The discrepancy in min-max statistics between two random matrices with finite third moments

TL;DR

This paper introduces a new coupling inequality for min-max statistics of two random matrices with finite third moments, extending previous Gaussian-specific bounds to more general matrices using advanced probabilistic methods.

Contribution

It develops a novel coupling inequality that generalizes existing Gaussian bounds to matrices with finite third moments, broadening the scope of min-max statistical analysis.

Findings

Established a new coupling inequality for matrices with finite third moments.

Extended bounds previously limited to Gaussian matrices to more general cases.

Utilized Stein's method of exchangeable pairs for multivariate normal approximation.

Abstract

We propose a novel coupling inequality of the min-max type for two random matrices with finite absolute third moments, which generalizes the quantitative versions of the well-known inequalities by Gordon. Previous results have calculated the quantitative bounds for pairs of Gaussian random matrices. Through integrating the methods utilized by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of exchangeable pairs for multivariate normal approximation, this study eliminates the Gaussian restriction on random matrices, enabling us to achieve more extensive results.