Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields

Abstract

We introduce a boundedness condition on the Malliavin derivative of a random variable to study sub-Gaussian and other non-Gaussian properties of functionals of random fields, with particular attention to the estimation of suprema. We relate the boundedness of the nth Malliavin derivative to a new class of “sub-nth-Gaussian chaos” processes. An expected supremum estimation, extending the Dudley theorem, is proved for such processes. Sub-nth-Gaussian chaos concentration inequalities for the supremum are obtained, using Malliavin derivative conditions; for n = 1, this generalizes the Borell-Sudakov inequality to a class of sub-Gaussian processes, with a particularly simple and efficient proof; for n = 2 a natural extension to sub-2nd-Gaussian chaos processes is established; for n ≥ 3 a slightly less efficient Malliavin derivative condition is needed.