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August 2018 On Gaussian comparison inequality and its application to spectral analysis of large random matrices
Fang Han, Sheng Xu, Wen-Xin Zhou
Bernoulli 24(3): 1787-1833 (August 2018). DOI: 10.3150/16-BEJ912

Abstract

Recently, Chernozhukov, Chetverikov, and Kato (Ann. Statist. 42 (2014) 1564–1597) developed a new Gaussian comparison inequality for approximating the suprema of empirical processes. This paper exploits this technique to devise sharp inference on spectra of large random matrices. In particular, we show that two long-standing problems in random matrix theory can be solved: (i) simple bootstrap inference on sample eigenvalues when true eigenvalues are tied; (ii) conducting two-sample Roy’s covariance test in high dimensions. To establish the asymptotic results, a generalized $\varepsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent interest.

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Fang Han. Sheng Xu. Wen-Xin Zhou. "On Gaussian comparison inequality and its application to spectral analysis of large random matrices." Bernoulli 24 (3) 1787 - 1833, August 2018. https://doi.org/10.3150/16-BEJ912

Information

Received: 1 November 2015; Revised: 1 July 2016; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839252
MathSciNet: MR3757515
Digital Object Identifier: 10.3150/16-BEJ912

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 3 • August 2018
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