Density estimates and concentration inequalities with Malliavin calculus

TL;DR

This paper leverages Malliavin calculus to derive density estimates and concentration inequalities for Gaussian-related random variables, providing new bounds and improvements on classical inequalities like Borell-Sudakov.

Contribution

It introduces a novel density formula for Gaussian functionals and applies it to improve bounds on the maximum of Gaussian processes and related concentration inequalities.

Findings

New density bounds for Gaussian process maxima

Improved Borell-Sudakov inequalities

Applicability to tail probability estimates

Abstract

We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a non-degeneracy condition, we prove and use a new formula for the density of a random variable which is measurable and differentiable with respect to a given isonormal Gaussian process. Among other results, we apply our techniques to bound the density of the maximum of a general Gaussian process from above and below; several new results ensue, including improvements on the so-called Borell-Sudakov inequality. We then explain what can be done when one is only interested in or capable of deriving concentration inequalities, i.e. tail bounds from above or below but not necessarily both simultaneously.