The Asymptotic Distribution For The Autocovariances Of Linear Stationary Processes

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, when{xt}t=1∞is 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 withχ2(m) distribution. We found that when m is small, the distribution function of Q(m) isχ2(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 notχ2(m).