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June 2020 Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices
T. Tony Cai, Xiao Han, Guangming Pan
Ann. Statist. 48(3): 1255-1280 (June 2020). DOI: 10.1214/18-AOS1798


We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy–Widom law for the largest nonspiked sample eigenvalue is obtained.

Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a central limit theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.


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T. Tony Cai. Xiao Han. Guangming Pan. "Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices." Ann. Statist. 48 (3) 1255 - 1280, June 2020.


Received: 1 November 2017; Revised: 1 August 2018; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241590
MathSciNet: MR4124322
Digital Object Identifier: 10.1214/18-AOS1798

Primary: 60B20, 62H25
Secondary: 60F05, 62H10

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 3 • June 2020
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