Strong Consistency Of M-Estimator In Linear Model For Dependent Samples

Let us consider the linear model: (1)where is an unknown -dimension vector of regressive parameter and are known -dimension vectors and are random errors. Assume that is a function on , is said to be -Estimator of if (2) hold.When is independent ,Chen Xiru & Zhao Lincheng[1] give a systematic account on the asymptotic theory of -methods in linear model.It consists of a discussion on the definition of the -estimate , on the weak and strong consistency and asymptotic normality, weak and strong linear representation of -estimate and linear hypothesis testing based on -estimate.But the moment condition of the theorem 3.1 in the monograph is so highly. Yang Shanchao(2002)[2] has improved the moment conditions.This paper has discussed the strong consistency of - estimator in linear model for dependent samples and has obtained some more suitable suficiency conditions. These results greatly improve the corresponding ones by Chen Xiru & Zhao Lincheng[1] and Yang Shanchao(2002)[2]. 　Let us note the model(1),marking,marking and on the derivative on the left and it on the right of. 　　Theorem 1. Suppose be sequence of negatively associated , be a convex function and exists constants > 0, >0,for all satisfying (3) when <,and (4)for some .Assume 　 (5) for some >0,and .Then a.s. () Let us define by 　Theorem 2 Suppose be a sequence of positively associated and be a con-vex function. If it satisfying (3), exists constant satisfying (4) and exists >0,, satisfying , (6)when . Assume , (7) for and (8)where is a measure zero set.Then a.s. () 　Theorem 3 Suppose be a sequence of positively associated and be a con-vex function satisfying (3), exists constant satisfying (4) and satisfying either condition for : (i) for (ii) is bounded or equally to satisfies Lipschitz condition ,assume (9) for some .Then a.s. ()...