT FROM AUTHOR]

Abstract: We investigate a one-parameter family of infinite generalised continued fractions. The fractions converge to rational values which can be explicitly evaluated. The sequences of numerators and denominators are already to be found in the Online Encyclopedia of Integer Sequences with reference to various topics – but not to continued fractions. [Copyright &y& Elsevier]

Abstract: The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier–Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 155 (2004) 81–161] for Maxwell molecules. [Copyright &y& Elsevier]

We prove that, for any sequence of positive real numbers $(g_n)_{n ge 1}$ satisfying $g_n ge 1$ for $n ge 1$ and $lim _{n to + infty } g_n = + infty $, for any real number $theta $ in $[0, 1]$ and any irrational real number $xi $, there exists an increasing sequence of positive integers $(a_n)_{n ge 1}$ satisfying $a_n le n g_n$ for $n ge 1$ and such that the sequence of fractional parts $({a_n xi })_{n ge 1}$ tends to $theta $ as $n$ tends to infinity. This result is best possible in the sense that the condition $lim _{n to + infty } g_n = + infty $ cannot be weakened, as recently proved by Dubickas. [ABSTRACT FROM AUTHOR]

In this article, we introduce the concepts of strongly statistically convergent sequence and strong statistically Cauchy sequence in a probabilistic metric (PM) space endowed with the strong topology, and establish some basic facts. Next, we define the strong statistical limit points and the strong statistical cluster points of a sequence in this space and investigate the relations between these concepts. [ABSTRACT FROM AUTHOR]

Abstract: We prove that certain classes of sequences of positive real numbers satisfy some selection principles related to a special kind of convergence. [Copyright &y& Elsevier]