The Strong Laws Of Large Numbers For Maximum Value Of Weighted Sums Of WOD Random Variables And Application In Nonparametric Regression Model

Dependent structure is an important research content in probability and statistics,and a lot of abundant scientific achievements have been accumulated up to now.Among them,the widely orthant dependent structure(WOD,in short)is a structure of great concern.Many scholars have made remarkable contributions to the complete convergence,the strong limit theorem and related inequalities of weighted sums of widely orthant dependent random variables.The strong law of large numbers is a well-known conclusion in probability theory.The strong law of large numbers has been studied deeply.Lots of scholars have investigated the strong laws of large numbers of random variables with different distributions under certain conditions,and applied the main conclusions to other theories and models.Therefore,it is pretty essential to study the strong laws of large numbers under the widely orthant dependent condition.In this paper,the strong laws of large numbers for maximum value of weighted sums of widely orthant dependent random variables are further obtained,which extend the correspond-ing results under the independent condition of Bai and Cheng(2000)^[1]and Li et al.(1995)^[6].In order to make the main conclusions more applicable,using the theorem that we established,we present some appropriate conditions to prove the strong consistency for the weighted esti-mator of nonparametric regression model based on WOD errors.In addition,we also provide a numerical simulation to verify the validity of the results.