Some Strong Limit Theory For END Random Variables And Its Applications

The class of extended negatively dependent (END) random variables includes in-dependent random variables, negatively associated (NA) random variables, negatively orthant dependent (NOD) random vriables as special cases. It has wide applications in insurance and financial mathematics, complex systems, reliability theory, survival analy-sis and other fields. This thesis focuses mainly on probability inequalities and moment inequalities for END random variables, which will be applied to study some strong con-vergence properties for END random variables, such as complete convergence, complete moment convergence, almost sure convergence, and establish the asymptotic approxima-tion of inverse moments and the consistency of the estimator of nonparametric regression model.In Chapter2, we study the exponential probabilities and moment inequalites for END random variables. By using the definition of END random variables and some basic properties, we establish some exponential probabilities for END random variables, such as Bernstein type inequality, Hoeffding type inequality, Kolmogorov type inequality, and so on. In addition, we obtain the moment inequalities for END random variables, such as Marcinkiewicz-Zygmund type moment inequality and Rosenthal type moment inequality. The exponential probabilities and moment inequalites will be regarded as key tools to study the strong limit theorems for END random variables and its applications.In Chapter3, we investigate the complete convergence and complete moment conver-gence for sums and weighteds sums of END random variables. By using the Marcinkiewicz-Zygmund type moment inequality and the method of truncation, we establish the the complete convergence for sums and weighteds sums of END random variables. As an ap-plication, we get the complete moment convergence for sums of END random variables. The results generalize the corresponding ones of Qiu et al for NOD random variables, generalize and improve the corresponding ones of Wang et al for END random vriables. In addition, we investigate the complete convergence and complete moment convergence for weighteds sums of END random variables based on the slowly varying function. As an application, we get the Baum-Katz type strong law of large numbers.In Chapter4, we study the complete convergence and complete moment convergence for maximal partial sums of END random variables. By using the Rosenthal type moment inequality and the method of truncation, we establish the the complete convergence for maximal partial sums of END random variables based on nonidentical distribution. As applications, we get the Baum-Katz type strong law of large numbers and Marcinkiewicz-Zygmund type strong law of large numbers. In addition, the complete moment convergence for maximal partial sums of END random variables is also obtained based on nonidentical distribution. The results established in this chapter generalize some corresponding ones for independent random variables and some dependent random variables.The asymptotic approximation of inverse moments for nonnegative END random variables is the main target of Chapter5. Let{Zn, n≥1} be a sequence of nonnegative END random variables. Denote Xn=Mn-1,Zi,where{Mn,n≥1} is a sequence of positive real numbers. Under some suitable conditions, the inverse moment can be asymptotically approximated by for all a>0and α>0. As a special case, the asymptotic approximation of inverse moments for partial sums of nonnegative END random variables is obtained. Under some suitable conditions, the growth rate for the asymptotic approximation of inverse moments is established. In addition, the asymptotic approximation of inverse moments for a class of random variables is also obtained. The results obtained in this chapter generalize and improve some correspongding ones of Wu et al. Wang et al and Sung.In the lase chapter of this thesis, we study the complete consistency and mean con-sistency for the estimator of nonparametric regression model under END errors. Consider the following nonparametric regression model: where xni are known fixed design points from A, where A(?)Rp is a given compact set for some p≥1, g(·) is an unknown regression function defined on A and εni are random errors. As an estimator of g(·), the following weighted regression estimator will be considered:where Wni{x)=Wni(x;xni,xn2,…,xnn), i=1,2,…,n are the weight function. By using the Rosenthal type moment inequality and the method of truncation, we study the complete consistency of estimator of nonparametric regression model based on END er-rors. As an application, the complete consistency for the nearest neighbor estimator is obtained. In addition, under some suitable moment condition and weight condition, we study the mean consistency of estimator of nonparametric regression model based on END errors. The results obtained in this chapter extend the corresponding ones of Liang and for NA random variables and Wang et al. for NOD random vriables.