Displaying similar documents to “A correction to “Some mean convergence and complete convergence theorems for sequences of m -linearly negative quadrant dependent random variables””

Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages 1 / N n = 1 N U ( f · R ( g ) ) , where U and R are positive operators. We also study the pointwise convergence of the averages 1 / N n = 1 N f ( S x ) g ( R x ) when T and S are measure preserving transformations.

A note on necessary and sufficient conditions for convergence of the finite element method

Kučera, Václav

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In this short note, we present several ideas and observations concerning finite element convergence and the role of the maximum angle condition. Based on previous work, we formulate a hypothesis concerning a necessary condition for O ( h ) convergence and show a simple relation to classical problems in measure theory and differential geometry which could lead to new insights in the area.

On the convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2007)

Annales Polonici Mathematici

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Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates f : X × Ω X defined by f¹(x,ω ) = f(x,ω₁), f n + 1 ( x , ω ) = f ( f ( x , ω ) , ω n + 1 ) .

Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus, Ferenc Móricz (2010)

Studia Mathematica

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We investigate the convergence behavior of the family of double sine integrals of the form 0 0 f ( x , y ) s i n u x s i n v y d x d y , where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals a b a b to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and b j > a j 0 , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial...

Convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2003)

Colloquium Mathematicae

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Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates f : X × Ω X defined by f¹(x,ω) = f(x,ω₁), f n + 1 ( x , ω ) = f ( f ( x , ω ) , ω n + 1 ) , and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Andrew Rosalsky, Yongfeng Wu (2015)

Applications of Mathematics

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Let { X n , j , 1 j m ( n ) , n 1 } be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let 0 < b n . Conditions are given for j = 1 m ( n ) X n , j / b n 0 completely and for max 1 k m ( n ) | j = 1 k X n , j | / b n 0 completely. As an application of these results, we obtain a complete convergence theorem for the row sums j = 1 m ( n ) X n , j * of the dependent bootstrap samples { { X n , j * , 1 j m ( n ) } , n 1 } arising from a sequence of i.i.d. random variables { X n , n 1 } .

Linearized plasticity is the evolutionary Γ -limit of finite plasticity

Alexander Mielke, Ulisse Stefanelli (2013)

Journal of the European Mathematical Society

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We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Γ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula π = n = 0 ( ( 5 n + 3 ) n ! ( 2 n ) ! ) / ( 2 n - 1 ( 3 n + 2 ) ! ) with convergence as 13 . 5 - n , in much the same way as the Euler transformation gives π = n = 0 ( 2 n + 1 n ! n ! ) / ( 2 n + 1 ) ! with convergence as 2 - n . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

Regular statistical convergence of double sequences

Ferenc Móricz (2005)

Colloquium Mathematicae

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The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence x j k : ( j , k ) ² is said to be regularly statistically convergent if (i) the double sequence x j k is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence x j k : k is statistically convergent to some ξ j for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence x j k : j is statistically convergent...

Limit theorems for sums of dependent random vectors in R d

Andrzej Kłopotowski

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CONTENTSIntroduction.......................................................................................................................................................................... 5 I. Infinitely divisible probability measures on R d ....................................................................................... 6 II. The classical limit theorems for sums of independent random vectors................................................ 14 III. Convergence in law to ℒ ( a ,...

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) k = 1 a k s i n k x be given with nonnegative coefficients a k . If a k is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that k a k 0 as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) k = 1 l = 1 c k l s i n k x s i n l y , even with complex coefficients c k l . We also...

A-Statistical Convergence of Subsequence of Double Sequences

Harry I. Miller (2007)

Bollettino dell'Unione Matematica Italiana

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The concept of statistical convergence of a sequence was first introduced by H. Fast [7] in 1951. Recently, in the literature, the concept of statistical convergence of double sequences has been studied. The main result in this paper is a theorem that gives meaning to the statement: s = s i j converges statistically A to L if and only if "most" of the "subsequences" of s converge to L in the ordinary sense. The results presented here are analogue of theorems in [12], [13] and [6] and are concerned...

On the rate of convergence of the Bézier-type operators

Grażyna Anioł (2006)

Bollettino dell'Unione Matematica Italiana

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For bounded functions f on an interval I , in particular, for functions of bounded p-th power variation on I there is estimated the rate of pointwise convergence of the Bezier-type modification of the discrete Feller operators. In the main theorem the Chanturiya modulus of variation is used.

Orthogonal series estimation of band-limited regression functions

Waldemar Popiński (2014)

Applicationes Mathematicae

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The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions s k , k = 0,±1,..., for the observation model y j = f ( u j ) + η j , j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, u j are independent random variables uniformly distributed in the observation interval [-T,T], η j are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions...

Convergence of Taylor series in Fock spaces

Haiying Li (2014)

Studia Mathematica

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It is well known that the Taylor series of every function in the Fock space F α p converges in norm when 1 < p < ∞. It is also known that this is no longer true when p = 1. In this note we consider the case 0 < p < 1 and show that the Taylor series of functions in F α p do not necessarily converge “in norm”.

Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

Boaz Tsaban, Lubomyr Zdomsky (2012)

Journal of the European Mathematical Society

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A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel’skii α 1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C p ( X ) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our...

The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

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We show that if a decreasing sequence of subharmonic functions converges to a function in W l o c 1 , 2 then the convergence is in W l o c 1 , 2 .

Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality

Mark A. Peletier, Robert Planqué, Matthias Röger (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We derive a new criterion for a real-valued function u to be in the Sobolev space W 1 , 2 ( n ) . This criterion consists of comparing the value of a functional f ( u ) with the values of the same functional applied to convolutions of u with a Dirac sequence. The difference of these values converges to zero as the convolutions approach u , and we prove that the rate of convergence to zero is connected to regularity: u W 1 , 2 if and only if the convergence is sufficiently fast. We finally apply our criterium to...