Density estimates and concentration inequalities with Malliavin calculus

Ivan Nourdin (1) , Frederi Viens (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 Department of Statistics

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 Z 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.

Mots clés: Malliavin calculus; density estimates; concentration inequalities; fractional Brownian motion; Borell-Sudakov inequality; suprema of Gaussian processes

Domaines: Probabilités [math.PR]

Ivan Nourdin, Frederi Viens. Density estimates and concentration inequalities with Malliavin calculus. 2008. ⟨hal-00311755⟩