The Annals of Statistics

Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix

Ningning Xia, Yingli Qin, and Zhidong Bai

Full-text: Open access

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.

Article information

Source
Ann. Statist., Volume 41, Number 5 (2013), 2572-2607.

Dates
First available in Project Euclid: 19 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1384871346

Digital Object Identifier
doi:10.1214/13-AOS1154

Mathematical Reviews number (MathSciNet)
MR3161438

Zentralblatt MATH identifier
1285.15018

Subjects
Primary: 15A52 60F15: Strong theorems 62E20: Asymptotic distribution theory
Secondary: 60F17: Functional limit theorems; invariance principles 62H99: None of the above, but in this section

Keywords
Eigenvector empirical spectral distribution empirical spectral distribution Marčenko–Pastur distribution sample covariance matrix Stieltjes transform

Citation

Xia, Ningning; Qin, Yingli; Bai, Zhidong. Convergence rates of eigenvector empirical spectral distribution of large dimensional sample covariance matrix. Ann. Statist. 41 (2013), no. 5, 2572--2607. doi:10.1214/13-AOS1154. https://projecteuclid.org/euclid.aos/1384871346


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