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

We compute quantitative bounds for measuring the discrepancy between the distribution of two min–max statistics involving either one pair 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 generalizing 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–Itô integrals, that we illustrate with an example having a statistical flavor.