Complete Convergence For  Mixing Random Variables And Its Application In Partially Linear Regression Model Sequences

In recent years, one of the hot point with respect to probability theory and the mathematical statistics is to weaken the restriction of independence. In order to make the application more practical, concepts of mixing sequence and dependent sequence have been brought in. Limit theory of mixing sequence and dependent sequence are widely used in probability and statistics, finance, reliability theory, econometrics and so on. In this thesis, we study the com-plete convergence for ρ mixing random variable sequence and its application in partially linear regression model.In chapter 2, firstly we discuss the complete convergence for p mixing random variables under the condition of EXn=0, E|X|pP<∞, p> 1/α, 1/2< α≤1, and the array of real numbers {ani,1≤i≤n, n≥ 1} satisfying|ani|≤C,1≤i≤n, n≥ 1. Secondly, we study the complete convergence for p mixing random variables under the condition of |ani|> 1 or ani=0. Thirdly, we combine the two situations above and give the complete convergence for ρ mixing random variables under the condition of ∑i=1n|ani|q= O(n),(?)q> p. In the last, we give a necessary and sufficient condition for the complete convergence for p mixing random variables. In this part, under some different conditions we give the same result which has much important theoretical significance.In chapter 3, we study the semiparametric regression model:Y(j)(xin, tin) = tinβ+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(j) (xin) are unmeasurable mean zero ρ mixing random variables and Y(j)(xin, tin) is random variables which is observable at points xin and tin. In the paper, we give the strong consistency, r-th (r> 2) mean consistency and complete consistency for estimators βm,n and gm,n(x) of β and g. The study of application for p mixing random variables is very important and have much pratical application value.