Some Convergence Properties Of φ-mixing Random Variables

Probability limit theory is a very essentially theoretical basis of probability and mathematical statistics. Currently, the classical probability limit theory of independent sequence has been well developed. However, although in some cases the independence assumption is reasonable, the sample is not possible to satisfy the independence in practical problems. The φ-mixing sequence is the sequence whose sample datas are fit for a relationship, and compared with independent sequence, the φ-mixing sequence has more practical significance.This dissertation is based on the basic properties and some inequalities of φ-mixing sequence and the limit theory research technique, mainly from the following several aspects to research work:Firstly, we study the complete moment convergence of φ-mixing random variables. By using some methods such as probability inequalities and moment inequality, we obtain the complete convergence of φ-mixing random variables under more weak conditions. Then we generalize the corresponding results of independence sequence. At the same time, we reveal that complete moment convergence imply complete convergence.Secondly, we mainly use the moment inequalities and the truncation method to study and obtain the Lr convergence of φ-mrxing random vari-ables. The obtained results extend and improve the Theorem 6 in Gan et al.[1].Finally, we consider the fixed design nonparametric regression model Yni= g(xni)+εni,1<i<n, and give the definition of weighted regression estimator gn(x). Then under some weight function constraints, through the establish-ment and use of relevant inequalities, we obtain the asymptotic normality for estimator of nonparametric regression models under φ-mixing errors condition.