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October 2013 Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix
Ningning Xia, Yingli Qin, Zhidong Bai
Ann. Statist. 41(5): 2572-2607 (October 2013). DOI: 10.1214/13-AOS1154

Abstract

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of sample covariance matrix and the Marčenko–Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution.

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Ningning Xia. Yingli Qin. Zhidong Bai. "Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix." Ann. Statist. 41 (5) 2572 - 2607, October 2013. https://doi.org/10.1214/13-AOS1154

Information

Published: October 2013
First available in Project Euclid: 19 November 2013

zbMATH: 1285.15018
MathSciNet: MR3161438
Digital Object Identifier: 10.1214/13-AOS1154

Subjects:
Primary: 15A52, 60F15, 62E20
Secondary: 60F17, 62H99

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 5 • October 2013
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