Conditional independence,conditional mixing and conditional association

Some properties of conditionally independent random variables are studied. Conditional versions of generalized Borel-Cantelli lemma, generalized Kolmogorov’s inequality and generalized Hájek-Rényi inequality are proved. As applications, a conditional version of the strong law of large numbers for conditionally independent random variables and a conditional version of the Kolmogorov’s strong law of large numbers for conditionally independent random variables with identical conditional distributions are obtained. The notions of conditional strong mixing and conditional association for a sequence of random variables are introduced. Some covariance inequalities and a central limit theorem for such sequences are mentioned.     

《概率论与数理统计》中有关独立性一反例的深入分析

独立性是《概率论与数理统计》是的一个非常重要的概念.教学中在说明随机变量函数独立性时会涉及许多反例.本文就有关随机变量函数独立性的一个反例作了进一步的推广分析.     

条件期望与三变量独立性的两个充要条件

随机事件的独立性、随机变量的独立性是概率统计中的重要概念,不少学者都在这方面有所讨论.本文作者讨论了三维连续型随机变量（X,Y,Z）中三个分量X,Y,Z的相互独立性、条件独立性,得到三个引理.利用条件期望及三个引理作者给出了三变量相互独立的两个充要条件.     

离散型随机变量的几种独立性及相互关系

随机变量的独立性在概率论中有着十分重要的意义.本文给出了离散型随机变量与离散型随机向量相互独立的概念,条件独立的概念,以及几种独立性的相互关系.     

双变量参数联合函数的条件均值和条件方差(英文)

联合函数是指连接单变量边际分布的多变量函数.联合函数由Sklar(1959)在概率测度空间的内容时引入的.本文主要对边际分布是标准正态分布函数U(0,1)的Farlie-Gum-bel-Morgenstern和Gumbel-Hougaard这两个双变量参数联合函数进行研究,我们得到了他们密度函数的基本性质并导出了他们的条件均值和条件方差.另外,本文还给出了不同参数的条件均值和条件方差的相应图示,并进行了对比和解释.     

On Bounds for the Concentration Function II

We derive an upper bound for the concentration of the sum of i.i.d. random variables with values in by appealing to functions of positive type and the structure theory of set addition.     

On the asymptotic normality of sequences of weak dependent random variables

The aim of this paper is to investigate the asymptotic normality for strong mixing sequences of random variables in the absense of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences. The class of applications include filters of certain Gaussian sequences.Supported in part by an NSF grant, cost-sharing from the University of Cincinnati, and a Taft research grant.     

On Normal Approximation for Strongly Mixing Random Variables

In this paper, we estimate the difference , where Z _n is the sum of n centered and normalized random variables (without the stationarity assumption) satisfying the strong mixing condition, N is a standard normal random variable, and h:ℝ→ℝ is a Lipschitz function. In particular cases, the obtained upper bounds are of order O(n ^−1/2). The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-10/06.     

随机变量的和、积、商分布的一种新表示

应用Feller提出的点-集函数并结合二元copula,对二元连续型正值随机变量的和、积、商的分布进行了研究,得到了和、积、商分布的一种新的计算方法.最后给出一个应用实例.     

Symmetric sum and symmetric product of two independent random variables

The symmetry of the sum and product of two independent random variablesX andY, whenX orY is not symmetric, is studied.     

On computing distributions of products of non-negative independent random variables

We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative independent random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.     

A Strong Law for Mixing Random Variables

A sharp almost sure bound is derived for limit points of average sum of weakly dependent random variables, which ensures strong laws of large numbers for and -mixing random variables, without assumptions on rate of tending to zero of and -mixing parameters _n and _n.     

Bivariate gamma distributions, sums and ratios

Exact distributions of R = X +Y and W = X/(X +Y ) and the corresponding moment properties are derived when X and Y follow five flexible bivariate gamma distributions. The expressions turn out to involve several special functions.     

Extending Mosaic Displays: Marginal,Conditional, and Partial Views of Categorical Data

Abstract

This article first illustrates the use of mosaic displays for the analysis of multiway contingency tables. We then introduce several extensions of mosaic displays designed to integrate graphical methods for categorical data with those used for quantitative data. The scatterplot matrix shows all pairwise (bivariate marginal) views of a set of variables in a coherent display. One analog for categorical data is a matrix of mosaic displays showing some aspect of the bivariate relation between all pairs of variables. The simplest case shows the bivariate marginal relation for each pair of variables. Another case shows the conditional relation between each pair, with all other variables partialled out. For quantitative data this represents (a) a visualization of the conditional independence relations studied by graphical models, and (b) a generalization of partial residual plots. The conditioning plot, or coplot shows a collection of partial views of several quantitative variables, conditioned by the values of one or more other variables. A direct analog of the coplot for categorical data is an array of mosaic plots of the dependence among two or more variables, stratified by the values of one or more given variables. Each such panel then shows the partial associations among the foreground variables; the collection of such plots shows how these associations change as the given variables vary.     

Association of probability measures on partially ordered spaces

A notion of association of probability measures on partially ordered (Polish) spaces is introduced and its basic properties are investigated. This generalizes the classical notion of association among random variables, due to Esary, Proschan, and Walkup. The relation between association and monotone stochastic kernels is investigated and a theorem of Jogdeo is generalized. The general theory is applied to stochastic processes with both discrete and continuous time parameter and partially ordered state spaces. Also, an application to mixtures of statistical experiments is included.     

p-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions

In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.     

随机变量函数概率密度的定点算法

利用曲线积分和曲面积分作为工具,导出计算随机变量函数的密度函数的一种定点算法,并借助实例说明相应计算公式的应用.     

随机变量和的模函数分布及其应用

介绍了一个在计算机科学、信息科学等学科中具有广泛应用的随机变量和的模函数,计算了其分布,并提供了该函数在图像信息安全领域的一个应用例子,验证了理论结果.     

The Stein--Tikhomirov Method and a Nonclassical Central Limit Theorem

We present a simplified version of the Stein--Tikhomirov method realized by defining a certain operator in the class of twice differentiable characteristic functions. Using this method, we establish a criterion for the validity of a nonclassical central limit theorem in terms of characteristic functions.