Exponential inequalities for N-demimartingales and negatively associated random variables

The class of N-demimartingales generalizes in a natural way the concept of negative association and includes as special cases martingales with respect to the natural choice of σ-algebras. For this class of random variables, a number of maximal and other inequalities were obtained by [Christofides, T.C., 2003. Maximal inequalities for N-demimartingales. Archives of Inequalities and Applications 50, 397–408] and [Prakasa Rao, B.L.S., 2004. On some inequalities for N-demimartingales. J. Indian Soc. Agricultural Statist. 57, 208–216; Prakasa Rao, B.L.S., 2007. On some maximal inequalities for demisubmartingales and N-demisupermartingales. J. Inequal. Pure Appl. Math. 8, 17]. In this paper we prove Azuma’s inequality for N-demimartingales and as a corollary we obtain an exponential inequality for negatively associated random variables.     

Some new results for demimartingales

Harremoёs obtained some new maximal inequalities for non-negative martingales. In this paper, we get some new maximal and minimal inequalities for non-negative demimartingales which generalize the results of Harremoёs. We also obtain an inequality for non-negative demimartingales which generalizes the result of Iksanov and Marynych. Finally we obtain a strong law of large numbers, strong growth rate and integrability of supremum for demimartingales which generalize and improve the result of Chow.     

Maximal inequalities for demimartingales and their applications

In this paper, we establish some maximal inequalities for demimartingales which generalize and improve the results of Christofides. The maximal inequalities for demimartingales are used as key inequalities to establish other results including Doob’s type maximal inequality for demimartingales, strong laws of large numbers and growth rate for demimartingales and associated random variables. At last, we give an equivalent condition of uniform integrability for demisubmartingales.     

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.     

B值拟鞅不等式及其应用

本文给出B值拟鞅的概率不等式与集合不等式,并用它们刻划了B空间的p可光滑性及q可凸性,作为应用,还证明了B值拟鞅的强大数律,收敛速度及极大值函数的可积性。     

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.     

On One Sufficient Condition for the Validity of the Strong Law of Large Numbers for Martingales

We prove a theorem on the strong law of large numbers for martingales. The existence of higher moments is not assumed. From the theorem proved, we deduce numerous well-known results on the strong law of large numbers both for martingales and for sequences of sums of independent random variables.     

一种新的弱相依系数和应用(英文)

本文在积分概率距离意义下提出了两个随机变量之间一种新的弱相依系数,并证明了此系数可获得协方差不等式和强大数定律,而且对于相关随机变量序列,我们还可以进一步研究矩不等式.     

强大数定律的若干新结果

本文利用Hajek-Renyi型最大值不等式,获得了随机变量和的强大数定律和 收敛速度.作为应用,给出了某些相依随机变量和新的强大数定律.     

两两NQD序列部分和的强大数定律

By using the moment inequality, maximal inequality and the truncated method of random variables, we establish the strong law of large numbers of partial sums for pairwise NQD sequences, which extends the corresponding result of pairwise NQD random variables.     

Maximal inequalities for N-demimartingale and strong law of large numbers

In this paper, we establish some maximal inequalities for N-demimartingale. The maximal inequality for N-demimartingale is used as a key inequality to establish other results including the strong law of large numbers, strong growth rate and the integrability of supremum for N-demimartingale.     

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.     

NA随机变量加权和的收敛性

利用Rosenthal型最大值不等式,得到了NA随机变量加权和的Marcinkiewicz-Zygmund强大数定律和完全收敛性,所获结果推广和改进了一些文献中相应的结果.     

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.     

GENERALIZED ROSENTHAL'S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal's inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and character-izes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.     

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.     

Rosenthal’s Inequalities for Asymptotically Almost Negatively Associated Random Variables Under Upper Expectations

In this paper, the authors generalize the concept of asymptotically almost \linebreak negatively associated random variables from the classic probability space to the upper expectation space. Within the framework, the authors prove some different types of Rosenthal''s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal''s inequalities.     

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.     

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.     

A limit theorem for sums of independent,nonidentically distributed random variables

A generalization and refinement of Chen's theorem related to a strong law of large numbers for sums of independent, nonidentically distributed random variables are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta in im. V. A. Steklova AN SSSR, Vol. 85, pp. 188–192, 1979.