Conditions For Weak Convergence Research Articles

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ be the empirical process associated to an ℝ d -valued stationary process (X i ) i≥0. In the present paper, we introduce very general conditions for weak convergence of $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ , which only involve properties of processes (f(X i )) i≥0 for a restricted class of functions $f\in\mathcal{G}$ . Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications. The central interest in our approach is that it does not need the indicator functions which define the empirical process $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ to belong to the class $\mathcal{G}$ . This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011). Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).

Directional Secant-Type Methods for Solving Equations

A semilocal convergence analysis for directional Secant-type methods in multidimensional space is provided. Using weaker hypotheses than the ones exploited by An and Bai, we provide a semilocal convergence analysis with the following advantages: weaker convergence conditions, larger convergence domain, finer error estimates on the distances involved, and more precise information on the location of the solution. A numerical example, where our results apply to solve an equation but not the ones of An and Bai, is also provided. In a second example, we show how to implement the method.

Majorizing sequences for iterative procedures in Banach spaces

Newton-like methods are often used for solving nonlinear equations. In the present paper, we introduce very general majorizing sequences for Newton-like methods. Then, we provide semi-local convergence results for these methods. The new convergence results can be weaker than in earlier studies. These new results are illustrated by several numerical examples and special cases of Newton-like methods, for which the older convergence conditions do not hold but for which our weaker convergence conditions are satisfied.

Weak convergence conditions for Inexact Newton-type methods

We establish a new semilocal convergence results for Inexact Newton-type methods for approximating a locally unique solution of a nonlinear equation in a Banach spaces setting. We show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works [15–31]. Special cases and numerical examples are also provided in this study.

ON THE CONVERGENCE OF INEXACT TWO-STEP NEWTON-TYPE METHODS USING RECURRENT FUNCTIONS

We approximate a locally unique solution of a nonlinear equa- tion in a Banach space setting using an inexact two{step Newton{type method. It turn out that under our new idea of recurrent functions, our semilocal analysis provides tighter error bounds than before, and in many interesting cases, weaker sucient convergence conditions. Appli- cations including the solution of nonlinear Chandrasekhar{type integral equations appearing in radiative transfer and two point boundary value problems are also provided in this study.

A semilocal convergence analysis for directional Newton methods

A semilocal convergence analysis for directional Newton methods in n-variables is provided in this study. Using weaker hypotheses than in the elegant related work by Y. Levin and A. Ben-Israel and introducing the center-Lipschitz condition we provide under the same computational cost as in Levin and Ben-Israel a semilocal convergence analysis with the following advantages: weaker convergence conditions; larger convergence domain; finer error estimates on the distances involved, and an at least as precise information on the location of the zero of the function. A numerical example where our results apply to solve an equation but not the ones in Levin and Ben-Israel is also provided in this study.

Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.

New cooperative projection neural network for nonlinearly constrained variational inequality

This paper proposes a new cooperative projection neural network (CPNN), which combines automatically three individual neural network models with a common projection term. As a special case, the proposed CPNN can include three recent recurrent neural networks for solving monotone variational inequality problems with limit or linear constraints, respectively. Under the monotonicity condition of the corresponding Lagrangian mapping, the proposed CPNN is theoretically guaranteed to solve monotone variational inequality problems and a class of nonmonotone variational inequality problems with linear and nonlinear constraints. Unlike the extended projection neural network, the proposed CPNN has no limitation on the initial point for global convergence. Compared with other related cooperative neural networks and numerical optimization algorithms, the proposed CPNN has a low computational complexity and requires weak convergence conditions. An application in real-time grasping force optimization and examples demonstrate good performance of the proposed CPNN.

Solving equations using Newton's method under weak conditions on Banach spaces with a convergence structure

We provide new semilocal results for Newton's method on Banach spaces with a convergence structure. Using more precise majorizing sequence we show that, under weaker convergence conditions than before, we can obtain finer error bounds on the distances involved and a more precise information on the location of the solution.

General Study of Iterative Processes of R-Order at Least Three under Weak Convergence Conditions

We consider a family of Newton-type iterative processes solving nonlinear equations in Banach spaces, that generalizes the usually iterative methods of R-order at least three. The convergence of this family in Banach spaces is usually studied when the second derivative of the operator involved is Lipschitz continuous and bounded. In this paper, we relax the first condition, assuming that ?F?(x)?F?(y)?≤?(?x?y?), where ? is a nondecreasing continuous real function. We prove that the different R-orders of convergence that we can obtain depend on the quasihomogeneity of the function ?. We end the paper by applying the study to some nonlinear integral equations.

