多维平稳序列相关阵估计量的渐近分布

THE ASYMPTOTIC DISTRIBUTION OF THE COVARIANCE MATRIX ESTEMATOR FOR MULTIDIMENSIONAL STATIONARY TIME SERIES

若X_t是线性平稳序列、可表示为X_t=sum from j=-∞ to +∞(b_(t-j)ζ_j的形式、其中{ζ_j}j=0,±1,……是独立同分布的随机序列:Eζ_j=0,Eζ_j^2=σ~2>0。对于这种平稳随机序列,T.W.Anderson讨论了其相关系数估计量的渐近分布问题。本文将要讨论{ζ_j}是M维实四阶鞅差序列时,多维线性平稳序列(1)的相关系数组成的协方差阵的估计量的渐近分布问题。为此目的,我们研究了鞅差序列二次型的渐近分布,改进了作者在[2]中所得到的结果。並求出了此种协方差阵估计的渐近分布。

Let Xt be the multidimensional stationary sequence which can be written in the form:where ξj is a M-dimensional martingle difference sequence with the fourth moments.The author considered the asymptotic distribution of the covariance matrix estimator for Xt.. Two theorems about the asymptotic distribution of quatratic form of {ξj} and {Xj} are proved. In theorem 3, we proved that the covariance matrix estimator would be asymptotically normally distributed under some sufficient conditions.