Asymptotic behavior of posterior distributions for random processes under incorrect models

In this paper, the asymptotic behavior of posterior distributions on parameters contained in random processes is examined when the specified model for the densities is not necessarily correct. Uniform convergence of likelihood functions in some way is shown to be a sufficient condition for the posterior distributions to be asymptotically confined to a set (Theorem 1). For ergodic stationary Markov processes uniform convergence of likelihood functions is established by the ergodic theorem for Banach-valued stationary processes (Proposition 1). A sufficient condition for the uniform convergence is also shown for general random processes (Proposition 2). These results are used to analyze the asymptotic behavior of posterior distributions on parameters contained in linear systems under incorrect models (Example 1 and 2).     

Almost uniform convergence

The almost uniform convergence is between uniform and quasi-uniform one. We give some necessary and sufficient conditions under which the almost uniform convergence coincides on compact sets with uniform, quasi-uniform or uniform convergence, respectively. In the second section continuity of almost uniform limits is considered. Finally we characterize the set of all points at which a net of functions is almost uniformly convergent to a given function.     

A note on almost sure uniform and complete convergences of a sequence of random variables

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.     

Convergence of Polynomial Level Sets

In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.

     

函数列在不同区间上一致收敛性的研究

函数列的一致收敛性与所讨论的区间有关.在区间的子区间或不同的区间上,函数列的一致收敛性表现如何呢?通过几个命题和几个实例,并利用几何画板可以帮助我们辨别函数列在不同区间上的一致收敛性.     

有界闭区间上连续函数列一致收敛的充要条件

判别函数列一致收敛的方法有函数列一致收敛定义、Cauchy一致收敛准则、limn→∞supx∈D|fn(x)-f(x)|=0及Dini定理,本文由函数列的等度连续性,可得出几个有界闭区间上连续函数列一致收敛的充要条件,推广了Dini定理.     

Nyström method for Cauchy singular integral equations with negative index

In this paper, the authors propose a Nyström method to approximate the solutions of Cauchy singular integral equations with constant coefficients having a negative index. They consider the equations in spaces of continuous functions with weighted uniform norm. They prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates.     

Isoinertial family of operators and convergence of KdV cnoidal waves to solitons

In this paper we show the convergence of Korteweg-de Vries cnoidal waves to the limit soliton. It is proved that the convergence is uniform and in H²-norm, as the period of the solutions tends to infinity. Families of Hill operators are also studied. We obtain a condition under which families of operators are isoinertial. This condition is satisfied for classes of Hill operators that are obtained by linearization. Our application is to the family of linearized operators at the KdV cnoidal waves. It is proved that this family is isoinertial and also the value of the inertial index is calculated.     

About the statistical uniform convergence

In this work we study the concept of statistical uniform convergence. We generalize some results of uniform convergence in double sequences to the case of statistical convergence. We also prove a basic matrix theorem with statistical convergence.     

GLIVENKO-CANTELLI TYPE THEOREMS ANDTHEIR APPLICATIONS

1IntroductionLet(S,yi)beameasurablespacealldletPbeaprobabilitymeasureon(S,yi).I,et{Xi,i21}bei.i.d.S--valuedrandolllvariableswitllcommonlawP,Pc,istheempiricalmeasurebasedonXI,'',X,,.OneofthelllostpersistentproblemsinprobabilityandstatisticsisthatoffindingoutthelawPfi,onltherandomlawP,,.Thea.s.convergenceofPntoPuniformlyoverclassesoffunctionsisasubjectofparticularinterest,thefundamentalresultisthecelebratedGlivenko-Cantellitlleorm.TheworkofVapnikandC.,..u..ki,[5](1971)vitalizestilestudy…     

Convergence analysis of sectional methods for solving breakage population balance equations-I: the fixed pivot technique

In this work we study the convergence of the fixed pivot techniques (Kumar and Ramkrishna Chem. Eng. Sci. 51, 1311–1332, 1996) for breakage problems. In particular, the convergence is investigated on four different types of uniform and non-uniform meshes. It is shown that the fixed pivot technique is second order convergent on a uniform and non-uniform smooth meshes. Furthermore, it gives first order convergence on a locally uniform mesh. Finally the analysis shows that the method does not converge on a non-uniform random mesh. The mathematical results of convergence analysis are also validated numerically.     

Numerical solution of cauchy-type integral equations of index −1 by collocation methods

This paper describes a collocation method for solving constant coefficient Cauchy-type singular integral equations of index –1. A technique for reducing the set of linear equations resulting from collocation to match the number of unknows is described. The uniform convergence analysis of the resulting method is presented and convergence rates based on the smoothness of the data are given.This work was partially supported by CNPq under grant #300105/88-6(RN).     

The best uniform convergence rate of linear empirical bayes estimators

In this paper the thought on the uniform convergence of an empirical Bayes estimator or linear empirical Bayes (l.e.B.) estimator is advanced. Under two different models the l.e.B. estimators of the parameter are constructed respectively. It is proved that the convergence rates and uniform convergence rates of these l.e.B. estimators are all one with respect to the corresponding prior families. It is shown that the uniform convergence rate one is the best, under mild assumptions imposed on the conditional density of the sample.     

Asymptotic properties of conditional quantile estimator for censored dependent observations

In this paper, we establish strong uniform convergence and asymptotic normality of the conditional quantile estimator for the censorship model when the data exhibit some kind of dependence. It is assumed that the observations form a stationary α-mixing sequence. The strong uniform convergence in iid framework has recently been discussed by Ould-Saïd (Stat Probab Lett 76:579–586, 2006). As a by-product, we also obtain a uniform weak convergence rate for the product-limit estimator of the lifetime and censoring distributions under dependence, which is interesting independently.     

On the uniform convergence of a finite difference scheme for time dependent singularly perturbed reaction-diffusion problems

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction-diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method.     

多维参数线性经验Bayes估计一致收敛速度的最优性

证明出任何一个多维参数性经验Ｂａｙｅｓ估计的一致收敛速度不可能超过１，从而说明文［１］中构造的线性经验Ｂａｙｅｓ估计的一致收敛速度１是最优的。     

On conditional weak convergence

In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.     

关于概率赋范线性空间上的线性算子的一致收敛

本文引入了概率赋范线性空间上线性算子的一致收敛和可完全刻划这种收敛的算子间的概率距离概念,并利用这些概念获得了算子连续和算子列一致收敛的本质特征,及其连续性和全连续性对于一致收敛极限运算的封闭性.     

关于Dini-Lipschitz型逼近定理的统一处理方法方法

This survey paper studies the approximation of (polynomial) processes for which the operator norms do not form a bounded sequence. In view of familiar direct estimates and quantitative uniform boundedness principles, a unified approach is given to results concerning the equivalence of Dini-Lipschitz-type conditions with (strong) convergence on (smoothness) classes. Emphasis is laid upon the necessity of these conditions, essential ingredients of the proofs are suitable modifications of the familiar gliding hump method. Apart from the classical results concerned with Fourier partial sums, explicit applications are treated for (trigonometric as well as algebralc) Lagrange interpolation, interpolatory quadrature rules based upon Jacobl knots, multipliers or strong convergence, and for Bochner-Riesz means of multivariate Fourier series for parameter values below the critical index.     

Some uniform convergence results for kernel estimators

This paper derives some uniform convergence rates for kernel regression of some index functions that may depend on infinite dimensional parameter. The rates of convergence are computed for independent, strongly mixing and weakly dependent data respectively. These results extend the existing literature and are useful for the derivation of large sample properties of the estimators in some semiparametric and nonparametric models.