Strong Convergence Of (?)Mixing Sequence And Its Application In Semiparametric Regression Model

We have studied the probability limit theory of independent random vari-ables. But in many practical problems, the samples often are not independent. The limit theory of dependent sequences is one of the central issues for study-ing probability, which has very wide application in fields such as reliability, penetration theory, multivariate statistical analysis, risk assessment, weather forecast, stocks and bonds. In this article, we investigate strong convergence properties for weighted sums of(?)-mixing random variables by several tech-niques such as Cr inequality, Markov’s inequality, Jensen’s inequality, moment inequality, random control for random variables, etc., and obtain situation of semiparametric regression model based on(?)-mixing errors.In chapter 1, first of all, the author introduce the background and current studied result, secondly, some used inequalities are given, finally, the structure of the article is given.In chapter 2, the strong convergence for weighted sums of(?)-mixing se-quence is studied by using the moment of inequalities and the condition of random control fo（?）-mixing random variables, and some sufficient conditions to prove the strong convergence for weighted sums of(?)-mixing sequence are given. The results extend some corresponding ones for independent sequences.In chapter 3, we study parametric component and nonparametric com-ponent estimators in a semiparametric regression model based on(?)-mixing random variables; their r-th mean consistency, complete consistency and uni-form consistency are obtained under suitable conditions. The results improve the conditions in the literature and generalize the existing results of indepen-dent random errors to the case of(?)mixing random errors.