Goodness‐of‐fit Procedures for Copula Models Based on the Probability Integral Transformation

Abstract. Wang & Wells [J. Amer. Statist. Assoc. 95 (2000) 62] describe a non‐parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [J. Amer. Statist. Assoc. 88 (1993) 1034] and later studied by Barbe et al. [J. Multivariate Anal. 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its L2‐norm. This paper shows how to compute asymptotic p‐values for various goodness‐of‐fit test statistics based on a non‐truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness‐of‐fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih [Biometrika 85 (1998) 189] for the gamma frailty family.

Limit theory of generalized order statistics

Generalized order statistics (gos) were introduced by Kamps [1995. A Concept of Generalized Order Statistics. Teubner, Stuttgart] to unify several models of ordered random variables (rv's), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos that includes oos and sos, the possible limit distribution functions (df's) of the maximum gos are obtained in Nasri-Roudsari [1996. Extreme value theory of generalized order statistics. J. Statist. Plann. Inference 55, 281–297]. In this paper, for this subclass, necessary and sufficient conditions of weak convergence, as well as the form of the possible limit df's of extreme, intermediate and central gos are derived. These results are extended to a wider subclass.

Random Broken Lines That Weakly Converge to a Fractional Ornstein-Uhlenbeck Process

We study the behavior of sequences of continuous random broken lines that are constructed from increased sums of stationary sequences. We obtain conditions of weak convergence of these random broken lines to Gaussian processes of types of a fractional Ornstein-Uhlenbeck process and the fractional Brownian motion. Sequences of moving averages are considered as the main examples of the investigated stationary sequences. Bibliography: 8 titles.

Stability and Convergence of Finite-Element Approximation Schemes for Harmonic Maps

This article discusses stability and convergence of approximation schemes for harmonic maps. A finite-element discretization of an iterative algorithm due to F. Alouges is introduced and shown to be stable and convergent in general only on acute-type triangulations. An\break a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement. Numerical\break experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency.

수준 연속인 퍼지 랜덤 변수의 가중 합에 대한 약 수렴성

이 논문에서는 퍼지 랜덤 변수의 합에 대한 약한 대수의 법칙을 일반화로서, 컴팩트 일양 적분 가능한 수준 연속 퍼지 랜덤 변수의 가중 합이 약 수렴하기 위한 동치 조건을 구하였다. The present paper establishes a necessary and sufficient condition for weak convergence for weighted sums of compactly uniformly integrable level-continuous fuzzy random variables as a generalization of weak laws of large numbers for sums of fuzzy random variables.

Weak sufficient convergence conditions and applications for newton methods

The famous Newton—Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton—Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton— Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.

Normal and Poisson Convergencies

This paper provides necessary and sufficient conditions for weak convergence of the distributions of sums of independent random variables to normal and Poisson distributions. The conditions are given in terms of the first through fourth moments of the truncated components and some condition guaranteeing the asymptotic equality between the distribution of the sums mentioned above and the distribution of the sums of the truncated components. The theorems proved in this work are similar to the well-known criteria for normal and Poisson convergencies.

Necessary and sufficient conditions for weak convergence of smoothed empirical processes

Let X 1,…, X n be a sequence of i.i.d. random variables with common distribution P on the real line. Assuming that P has a smooth density, we construct a histogram based estimator P n, H and establish weak convergence of the empirical process n (P n,H−P)(f) f∈ F under sharp conditions. If F is a class of indicators of sets, then the conditions imposed are necessary and sufficient.

Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups

We give three equivalent conditions for weak convergence of almost orbits of an asymptotically nonexpansive commutative semigroup acting on a nonempty bounded closed convex subset of a uniformly convex Banach space whose dual has the Kadec property.

Limit Processes with Independent Increments for the Ewens Sampling Formula

The Ewens sampling formula in population genetics can be viewed as a probability measure on the group of permutations of a finite set of integers. Functional limit theory for processes defined through partial sums of dependent variables with respect to the Ewens sampling formula is developed. Techniques from probabilistic number theory are used to establish necessary and sufficient conditions for weak convergence of the associated dependent process to a process with independent increments. Not many results on the necessity part are known in the literature.