Non-asymptotic variance bounds and deviation inequalities by optimal transport

TL;DR

This paper demonstrates how optimal transport techniques on the real line can derive sharp non-asymptotic variance bounds and deviation inequalities for Gaussian functionals and other models, enhancing superconcentration theory.

Contribution

It introduces a simple, flexible optimal transport approach to obtain sharp variance bounds and deviation inequalities for various functionals of Gaussian vectors and other models.

Findings

Sharp non-asymptotic variance bounds for Gaussian functionals

Exponential deviation inequalities derived from variance bounds

Application to extreme value laws and Coulomb gases

Abstract

The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\R^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.