线性平稳过程论文：线性平稳序列自协方差函数的渐近分布.doc

线性平稳过程论文：线性平稳序列自协方差函数的渐近分布【中文摘要】时间序列分析的许多基本结果建立在其多个样本自协方差函数联合服从渐近正态分布的基础上。本文针对线性平稳序列的若干个样本自协方差函数,讨论其联合渐近分布问题。众所周知,在个数事先固定后,样本自协方差函数的联合渐近正态是一个著名的结果,是时间序列拟合优度检验,比如独立同分布白噪声检验,的一个基本理论依据。然而在实际应用中,我们经常是先得知样本的容量n,然后选取某个m,把m看成是固定,之后引用上面的经典结论。因此研究个数m随着样本容量n变化时样本自协方差函数的联合渐近分布问题是很有实际意义的。所以本文讨论的第一个问题是,对于线性平稳序列,对给定的观测数据个数n,应该选取什么样的m(n),能够保证其样本自协方差函数(?)n,j=0,1,m(n)(按照某种方式)渐近服从多元正态分布。正如Keenan,D.M.(1997)指出的那样,对于这种维数变化的随机向量的渐近分布,用定义在(R,R)上的传统弱收敛并不恰当。Keenan,D.M.(1997)提出的处理这种情形的方法是考虑一致弱收敛,得到了当xtt=1为一个严平稳强混合序列,并且满足一定的假设条件,在mlog(mlogm)=O(logn)时,样本自协方差函数(?)n,j=0,1,m(n)一致弱收敛到一个多元正态分布。本文在Keenan,D.M.(1997)的基础上,讨论了应用中更为广泛的线性平稳序列的m(n)维样本自协方差函数的一致弱收敛性问题,我们给出了不完全于Keenan,D.M.(1997)的证明。受近些年来人们广泛使用的线性组合的分布收敛做法,我们考虑了m(n)维样本自协方差函数线性组合的弱收敛问题,这是本文讨论的第二个问题。我们将Richad Lewis and Gregory C.Reinsel(1985)的方法运用到无穷维样本自协方差函数的渐近分布,得到了在一定条件下m(n)维样本自协方差函数的线性组合弱收敛到正态分布。本文讨论的第三个问题是,对于维数变化的随机向量序列,其联合分布的一致弱收敛与其线性组合的弱收敛之间有怎样的关系呢?我们通过一个例子表明,随机向量线性组合的弱收敛似乎要比其联合分布的一致弱收敛弱。但是,两者内在的关联还在进一步的探讨当中。本文的最后一部分进行了大量的模拟,通过选取不同的样本容量n和不同的维数m,给出了统计量Q(m),通过重复,我们得到了3000个Q(m)的值,得到Q(m)的经验分布,对其与自由度为m的2分布进行拟合比较。和理论上一致,我们发现确实当m较小时,Q(m)的经验分布函数近似为2(m)分布,从而样本自协方差函数的联合分布与正态分布接近,但是当m较大时,两者相差较远。【英文摘要】Lots of the fundamental results in time series analysis depend on the asymptotic normality of the sample autocovariances.This paper discusses the asymptotic distribution for the sample autocovariances of linear stationary processes.The joint asymptotic normality of a fixed number m of sample auto-covariances is a well-known result and has been fundamental criterion in the time series goodness-of-fit tests.However,in practice,m is often chosen after the number of observations n,then treated m as fixed.Therefore,the first ques-tion we discussed in this paper is how to select a appropriate function of n ,so that asymptotic normality of the sample autocovariances(?)cn(j)-r(j),j=0,1,m(n) can be verified for a large amount of n.For the asymptotic distribution of the random vector that dimension is varying with n,the traditional weak convergence in (R,R) tends to be not appropri-ate.Keenan,D.M.(1997) proposed that uniform weak convergence was a rea-sonable criterion for this type of applications and proved that the sample autocovariances(?)cn(j)-r(j),j=0,1,m(n) have uniform weak convergence, whenxtt=1is a strictly stationary process satisfying a strong mixing condition and m(n) satisfying m log(m log m)= O(logn).This paper discusses the uniform weak convergence of the sample autocovariances of lin-ear stationary processes. The second question of this paper discusses the weak convergence for the linear combination of the m(n) sample autocovariances of linear stationary processes.This paper applies the method of Richad Lewis and Gregory C.Reinsel(1985) to the weak convergence of the linear combination of the m(n) sample autocovariances, and gains the asymptotic normality where the dimension of the random vector is growing with n.The third question this paper discussed is the relationship between the uniform weak convergence for the joint distribution and the weak convergence for the linear combination of the m(n) sample autocovariances.Through an example,we gain that the weak convergence for the linear combination of the sample autocovariances is weaker than the uniform weak convergence for the joint distribution of the sample autocovariances in this rather unusual situation where the dimension of the random vector is growing with n.At last,by a large number of simulations,we define the statistics Q(m).By loop,we get 3000 Q(m),then we fit them with2(m) distribution. We found that when m is small, the distribution function of Q(m) is2(m).Therefore,the sample autocovariance function of the joint distribution is similar to asymptotically normality. But when m is too large, the distribution function of Q(m) is not2(m).【关键词】线性平稳过程 自协方差 渐近分布 Kolmogorov检验【英文关键词】linear stationary process autocovariances asymptotic distribution