两种混合随机变量序列的收敛性及应用

【作者】 王聪；

【导师】 张立新；

【作者基本信息】 浙江大学 ， 概率论与数理统计， 2005， 硕士

【摘要】 本文是在攻读硕士学位期间完成的。在实际问题中,我们研究的随机变量或随机过程通常是不独立的。在非独立系统建模中混合相依是广泛应用的概念。混合相依是指系统中随机变量间的相依关系是以时间或空间的距离衰退的,也就是说,混合相依是指我们所考虑的随机变量是渐近独立的。有许多描述非独立随机变量相依情况的混合概念,如α-混合,β-混合,ρ-混合,φ-混合等等。本文考虑两种混合相依随机变量:φ-混合随机变量和ρ^*-混合随机变量。φ-混合随机变量这一概念的提出已有很长的历史,许多学者对它进行过研究。随机变量序列加权和的收敛性质在统计理论中有广泛的应用,如线性模型的大样本性质。本文首先研究φ-混合组列加权和的收敛性质。ρ^*-混合的概念是在1990年提出来的。从定义看,ρ^*-混合比我们已熟知的ρ-混合稍强。然而,由于在研究ρ^*-混合随机变量序列的极限定理时对混合系数的收敛速度几乎不要加什么条件,实际上许多ρ-混合序列也是ρ^*-混合序列。ρ^*-混合序列的性质已经引起了许多学者的注意。本文接下来要研究的是ρ^*-混合随机变量序列加权和的收敛性及平稳ρ^*-混合序列部分和估计的大样本性质。 全文共分三章。 第一章是关于φ-混合随机变量序列的收敛性质。自从Dobrushin（1965）对马氏过程引入了φ-混合的定义之后,有许多学者对φ-混合随机变量序列的性质作了研究。Ibragimov（1962）给出了φ-混合序列的中心极限定理,Herrndoff（1983）,Paligrad（1985）,Shao（1993）和杜初午（1993）等研究了它的弱不变原理,邵启满（1988）研究了完全收敛性。关于φ-混合随机变量序列的极限定理的更多细致结果参见陆传荣和林正炎（1997）的专著。本文的第一章主要考虑逐行φ-混合随机变量组列的加权和。Hu和Taylor（1997）对于逐行独立的随机变量组列{X_ni,1≤i≤n,n≥1}给出了强大数律的结果,在第一章的第二节,我们在比Hu和Taylor（1997）中更广的一类函数ψ（定义见1.2.3）下给出了逐行φ-混合序列加权和的L¹收敛,a.s.收敛,依概率收敛及完全收敛性之间的等价关系,并在另一组条件下证明上述几种收敛性对于φ-混合序列总成立,从而推广了Hu和Taylor（1997）的结果。 定理0.1 令k_n→∞是正整数列。{X_ni,1≤i≤k_n,n≥1}为逐行φ-混合随机变量组列, ∑_n=1^∞φ^1/2（n）<∞,{a_ni,1≤i≤k_n,n≥1}为常数数组,max_1≤i≤kn|a_ni|→0。假设更多还原

