The eigenvector empirical spectral distribution (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the deformed Mar\v{c}enko-Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma^{1/2} X X^* \Sigma^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite matrix. We prove that the Kolmogorov distance between the expected VESD and the deformed MP distribution is bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that the entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moments and $|N/M-1|\ge \tau$ for some constant $\tau>0$. This result improves the previous one obtained in \cite{XYZ2013}, which gave the convergence rate $O(N^{-1/2})$ assuming $i.i.d.$ $X$ entries, bounded 10th moment, $\Sigma=I$ and $M<N$. Moreover, we also prove that under the finite $8$th moment assumption, the convergence rate of the VESD is $O(N^{-1/2+\epsilon})$ almost surely for any fixed $\epsilon>0$, which improves the previous bound $N^{-1/4+\epsilon}$ in \cite{XYZ2013}.Comment: To appear Annals of Statistic

Topics:
Mathematics - Probability

Publisher: 'Institute of Mathematical Statistics'

Year: 2020

DOI identifier: 10.1214/19-AOS1832

OAI identifier:
oai:arXiv.org:1705.03954

Provided by:
arXiv.org e-Print Archive

Downloaded from
http://arxiv.org/abs/1705.03954

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.