Weak Convergence Results Research Articles

An asymptotic plate model for magneto-electro-thermo-elastic sensors and actuators

We present an asymptotic two-dimensional plate model for linear magneto-electro-thermo-elastic sensors and actuators, under the hypotheses of anisotropy and homogeneity. Four different boundary conditions pertaining to electromagnetic quantities are considered, leading to four different models: the sensor–actuator model, the actuator–sensor model, the actuator model and the sensor model. We validate the obtained two-dimensional models by proving weak convergence results. Each of the four plate problems turns out to be decoupled into a flexural problem, involving the transversal displacement of the plate, and a certain partially or totally coupled membrane problem.

Mixed iterative algorithms for the multiple-set split equality common fixed-point problems without prior knowledge of operator norms

Let , , be real Hilbert spaces, let , be two bounded linear operators. The multiple-set split equality common fixed-point problem (MSECFP) under consideration in this paper is to(Section.Display) where are integers, and are quasi-nonexpansive mappings with nonempty fixed-point sets. Note that, the above problem (1) allows asymmetric and partial relations between the variables and . If and , then the MSECFP (1) reduces to the multiple-set split common fixed-point problem proposed by Censor et al. In this paper, we introduce mixed cyclic and simultaneous iterative algorithms for the MSECFP (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. Weak convergence results are given.

Another look at the split common fixed point problem for demicontractive operators

In this article, we take a look at the recent results of Moudafi (Inverse Probl 26:587–600, 2010), Tang et al. (Math Model Anal 17:457–466, 2012) and Wang and Cui (Math Model Anal 18:537–542, 2013), which weak convergence results were obtained for the split common fixed point problem for demicontractive mappings. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings and then prove strong convergence of the sequence in real Hilbert spaces. We also apply our results to the split common null point problem in real Hilbert spaces. Finally, we give numerical results to demonstrate its convergence.

Joint asymptotics for semi-nonparametric regression models with partially linear structure

We consider a joint asymptotic framework for studying semi-nonparametric regression models where (finite-dimensional) Euclidean parameters and (infinite-dimensional) functional parameters are both of interest. The class of models in consideration share a partially linear structure and are estimated in two general contexts: (i) quasi-likelihood and (ii) true likelihood. We first show that the Euclidean estimator and (pointwise) functional estimator, which are re-scaled at different rates, jointly converge to a zero-mean Gaussian vector. This weak convergence result reveals a surprising joint asymptotics phenomenon: these two estimators are asymptotically independent. A major goal of this paper is to gain first-hand insights into the above phenomenon. Moreover, a likelihood ratio testing is proposed for a set of joint local hypotheses, where a new version of the Wilks phenomenon [Ann. Math. Stat. 9 (1938) 60-62; Ann. Statist. 1 (2001) 153-193] is unveiled. A novel technical tool, called a joint Bahadur representation, is developed for studying these joint asymptotics results.

A mean-variance control framework for platoon control problems: Weak convergence results and applications on reduction of complexity

This paper introduces a new approach of treating platoon systems using meanvariance control formulation. The underlying system is a controlled switching diffusion in which the random switching process is a continuous-time Markov chain. This switching process is used to represent random environment and other random factors that cannot be given by stochastic differential equations driven by a Brownian motion. The state space of the Markov chain is large in our setup, which renders practically infeasible a straightforward implementation of the mean-variance control strategy obtained in the literature. By partitioning the states of the Markov chain into sub-groups (or clusters) and then aggregating the states of each cluster as a super state, we are able to obtain a limit system of much reduced complexity. The justification of the limit system is rigorously supported by establishing certain weak convergence results.

Near equivalence on metric spaces and a nonstandard central limit theorem

This article proves a nonstandard Central Limit Theorem (CLT) in the sense of Nelson’s Radically Elementary Probability Theory. The CLT proved here is obtained by establishing the near equivalence of standardized averages obtained from L2 IID random variables to the standardized average resulting from a binomial CLT. A nonstandard model for near equivalence on metric spaces replaces conventional results of weak convergence. Statements and proofs remain radically elementary without applying the full Internal Set Theory. A nonstandard notion of normality is discussed.

Relaxed conditions for convergence of batch BPAP for feedforward neural networks

In this paper, the deterministic convergence of the batch back-propagation algorithm with penalty (BPAP) is proved under certain relaxed conditions for the activation function, the learning rate and the stationary point set of the error function. Both weak and strong convergence results are established. The boundedness of the weights in the training procedure is also proved in a simple and clear way. As a result, the usual requirements on the boundedness of the weights to guarantee the convergence are removed. Simulation results for an approximation problem are presented to support our theoretical findings.

A queueing model with independent arrivals, and its fluid and diffusion limits

We study a queueing model with ordered arrivals, which can be called the \(\varDelta _{(i)}/GI/1\) queue. Here, customers from a fixed, finite, population independently sample a time to arrive from some given distribution \(F\), and enter the queue in order of the sampled arrival times. Thus, the arrival times are order statistics, and the inter-arrival times are differences of consecutive order statistics. They are served by a single server with independent and identically distributed service times, with general service distribution \(G\). The discrete event model is analytically intractable. Thus, we develop fluid and diffusion limits for the performance metrics of the queue. The fluid limit of the queue length is observed to be a reflection of a ‘fluid netput’ process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge process or ‘diffusion netput’ process. The diffusion limit can be seen as being reflected through the directional derivative of the Skorokhod regulator of the fluid netput process in the direction of the diffusion netput process. We also observe what may be interpreted as a sample path Little’s Law. Sample path analysis reveals various operating regimes where the diffusion limit switches between a free diffusion, a reflected diffusion process, and the zero process, with possible discontinuities during regime switches. The weak convergence results are established in the \(M_1\) topology.

