Convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables

In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T.C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.     

Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of m-linearly negative quadrant dependent random variables (m = 1, 2, …). For a sequence of m-linearly negative quadrant dependent random variables {X _{n, n} ? 1} and 1 < p < 2 (resp. 1 ? p < 2), conditions are provided under which $n^{ - 1/p} \sum\limits_{k = 1}^n {\left( {\left. {X_k - } \right|EX_k } \right) \to } 0$ in L ¹. Moreover, for 1 ? p < 2, conditions are provided under which $n^{ - 1/p} \sum\limits_{k = 1}^n {\left( {X_k - EX_k } \right)}$ converges completely to 0. The current work extends some results of Pyke and Root (1968) and it extends and improves some results of Wu, Wang, and Wu (2006). An open problem is posed.     

Strong law of large numbers for pairwise positive quadrant dependent random variables

We give the rate of convergence in the strong law of large numbers for pairwise positive quadrant dependent random variables and contemporaneous functions of these variables. Several examples of applications are given.      

The strong law of large numbers for pairwise negatively dependent random variables

Necessary and sufficient conditions for the validity of the strong law of large numbers for pairwise negatively dependent random variables with infinite means are formulated.     

Strong convergence for sequences of asymptotically almost negatively associated random variables

In this article, the complete convergence for sequences of asymptotically almost negatively associated (AANA) random variables is studied. As applications, the Baum–Katz-type theorem, Hsu–Robbins-type theorem and Marcinkiewicz–Zygmund strong law of large numbers for sequences of AANA random variables are obtained.     

On the strong convergence properties for weighted sums of negatively orthant dependent random variables

In the paper, the strong convergence properties for two different weighted sums of negatively orthant dependent(NOD) random variables are investigated. Let {X_n, n ≥ 1}be a sequence of NOD random variables. The results obtained in the paper generalize the corresponding ones for i.i.d. random variables and identically distributed NA random variables to the case of NOD random variables, which are stochastically dominated by a random variable X. As a byproduct, the Marcinkiewicz-Zygmund type strong law of large numbers for NOD random variables is also obtained.     

NA列与两两NQD列的L^p收敛性

设{Xn,n≥1）是NA列或两两NQD列,{ank;1≤k≤n,n∈N）是实数阵列．利用矩不等式和截尾方法,研究了∑k=1^n ankXk的L^p收敛性,所获结论推广和改进了前人的相应结果．     

On the exponential inequalities for negatively dependent random variables

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.     

Strong convergence of pairwise NQD random sequences

Strong limit theory is one of the most important problems in probability theory. Some results on the convergence of pairwise NQD random sequences have been presented. This paper further analyzes the strong convergence of pairwise NQD sequences and generalizes partial results of Wu [Q.Y. Wu, Convergence properties of pairwise NQD random sequences, Acta Math. Sinica 45 (3) (2002) 617-624 (in Chinese)]. Since no general moment inequalities are given as so far, we avoid this problem and obtain a class of strong limit theorem for NQD sequences and some corresponding conclusions by use of truncation methods and generalized three series theorem, which are the supplements to the previous fruits.     

Complete moment convergence of pairwise NQD random variables

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.     

END随机变量阵列加权和的完全收敛性

利用END变量的R0senthal型矩不等式,研究了END随机阵列加权和的完全收敛性,给出了证明完全收敛性的一些充分条件.另外,还给出了证明完全收敛性的一个必要条件.所得结果推广了独立变量和若干相依变量的相应结果.     

On convergence of the maximum of dependent random variables

Kaunas University of Technology, Studentu 50, 3028 Kaunas, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 32, No. 1, pp. 3–6, January–March, 1992.