Convergence of eigenvector empirical spectral distribution of sample covariance matrices

Abstract

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 Marčenko–Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma ^{1/2}XX^{*}\Sigma ^{1/2}$, 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 (Ann. Statist. 41 (2013) 2572–2607), 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 (Ann. Statist. 41 (2013) 2572–2607).