Open Access
April 2020 Convergence of eigenvector empirical spectral distribution of sample covariance matrices
Haokai Xi, Fan Yang, Jun Yin
Ann. Statist. 48(2): 953-982 (April 2020). DOI: 10.1214/19-AOS1832


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).


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Haokai Xi. Fan Yang. Jun Yin. "Convergence of eigenvector empirical spectral distribution of sample covariance matrices." Ann. Statist. 48 (2) 953 - 982, April 2020.


Received: 1 November 2017; Revised: 1 January 2019; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241576
MathSciNet: MR4102683
Digital Object Identifier: 10.1214/19-AOS1832

Primary: 15B52 , 62E20
Secondary: 62H99

Keywords: Eigenvector empirical spectral distribution , Empirical spectral distribution , Marčenko–Pastur distribution , Sample covariance matrix

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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