Estimation Of The Variance For The Partial Sums Of Some Mixing Random Variable Sequences

The random variables or stochastic processes that come from practical problems are usually not independent ,we remark that there has been a great amount of work on the properties of mixing random variables ,and some significant results have been reached through deep research by a lot of scholars. The mixing dependence means that the considered random variables are asymptotic independent, the dependence of random variables as a concept is developed in some branches of Probability theory and Mathematical Statistics,such as Markov chains,random field theory and time series analysis,etc.There is a lot of concept of mixing to measure the dependence of non-independent random variables,for example,ρ-mixing ,ρ*-mixing ,α-mixing(strong mixing),β-mixing,φ-mixing, etc.This thesis considers three kinds of mixing dependent random variables,ρ-mixing ,ρ*-mixing ,α-mixing random variables,since this three kinds of mixing dependent random variables has applications in the theory of statistics,and the properties of that have drawn many scholars'attentions,Hence one can see that, the study on the partical sums of dependent random variables has momentous significance.Based on reading and understanding the literacture on consistency and the asymptotic normalities of the estimation of the variance for the partial sums of some mixing random variable sequences ,we synthesize and think the research results of the question.The limit theory for the partial sums of some mixing random variable sequences has an important status in Probability and Mathematical Statistics,it also has certain research value and practical significance .For example ,for a stationary sequence be its partical sum process, let { X n(t );n∈N} be the influence of the nth investor on the return rate of a stock ,then the process describes the flucturation of the rate of return of a stock. So ,many scholars devote themselves to the research of this properties.Such as Xiao and Zheng gave the convergence of the partial sums of random processes;Lin and Lu proved that limit theory for mixing dependent random variables;Pligrad.M. gave on the central limit theorem forρ-mixing sequences of random variables;Dong and Yang proved that an almost sure central limit theorem for NA and LNQD random variables;Yang gave the strong law of large numbers forα-mixing sequences of random variables,etc.It is well known ,in the research of limit theorems of Probability theory for a sequence of random variables,let { X n, n≥1} be a sequence of i.i.d. random variables ,σ2 is the variance of X n,there has already a lot of methods used in the estimation of the variance so far.The most usual estimation is the variance of sample ,and X = ( 1n )∑i=n1 Xi is satisfied;let { X n, n≥1} be a mixing dependent random variables, the variance of sample is not the consistent estimation ofσ2,Under some mild conditions, we have i.e.,if l is large enough,Var( Sll ) is the approximateσ2, is the estimator of Var( Sll ),let {l n , n≥1} be a sequence of positive integers ,with Bn ,2 is the variance of sample ??? s j(l ) l; j = 0,1, ???, n ? l???.Paligrad and Shao defined two sample estimators Bn ,p and B?n ,pofσ(for definition see chapterⅡ),and studied their asymptotic properties forρ-mixing random variables;In chapterⅢwe study the large sample properties of the estimators of the partical sum of the stationaryρ*-mixing sequence and get the consistency as well as the asymptotic normality;The chapterⅣanalyzes and surveys the results about the consistency and the asymptotic normality of the estimators of the variance forα-mixing sequence of random variables.