ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL
ASYMPTOTICS OF SAMPLE EIGENSTRUCTURE FOR A LARGE DIMENSIONAL SPIKED COVARIANCE MODEL
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发表时间：
2007
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通讯作者：
D. Paul
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作者：
D. Paul
本文讨论了一个多元高斯观测模型，其中协方差矩阵的特征值除有限个较大的特征值外均为1。有趣的是当样本大小和观测值的维数都增长到无穷大时样本协方差矩阵的特征值的渐近行为，使得它们的比率收敛到一个正的常数。当总体特征值大于某一阈值且重数为1时，相应的样本特征值具有高斯极限分布。在相同的设置中，样本特征向量存在“相变”。本文的另一个贡献是研究了当相应的特征值简单且足够大时，样本特征向量的二阶渐近性。
This paper deals with a multivariate Gaussian observation model where the eigenvalues of the covariance matrix are all one, except for a finite number which are larger. Of interest is the asymptotic behavior of the eigenvalues of the sample covariance matrix when the sample size and the dimension of the obser- vations both grow to infinity so that their ratio converges to a positive constant. When a population eigenvalue is above a certain threshold and of multiplicity one, the corresponding sample eigenvalue has a Gaussian limiting distribution. There is a "phase transition" of the sample eigenvectors in the same setting. Another contribution here is a study of the second order asymptotics of sample eigenvectors when corresponding eigenvalues are simple and sufficiently l arge.