On Gaussian Comparison Inequality and Its Application to Spectral Analysis of Large Random Matrices

TL;DR

This paper applies a new Gaussian comparison inequality to improve spectral analysis of large random matrices, enabling bootstrap inference on eigenvalues and high-dimensional covariance testing.

Contribution

It introduces novel methods for spectral inference in large random matrices, including bootstrap techniques for tied eigenvalues and high-dimensional covariance tests.

Findings

Bootstrap inference on sample eigenvalues with tied true eigenvalues.

Two-sample Roy's covariance test in high dimensions.

Development of new empirical process bounds for spectral norms.

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 $\epsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent interest.