【Abstract】 This thesis is finished during my graduate student program to Master Degree of Science. The random variables or stochastic processes that come from practical problems are usually not independent. The mixing dependence is a widely used concept in modelling non-independent systems. The mixing dependence means that the dependence among random variables in a system decays with distance of time or space, i.e., the mixing dependence means that the considered random variables are asymptotic independent. There is a lot of concept of mixing to measure the dependence of non-independent random variables, for example, aα-mixing, β-mixing, p-mixing, φ-mixing, etc. This thesis considers two kinds of mixing dependent random variables, φ-mixing and p~*-mixing random variables. The concept of φ-mixing has a long history and has studied by a lot of scholars. The thesis put our major attention to the convergence of the weighted sums of an φ-mixing array. The convergence of the weighted sums has applications in the theory of statistics, for example the large sample properties of linear models. The concept of p~*-mixing is introduced in 1990s. p*-mixing is a little more rigid than the well-studied p-mixing. However, since almost no condition on the mixing rate is needed in the study of the limit theory of this kind of mixing dependent random variables and, many p-mixing random variables which appear in practice are also p~*-mixing, this kind of mixing have drawn many scholars’ attentions. This thesis majorally studies the convergence of weighted sums of p~*-mixing random variables, and the large sample properties of the estimators of the partial sum of a stationary p*-mixing sequence.The thesis consists of three chapters.Chapter I is on the convergence properties about φ-mixing random variables. Since Do-brushin(1956) first put forward the concept of φ-mixing random variables about Markov process, a lot of scholars had studied the properties of φ-mixing random variables. Ibragi-mov(1962) got the CLT, Herrndoff(1983) and Paligrad(1985), Shao(1993) and Du(1993) etc studied the weak invariance principle, and Shao(1998) studied the complete convergence. For more and detail results on the limit theorems, one can refer to the book of Lin and Lu: Limit Theory for Mixing Dependent Random Variables, Science Press & Kluwer Academic Publishers (1997). The Chapter I of the thesis majorally considers the weighted sums of an arrayof rowwise (^-mixing random variables. Hu and Taylor(1997) gave the strong law of numbers of an array of rowwise independent random variables. In the second section of chapter I, we get the equivalence between the convergence in L1, the convergence in probability, almost sure convergence and complete convergence of weighted rowwise (^-mixing random variables under the function ip (for definition see (1.2.3)) wider than in Hu and Taylor(1997), and show that these convergences are always true for (^-mixing random variables under another group of mild conditions, which generalizes the results in Hu and Taylor(1997).Theorem 0.1 Let kn —> oo be a sequence of positive integers. And let {Xni, 1 < i < kn,n > 1} be an array of rowwise (^-mixing random variables satisfying Ylc￡LilpHn) < °°> and {ani, 1 < i < kn,n > 1} an array of constants with maxK^^ |oni| -> 0. Assume thatIn Chapter II and Chapter III we study some properties of p*-mixing random variables. The concept of p*-mixing random variables was introduced by Bradley in 1990. Since its appearance in literature, the properties of this kind of dependent random variables had drawn many attentions from scholars for its extensive applications.In Chapter II, we mainly study the complete convergence and almost sure convergence of the weighted sums of p*-mixing random variables, and their applications to the large sample properties of the estimates of regression parameter in a linear model and the estimates of weighted function in nonparametric regression model.Theorem 0.3 Let {Xn,n > 1} be a sequence of /o*-mixing random variables. Assume that EXi = 0 and for p > 1sup E\Xi\p < oo.t>lFor 2 > p > 1 there is s € (-, 1] and for p > 2 there is s G (|, 1] such thatmax \ani\ = O(n~s),KKnthenTheorem 0.4 Let {Xn,n > 1} be a sequence of p*-mixing random variables. Assume that EXi = 0 and \Xi\ < D almost surely, where D is a positive constant and r > 2. If there is s € (￡ + |, 1] such thatmax \ani\ = O(n~s),thenn Sn = YlaniXi ---■> 0.t=lTheorem 0.5 Let {Xn, n > 1} be a sequence of p*-mixing random variables . Assume that EXi = 0 for i = 1,2, ■ ? ? and sup^ E\X{\T < oo for r > 2. If there is s e (± + i, 1] such thatmax |ani| = O(rTs),thennX0.Theorem 0.6 Let {Xn, n > 1} be a sequence of p*-mixing random variables satisfying EX{ = 0 for i = 1,2, ? ? ? and sup^ E\Xi\r < oo for r > 2. Suppose that for fixed i,lim ani = 0.n->ooIf there exists 6j > 0 such thatil > 60) —> 0 as nt=l and if there exist si(si > § +1), s2(s2 > 2) such that< k)} = O(n"*),l<i<nthenn5n = ^2,aniXi — ^ 0.?=iAs an application of Theorems 0.3-0.6, in Section 2.4, we give the strong consistency of both the least squares estimates of the regression parameter j3 in a linear model and the estimates of weighted function in a nonparametric regression model when the errors are p*-mixing. In Theorem 2.4.1 we give the strong consistency of the least squares estimates of the regression parameter /? in a linear model and in theorem 2.4.2 we give the strong consistency of the estimates of weighted function in a nonparametric regression model.For a stationary sequence {Xn;n > 1}, let Sn = YJk^i^k be its partial sum process. Under some mild conditions, the average variance Var{Sn)/n will converge to a constant a2. Paligrad and Shao (1995) defined two sample estimators Bn%v and Bn<p of a (for definition see (3.1.2) and (3.1.3)), and studied their asymptotic properties for p-mixing random variables. In chapter III we study their properties for p*-mixing random variables and get the consistency as well as the asymptotic normality.Theorem 0.7 Let {Xn, n > 1} be a stationary p*-mixing sequence of random variables satisfying EX\ = 0 and EX^Vp < oo where p>l. Assume that o2n -> oo as n -? oo. LetI = In -*? oo, I = o(n) as n —? oo. (0.1)thenL2Bn,P —> a as n —>■ oo,—------^> N(0,1) as n -> oo.Theorem 0.8 Let {Xn, n > 1} be a stationary p*-mixing sequence of random variables satisfying EXi — 0 and EXfp < oo where p> 1. Assume that a\ -+ oo as n -? oo and the condition (0.1) is satisfied. Thenp)h) -^ N(0, a2Ap). (0.2)where Ap = 2{cp/p)2 $ cov{\W{l)\p, \W{1 + t)- W{t)\p}dt and [W(t),t > 0} is a standard Wiener process. Moreover, if p > 1 thenJ2AP). (0.3)And EB\V in (0.2) and (0.3) can be replaced by EB*^ {EBn>p)p or (EBn>p)p.Paligrad and Shao (1995) showed that Bn<p and Bn%p were the consistent estimations of a and obtained the asymptotic normality of them for stationary p-mixing sequence of random variables. Paligrad and Shao (1995) only proved (0.3) in the case of p = 2 for p-mixing sequence of random variables. Following the method of this thesis one can show that for p-mixing sequence of random variables under the conditions of Theorem 3.2.2 in their paper (0.3) is always true for arbitrary p > 1, which generalizes the results in Paligrad and Shao (1995).更多还原