Complete And Complete Moment Convergence For Random Variables And Their Applications

Mathematics is the queen of science,and the probability limit theory is an indispensable part of mathematics.After some experts and scholars hard study,the probability limit theory has been relatively mature.In recent years,due to the importance of WOD in dependent sequences,more and more scholars have begun to pay attention to and study it.This paper further studies the convergence and application of WOD random variables on the basis of some existing properties and inequalities.Chapter 1,At the beginning of this paper,the research background,re-lated concepts and the inequality used in the proof of the main results are introduced.Chapter 2,By using the truncation method and the exponential inequality of WOD random variables,the complete convergence for the weighted sums of WOD random variables is established under weaker conditions of moment,and the M-Z type strong law of large numbers is obtained.Chapter 3,M-Z and Rosenthal moment inequalities of WOD random vari-ables are given.By using some properties of slow varying function,prove the complete convergence of weighted sums of WOD random variables.And the complete moment convergence is further obtained under changing conditions.Chapter 4,Based on the moment inequality of the maximum value of WOD random variables,some sufficient conditions for the complete conver-gence of q order moment for arrays of WOD random variables are established.These conditions are used to prove the complete convergence of q order moment for arrays of weighted sums of WOD random variables.Chapter 5,By using the results of complete convergence,the application of the proposed method in the nonparametric regression model is given.The complete consistency of the weighted estimators in the nonparametric regres-sion model under WOD error is established,and the numerical simulation is carried out.Chapter 6,At last,the whole paper is summarized.