Convergence For Weighted Sums Of I.i.d. Random Variables And Applications In Regression Estimation And EV Model

Probability limit theory is one of the key branches of probability theory, and also the important foundation of mathematics statistics. Let XnX}1,,{ 3nbe a sequence of independent distributed random variables and nnia3ï¿¡ï¿¡}1,1,{nian array of constants. The convergence results for weighted sums ?have been studied by many authors. Many useful linear statistics are expressed in a weighted sum of random variable, such as the least squares estimate of parameters, nonparametric regression estimator, jackknife estimates, etc. Thus, in this context, it is necessary to study the weighted sum of the sequence of random variables.This paper discusses from the point of complete convergence the convergence of random variable sequences. In chapter 1, we introduce some classical theories for weighted sums of i.i.d. random variables, which have been studied by Stout(1974)and Li et al.(1995). In this paper, with suitable moment condition, the sufficient conditions of complete convergence for Stout type weighted sums of i.i.d. random variables are obtained. This result improves and generalizes the results of Stout(1974)and Chen et al.(2014) and the given result is the particular cases of the result of this paper. A strong law is also obtained, which improve the results of Li et al.(1995) and Chen et al.(2014). The main idea in the proofs is to use the invariance principle. In chapter 3, as the applications of the strong law, the strong consistency of the nonparametric regression estimations, the rates of the strong consistency of the unknown parametric of the simple linear errors in variables(EV) model, are given.