On Estimation of Conditional Distribution Function under Dependent Random Right Censored Data

In this article we study simple integral-type estimator of distribution function under random right censored observations at fixed covariate values, where the dependence between a life time and a censoring variable may expressed by a given Archimedean copula. We prove an almost sure asymptotic representation which provides a key tool for obtaining weak convergence result for estimator.

An Exponential Integrator Scheme for Time Discretization of Nonlinear Stochastic Wave Equation

This work is devoted to convergence analysis of an exponential integrator scheme for semi-discretization in time of nonlinear stochastic wave equation. A unified framework is first set forth, which covers important cases of additive and multiplicative noises. Within this framework, the proposed scheme is shown to converge uniformly in the strong $L^p$-sense with precise convergence rates given. The abstract results are then applied to several concrete examples. Further, weak convergence rates of the scheme are examined for the case of additive noise. To analyze the weak error for the nonlinear case, techniques based on the Malliavin calculus were usually exploited in the literature. Under certain appropriate assumptions on the nonlinearity, this paper provides a weak error analysis, which does not rely on the Malliavin calculus. The rates of weak convergence can, as expected, be improved in comparison with the strong rates. Both strong and weak convergence results obtained here show that the proposed method achieves higher convergence rates than the implicit Euler and Crank-Nicolson time discretizations. Numerical results are finally reported to confirm our theoretical findings.

Logarithmic Quantile Estimation for Rank Statistics

We prove an almost sure weak limit theorem for simple linear rank statistics for samples with continuous distributions functions. As a corollary, the result extends to samples with ties and to the vector version of an almost sure (a.s.) central limit theorem for vectors of linear rank statistics. Moreover, we derive such a weak convergence result for some quadratic forms. These results are then applied to quantile estimation, and to hypothesis testing for nonparametric statistical designs, here demonstrated by the c-sample problem, where the samples may be dependent. In general, the method is known to be comparable to the bootstrap and other nonparametric methods (Thangavelu 2005; Fridline 2009), and we confirm this finding for the c-sample problem.

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (Xn,ηn) on Z+×S, where Z+ is the non-negative integers and S is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of Xn, and that, roughly speaking, ηn is close to being Markov when Xn is large. This departure from much of the literature, which assumes that ηn is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for Xn given ηn. We give a recurrence classification in terms of increment moment parameters for Xn and the stationary distribution for the large- X limit of ηn. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between Xn (rescaled) and ηn. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on Z+ (the case where S is a singleton). Motivation arises from modulated queues or processes with hidden variables where ηn tracks an internal state of the system.

Stochastic Liénard Equations with Random Switching and Two-time Scales

This article is devoted to the study of stochastic Liénard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Liénard equation and the discrete component is a Markov chain with a finite state space that is large. A distinct feature is that the processes under consideration are time inhomogeneous. Based on the idea of nearly decomposability and aggregation, the state space of the switching process can be viewed as “nearly decomposable” into l subspaces that are connected with weak interactions among the subspaces. Using the idea of aggregation, we lump the states in each subspace into a single state. Considering the pair of process (continuous state, discrete state), under suitable conditions, we derive a weak convergence result by means of martingale problem formulation. The significance of the limit process is that it is substantially simpler than that of the original system. Thus, it can be used in the approximation and computation work to reduce the computational complexity.

Fluctuation analysis for the loss from default

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated firm-by-firm default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy and computational efficiency of the approximation.

Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms

Let , and be real Hilbert spaces, let and be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Display where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that, by taking and , we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated.

Weighted approximations for Studentized $U$-statistics

In this article, we employ the jackknife method of estimation and the concept of Studentized $U$-statistics to derive a new weak convergence result for nondegenerate $U$-statistics on the space $D[0,1]$. We drop the classical condition that the second moment of the kernel of the underlying $U$-statistic exists and derive a weighted weak convergence result for these Studentized statistics. This weak convergence is concluded from a weighted approximation in sup-norm $\|/q\|$, in probability, of Studentized $U$-statistics by partial sums of i.i.d. projections, where $q$ is in an appropriate class of positive weight functions.

Convergence of Infinite Family of Multivalued Quasi-Nonexpansive Mappings Using Multistep Iterative Processes

We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of Zhang et al. (2013), Bunyawat and Suantai (2012), and some others from finite family, one countable family, and two countable families tok-number of countable families of multivalued quasi-nonexpansive mappings. Also we used a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

Almost sure asymptotics for extremes of non-stationary Gaussian random fields

In this paper, the authors prove an almost sure limit theorem for the maxima of non-stationary Gaussian random fields under some mild conditions related to the covariance functions of the Gaussian fields. As the by-products, the authors also obtain several weak convergence results which extended the existing results.

Approximating Solution Of ∈ T(x) for an H-Monotone Operator in Hilbert Spaces

The purpose of this paper is to study the solution of 0 ∈ T(x) for an H-monotone operator introduced in [Fang and Huang, Appl. Math. Comput. 145(2003)795-803] in Hilbert spaces, which is the first proposal of it's kind. Some strong and weak convergence results are presented and the relations between maximal monotone operators and H-monotone operators are analyzed. Simultaneously, we apply these results to the minimization problem for T = ∂f and provide some numerical examples to support the theoretical findings.

A Central Limit Theorem for a Nonparametric Maximum Conditional Hazard Rate Estimator in Presence of Right Censoring

In this paper, we consider a non-parametric kernel type estimator of the time where a hazard rate function is maximum in the presence of covariate and right censoring. Via a strong representation of the estimator, we establish weak convergence and asymptotic normality results.