Convergence Properties For NSD Sequences And Their Applications

As one of the main problems of probability limit study, many scholars pay more attention to convergence properties for the partial sums of random variables. The breakthrough and development from independent random vari-ables to dependent random variables well enrich and enhance probability limit theory. The proposal of NSD random variables makes the further expansion of dependent random variables. Then its moment inequality, Rosenthal-type maximal inequality, Kolmogorov-type inequality are constantly given. But now, the study about its convergence properties is relatively small. This paper aims to investigate the convergence properties for weighted sums of NSD ran-dom variables and to give their applications in the nonparametric regression model.This paper includes four chapters.The first chapter of this thesis is the research background, its aim is mainly to introduce the background and significance of studying convergence properties for dependent random variables, the starting point of the paper, related concepts, some useful lemmas and inequalities.In chapter 2, with techniques of truncated method and Rosenthal-type maximal inequality,we pay much attention to complete convergence of weight-ed sums for arrays of rowwise NSD random variables under the condition of stochastic domination. This part not only promotes the result of Volodin et al.[19]and Qiu[20], but also shows the result and proof under 1+α+β< 0.In chapter 3 of this paper, based on the above result of complete conver-gence, we first further study complete moment convergence of weighted sums for arrays of rowwise NSD random variables. Then we focus our discussion on the equivalent conditions of complete moment convergence for weighted sums of NSD random variables under another weights{ani ≈ (i/n)β(1/n),1≤ i≤ n,n≥ 1}. We extend the results from complete convergence for independent identical random variables to complete moment convergence for NSD random variables without the assumption of identical distribution, and improve the corresponding results of Li et al.[21] and Gut[22].In chapter 4, as applications, we investigate the complete consistency for the estimator of nonparametric regression model based on NSD errors by using the obtained complete convergence. This makes a contribution to statistical analysis problems such as the choice of estimators, hypothesis testing, interval estimation and so on.