The discrepancy between min-max statistics of Gaussian and Gaussian-subordinated matrices

TL;DR

This paper provides quantitative bounds on the difference between min-max statistics of Gaussian and Gaussian-subordinated matrices, extending classical inequalities and applying results to stochastic Wiener-Itô integrals.

Contribution

It introduces new bounds for comparing Gaussian and Gaussian-subordinated matrices' min-max statistics, generalizing classical inequalities and applying to stochastic integrals.

Findings

Quantitative bounds for Gaussian matrices recover classical inequalities.

Bounds extend to Gaussian-subordinated matrices, generalizing prior estimates.

Application to stochastic Wiener-Itô integrals with illustrative example.

Abstract

We compute quantitative bounds for measuring the discrepancy between the distribution of two min-max statistics involving either pairs of Gaussian random matrices, or one Gaussian and one Gaussian-subordinated random matrix. In the fully Gaussian setup, our approach allows us to recover quantitative versions of well-known inequalities by Gordon (1985, 1987, 1992), thus generalising the quantitative version of the Sudakov-Fernique inequality deduced in Chatterjee (2005). On the other hand, the Gaussian-subordinated case yields generalizations of estimates by Chernozhukov et al. (2015) and Koike (2019). As an application, we establish fourth moment bounds for matrices of multiple stochastic Wiener-It\^o integrals, that we illustrate with an example having a statistical flavour.