Convergence Properties Of AANA Random Variables And Its Applications

In this paper,we mainly study properties of asymptotically almost nega-tively associated(for short,A AN A)random variables which is one of significant dependent sequences.We introduce the concept of A AN A random variables and the related inequalities and lemmas,then research the strong convergence and complete convergence of A AN A random variables,and give the applica-tions of this variables on nonparametric and semiparametric regression modelFirstly,this paper studies the strong convergence properties of AANA sequences.Inspired by Qui and Yang[73]discussed the strong law of large numbers for weighted sums of negatively associated(for short,NA)sequences,We mainly prove the strong convergence properties for weighted sums of AANA sequences,and obtain the Marcinkiewicz-Zygmund type strong law of large numbers of this sequencesSecondly,the complete convergence for weighted sums of AANA random variables is studied.By using Marcinkiewicz-Zygmund type inequality and Rosenthal type inequality,we mainly prove the complete convergence for par-tial sums of AANA random variables,which conclusion improves the complete convergence properties of NA random variables in Baek et al.[74].In addi-tion,we prove the complete consistency for the estimator of nonparametric regression model based on AANA errorsFinally,the semiparametric regression model is considered:Y(i)(xin,tin)=tin?+g(xin)+e(j)(xin,1?j?m,1 ?i?n,where xin?Rp,tin?R are known to be nonrandom,g is an unknown continuous function on a compact set A in Rp,e(i)(xin)are AANA random errors with mean zero,Y(j)(xin,tin)are random variables which are observable at points xin and tin.These vari-ables are defined on the same probability space(?,F,P).In this paper,we study the strong consistency,r-th mean consistency and complete consistency for estimators ?m,n and gm,n of ? and g,respectively.The results generalize the corresponding ones for ?-mixing random variables in Hu[38].