## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 1787-1833.

### On Gaussian comparison inequality and its application to spectral analysis of large random matrices

Fang Han, Sheng Xu, and Wen-Xin Zhou

#### 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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 3 (2018), 1787-1833.

**Dates**

Received: November 2015

Revised: July 2016

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540460

**Digital Object Identifier**

doi:10.3150/16-BEJ912

**Mathematical Reviews number (MathSciNet)**

MR3757515

**Zentralblatt MATH identifier**

06839252

**Keywords**

extreme value theory Gaussian comparison inequality random matrix theory Roy’s largest root test spectral analysis

#### Citation

Han, Fang; Xu, Sheng; Zhou, Wen-Xin. On Gaussian comparison inequality and its application to spectral analysis of large random matrices. Bernoulli 24 (2018), no. 3, 1787--1833. doi:10.3150/16-BEJ912. https://projecteuclid.org/euclid.bj/1517540460