随机变量序列两种收敛性的一个注记

随机变量序列的依概率收敛必有依分布收敛,反之未必成立.若极限随机变量服从退化分布,则两种收敛等价.自然地,若这两种收敛等价,是否有极限随机变量服从退化分布?给予该问题肯定的回答.     

依概率收敛与依分布收敛的关系

本文探讨了随机变量序列依概率收敛与依分布收敛的关系 ,并给出了一个依分布收敛能保证依概率收敛的最弱的条件 ,即 :设分布函数列 { Fn(x) }弱收敛于连续的分布函数 F(x) ,则存在随机变量序列{ξn}和随机变量ξ,它们分别以 { Fn(x) }和 F(x)为其对应的分布函数列和分布函数 ,且 {ξn}依概率收敛于ξ.     

依概率收敛与依分布收敛的关系

本探讨了随机变量序列依概率收敛与依分布收敛的关系,并给出了一个依分布收敛能保证依概率收敛的最弱的条件,即：设分布函数列{Fn（x）}弱收敛于连续的分布函数F(x),则存在随机变量序列{ξn}和随机变量ξ,它们分别以{Fn（x）}和F（x）为其对应的分布函数和分面函数,且{ξn}依概率收敛于ξ。     

二层随机规划逼近解的收敛性

对二层随机规划的逼近解的收敛性作了探讨,证明了当随机向量序列{ζ(k)(w)}依分布收敛于ζ(w)时,相应于ζ(k)(w)的二层随机规划问题的任何最优解序列将收敛到原问题的最优解.     

多元q-Stancu多项式的收敛性及其收敛阶刻画

定义单纯形上的多元q-Stancu多项式,它是著名的Bernstein多项式,q-Bernstein多项式,Stancu多项式的推广.以多元函数的部分连续模及全连续模为度量,建立推广的多元q-Stancu多项式对连续函数的一致收敛定理与收敛阶估计,并以实例加以验证.     

与依测度收敛有关的若干反例

本文通过构造反例来进一步探究依测度收敛与处处收敛的关系，依测度收敛对乘法、除法、复合运算的封闭性，以及对可测集的可数可加性等性质.     

几种收敛间的关系

归纳总结可测函数列关于一致收敛、近一致收敛、几乎处处收敛、依测度收敛等情况之间在一定前提条件下的关系,反例说明条件的变化将影响结论的正误,从而使收敛及其相互关系更为清晰和透彻.     

关于修正的Lagrange插值多项式

1932年,S.Bernstein以第一类Chebyshev多项式的零点作为插值结点构造了f(x)∈C_(|-1,1)|的次数小于λ_n,1<λ<2,的修正的Lagrange插值多项式Q_n(f,x),证得了当n→∝时Q_n(f,x)在[-1,1]上一致收敛于f(x).本文得到了Bernstein这一结果的点态估计.     

依概率收敛性质探究

主要把数列收敛的一些性质引进到随机变量依概率收敛中来,并加以证明.     

实变函数课程教学中的反例应用

通过构造反例,辅助说明一致收敛和几乎处处收敛、依测度收敛和几乎处处收敛、可测函数和连续函数等概念间的关系,以加深学生对相关知识的理解.     

三角形上Bernstein多项式的导数逼近

本文研究了三角域上的非乘积型Ｂｅｒｎｓｔｅｉｎ多项式的导数逼近函数时的收敛阶估计及其迭代极限．     

The Convergence Rate to the Normal Law of a Certain Variable Defined on Random Polynomials

In this paper, we investigate the convergence rate to the normal law of the distribution of the logarithm of the degree of the splitting field of a random polynomial over a finite field F _q. The exact convergence rate is obtained. A similar result is proved for the distribution of the order of a random permutation.     

随机规划中的一些逼近结果

主要讨论了一类随机规划的目标函数分别在概率测度序列分布收敛、函数序列上图收敛以及随机变量序列均方可积收敛等收敛意义下目标函数序列的收敛情况。基于上述收敛情况给出了一些逼近思想，这些思想可应用于求解这类随机规划问题。     

Interpolation and convergence of Bernstein-Bézier coefficients

In this paper, two ways of the proof are given for the fact that the Bernstein-Bézier coefficients (BB-coefficients) of a multivariate polynomial converge uniformly to the polynomial under repeated degree elevation over the simplex. We show that the partial derivatives of the inverse Bernstein polynomial A _n (g) converge uniformly to the corresponding partial derivatives of g at the rate 1/n. We also consider multivariate interpolation for the BB-coefficients, and provide effective interpolation formulas by using Bernstein polynomials with ridge form which essentially possess the nature of univariate polynomials in computation, and show that Bernstein polynomials with ridge form with least degree can be constructed for interpolation purpose, and thus a computational algorithm is provided correspondingly.     

On the convergence of random polynomials and multilinear forms

We consider different kinds of convergence of homogeneous polynomials and multilinear forms in random variables. We show that for a variety of complex random variables, the almost sure convergence of the polynomial is equivalent to that of the multilinear form, and to the square summability of the coefficients. Also, we present polynomial Khintchine inequalities for complex gaussian and Steinhaus variables. All these results have no analogues in the real case. Moreover, we study the Lp-convergence of random polynomials and derive certain decoupling inequalities without the usual tetrahedral hypothesis. We also consider convergence on “full subspaces” in the sense of Sjögren, both for real and complex random variables, and relate it to domination properties of the polynomial or the multilinear form, establishing a link with the theory of homogeneous polynomials on Banach spaces.     

两两NQD阵列加权和的收敛性质

研究了不同分布行两两NQD阵列加权和的Lp收敛性和完全收敛性，改进了前人的结果．     

两两NQD列的Lp收敛性和完全收敛性

在较宽泛的条件下研究了不同分布两两NQD列加权和的收敛性质,利用矩不等式和截尾方法,获得了一般双下标加权系数的加权部分和的LP收敛性和完全收敛性定理,推广了前人的相应结果.     

行为混合阵列加权和的收敛性

本文研究了行为混合阵列加权和的收敛性.利用混合序列的Rosenthal型最大值不等式,讨论了混合阵列加权和的L1收敛性,依概率收敛性,几乎处处收敛性,及完全收敛性之间的等价关系,推广了行独立随机变量阵列相应的结果.     

SOME RESEARCHES ON WEAK CONVERGENCE OF KERGIN INTERPOLATION

The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted （partial） derivatives integral convergence of Kergin interpolation polynomial for the smooth functions on the unit disk were obtained in the paper. Those generalized Liang＇s main results were acquired in 1998 to the more extensive situation. At the same time, the estimation of convergence rate of Kergin interpolation polynomial is given by means of introducing a new kind of smooth norm.