Conditional limit theorems for conditionally linearly negative quadrant dependent random variables

From the ordinary notion of linearly negative quadrant dependence for a sequence of random variables, a new concept called conditionally linearly negative quadrant dependence is introduced. The relation between the two kinds of dependence is answered by examples, that is, the linearly negative quadrant dependence does not imply the conditionally linearly negative quadrant dependence, and vice versa. The fundamental properties of conditionally linearly negative quadrant dependence are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional exponential inequalities, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established.     

Conditional versions of limit theorems for conditionally associated random variables

Relation between association and conditional association is answered, several examples show that the association of random variables does not imply the conditional association, and vice versa. Several fundamental properties of conditional associated random variables are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional Hájek-Rényi type inequalities, a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established, which are conditional versions of the earlier results for associated random variables, respectively. In addition, some lemmas in the context are of independent interest.     

END随机变量的概率不等式及其应用(英文)

Some probability inequalities are established for extended negatively dependent(END) random variables. The inequalities extend some corresponding ones for negatively associated random variables and negatively orthant dependent random variables. By using these probability inequalities, we further study the complete convergence for END random variables. We also obtain the convergence rate O(n-1/2ln1/2n) for the strong law of large numbers, which generalizes and improves the corresponding ones for some known results.     

渐近负相关随机域强定律的收敛率

In this paper,a notion of negative side ρ-mixing (ρ--mixing) which can be regarded as asymptotic negative association is defined,and some Rosenthal type inequalities for ρ--mixing random fields are established. The complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are also discussed for ρ--mixing random fields. The results obtained extend those for negatively associated sequences and ρ*-mixing random fields.     

Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications

In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As applications of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability, we derive some results on L _p convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.     

NA随机变量列的有界重对数律

通过建立NA随机变量最大部分和的一些概率指数不等式，给出了具有不同分布的NA随机变量列有界重对数律的一些结果，因此推广了由R．Wittmann建立的独立随机变量的相关结果。     

Exponential Inequalities and Complete Convergence for Extended Negatively Dependent Random Variables

Some exponential inequalities and complete convergence are established for extended negatively dependent（END） random variables. The inequalities extend and improve the results of Kim and Kim（On the exponential inequality for negative dependent sequence. Communications of the Korean Mathematical Society, 2007, 22（2）： 315-321） and Nooghabi and Azarnoosh（Exponential inequality for negatively associated random variables. Statisti- cal Papers, 2009, 50 （2）： 419-428）. We also obtain the convergence rate O（n-1/2 In1/2 n） for the strong law of large numbers, which improves the corresponding ones of Kim and Kim, and Nooghabi and Azarnoosh.     

任意B值随机变量序列关于条件期望的一类强极限定理

利用随机变量的截尾方法和条件三级数定理,研究任意B值随机变量序列的极限性质,得到了一类关于条件期望的强极限定理和鞅差序列收敛定理,推广了与此相应的一些结果和若干经典的强大数定律.     

On the Kolmogorov inequalities for quadratic forms of dependent uniformly bounded random variables

Some Kolmogorov probability inequalities for quadratic forms and weighted quadratic forms of negative superadditive dependent (NSD) uniformly bounded random variables are provided. Using these inequalities, some complete convergence of randomized quadratic forms under some suitable conditions are evaluated. Moreover, various examples are presented in which the given conditions of our results are satisfied.     

Some conditional results for conditionally strong mixing sequences of random variables

The relation between strong mixing and conditionally strong mixing is answered by examples, that is, the strong mixing property of random variables does not imply the conditionally strong mixing property, and the opposite implication is also not true. Some equivalent definitions and basic properties of conditional strong mixing random variables are derived, and several conditional covariance inequalities are obtained. By means of these properties and conditional covariance inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established, which is a conditional version of the earlier result under non-conditional case.     

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.     

Equivalent Conditions of Complete Convergence and Complete Moment Convergence for END Random Variables

In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables.     

混合序列部分和的若干收敛性质

本文研究了混合序列部分和的若干收敛性质.利用Serfling不等式推广情形,证明了一类随机变量序列部分和的一个收敛性结果,获得了混合序列部分和的收敛性,并进一步得到了混合序列加权和的强收敛性和完全收敛性,推广并改进了文[2]中有关结果.     

Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables

In this paper,the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables are investigated.Some sufficient conditions for the convergence are provided.In addition,the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables is obtained.The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

Exponential inequalities for associated random variables and strong laws of large numbers

Some exponential inequalities for partial sums of associated random variables are established. These inequalities improve the corresponding results obtained by Ioannides and Roussas (1999), and Oliveira (2005). As application, some strong laws of large numbers are given. For the case of geometrically decreasing covariances, we obtain the rate of convergence n-1/2(log log n)1/2(logn) which is close to the optimal achievable convergence rate for independent random variables under an iterated logarithm, while Ioannides and Roussas (1999), and Oliveira (2005) only got n-1/3(logn)2/3 and n-1/3(logn)5/3, separately.     

Convergence rates in the strong laws of asymptotically negatively associated random fields

In this paper, a notion of negative side p-mixing (p -mixing) which can be regardedas asymptotic negative association is defined, and some Rosenthal type inequalities for p -mix-ing random fields are established. The complete convergence and almost sure summability onthe convergence rates with respect to the strong law of large numbers are also discussed for p--mixing random fields. The results obtained extend those for negatively associated sequences andp“ -mixing random fields.     

混合阵列的收敛性

该文引入了混合阵列的概念,讨论了混合阵列的完全收敛性与依概率收敛性.所得结果,推广了行独立随机变量阵列相应的结果.此外还得到了一般随机变量阵列的完全收敛性与依概率收敛性.     

Properties of uniform consistency of the kernel estimators of density and regression functions under dependence assumptions

The paper contains exponential inequalities for dependent random variables. As a measure of dependence we use φand ρ-mixing coefficients, the last one being based on the maximal coefficient of correlation. These results allow us to study the problem of uniform strong convergence for the kernel estimators of a density and for a kernel predictor for stochastic processes. Our uniform consistency theorems extend some known results     

m-NA随机变量部分和的若干强收敛性（英文）

In this paper, by the three series theorem of m-negatively associated(m-NA,in short) random variables and the truncation method of random variables, we mainly investigated the strong convergence properties for partial sums of m-NA random variables.In addition, the Khintchine-Kolmogorov convergence theorem and Kolmogorov-type strong law of large numbers for m-NA random variables are also obtained. The results obtained in the paper generalize some corresponding ones for independent random variables and some dependent random variables.     

ρ混合阵列的收敛性

该文引入了ρ～混合阵列的概念，讨论了ρ～混合阵列的完全收敛性与依概率收敛性. 所得结果，推广了行独立随机变量阵列相应的结果. 此外还得到了一般随机变量阵列的完全收敛性与依概率收敛性.