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.     

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.     

NA随机变量序列的Egorov型强大数律的等价条件

本文研究了NA随机变量的Egorov型强大数律.利用NA随机变量的概率不等式,得到了NA随机变量序列的Egorov型强大数律的一些等价条件,所获结果推广和改进了在独立随机变量序列的Egorov的结果和在NA随机变量序列已有的一些结果.     

Some inequalities and limit theorems under sublinear expectations

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob’s inequality for submartingale and Kolmogrov’s inequality. By Kolmogrov’s inequality, we obtain a special version of Kolmogrov’s law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.     

The essential equivalence of pairwise and mutual conditional independence

For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent — i.e., equivalent to up to null sets. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our result relies on an iterated extension of Bledsoe and Morse's completion of the product of two measure spaces. Part of this work was done while Peter Hammond was visiting the National University of Singapore in March–April 2004. The final version was completed while Yeneng Sun was on sabbatical leave at the University of Illinois at Urbana–Champaign and Stanford University in October 2004 – May 2005.     

The von Bahr–Esseen moment inequality for pairwise independent random variables and applications

In the paper, we generalize the von Bahr–Esseen moment inequality from independent random variables to pairwise independent random variables. As the applications, the moment convergence, the complete convergence and the strong law of large numbers are established for pairwise independent random variables.     

Deviation inequalities for an estimator of the conditional value-at-risk

In this paper, we present a deviation inequality for a common estimator of the conditional value-at-risk for bounded random variables. The result improves a deviation inequality which is obtained by Brown [D.B. Brown, Large deviations bounds for estimating conditional value-at-risk, Operations Research Letters 35 (2007) 722-730].     

ρ~-混合序列的H～jeck-Rènyi型不等式及其应用

利用ρ-混合序列的Rosenthal型最大值不等式,得到了ρ-混合随机变量序列的Hájeck-Rènyi型不等式,三级数定理和Chung型强大数律,所得结果达到了独立时一致的结果.     

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.     

可列非齐次马氏链泛函的一类强大数定律

本文用分析方法研究可列非齐次马氏链的泛函的极限性质，得到了一类不同形式的强大数定律，在其中通常形式的强大数定律中的数学期望被条件期望所代替，关于独立随机变量序列的若干经典强大数定律是本文结果的推论．     

An exponential inequality and strong limit theorems for conditional expectations

An exponential inequality for the tail of the conditional expectation of sums of centered independent random variables is obtained. This inequality is applied to prove analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers for conditional expectations. As corollaries we obtain certain strong theorems for the generalized allocation scheme and for the nonuniformly distributed allocation scheme.     

马氏环境中马氏链的一个概率不等式及其应用

本文研究了马氏环境中马氏链构成的随机变量之和的概率不等式问题.利用了结尾的方法,获得了马氏环境中马氏链构成的随机变量之和的尾部概率不等式,作为结果的应用,给出了将过程限制在(S,S∩F,PS)上的强大数定律.文中提出的方法和结果对研究独立的随机变量之和的大样本性质是十分有用的.     

Almost sure Cesàro and Euler summability of sequences of dependent random variables

For a sequence of real random variables C _α-summability is shown under conditions on the variances of weighted sums, comprehending and sharpening strong laws of large numbers (SLLN) of Rademacher-Menchoff and Cramér-Leadbetter, respectively. Further an analogue of Kolmogorov’s criterion for the SLNN is established for E _α-summability under moment and multiplicativity conditions of 4th order, which allows one to weaken Chow’s independence assumption for identically distributed square integrable random variables. The simple tool is a composition of Cesàro-type and of Euler summability methods, respectively. Received: 12 June 2006, Revised: 14 May 2007     

对称随机变量序列的Hájek-Rényi型不等式和强大数律

得到了对称随机变量序列的Hájek-Rényi型不等式,并利用它研究了对称随机变量序列的强大数律.     

Equivalent conditions of complete convergence for m-dimensional products of iid random variables and application to strong law of large numbers

Under very weak condition 0 × r(f) ↑ ∞, t→ ∞, we obtain a series of equivalent conditions of complete convergence for maxima of m-dimensional products of iid random variables, which provide a useful tool for researching this class of questions. Some results on strong law of large numbers are given such that our results are much stronger than the corresponding result of Gadidov’s.     

Asymptotical behavior of the variance of sums of random variables with random replacements

In this paper, we introduce a scheme of summation of independent random variables with random replacements. We consider a series of double arrays of identically distributed random variables that are row-wise independent, but such that neighboring rows contain a random common part of the repeating terms. By this-scheme we bscribe a model of strongly dependent noise. To investigate the sample mean of this noise, we consider the sum of random variables over the whole double array and its conditional variance with respect to replacements. For columns of the arrays we prove a covariance inequality. As a corollary of it, we demonstrate the law of large numbers for conditional variances. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 124–143. Translated by A. Sudakov.     

一类随机变量序列部分和的强大数定律

设{Xn,n≥1}是随机变量序列.文[4]在二阶矩限制下,获得了任意随机变量序列的Hajek-Renyi型不等式,并给出了随机变量序列的强大数定律.本文利用胡舒合等获得的强大数定律,给出了随机变量序列的一些几乎必然收敛性,并给出了结果在PA,NA和两两NQD序列场合下的应用.     

混合序列的Hájeck-Rènyi型不等式及强大数律

利用Utev S.和Peligrad M.不等式,得到了-混合随机变量序列的Hájeck-Rènyi不等式、三级数定理和Chung型强大数律,改进了甘师信与吴群英等人的结论,达到了与独立时一致的结果.     

关于AQSI序列的几乎处处收敛性及强大数定理

建立了关于AQSI序列的Kolmogorov型不等式和AQSI序列的三级数定理,讨论了AQSI序列的几乎处处收敛性并且得到了关于AQSI序列的chung型强大数定理.     

Laws of large numbers of negatively correlated random variables for capacities

Our aim is to present some limit theorems for capacities.We consider a sequence of pairwise negatively correlated random variables.We obtain laws of large numbers for upper probabilities and 2-alternating capacities,using some results in the classical probability theory and a non-additive version of Chebyshev’s inequality and Boral-Contelli lemma for capacities.