Lectures on Gaussian approximations with Malliavin calculus
arXiv:1203.4147
Abstract
In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the "Fourth Moment Theorem" in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor gave a multidimensional version of this characterization. Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and Ortiz-Latorre, giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper "Stein's method on Wiener chaos" (written in collaboration with Peccati) in which, by bringing together Stein's method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the Fourth Moment Theorem. It turns out that Stein's method and Malliavin calculus fit together admirably well. Their interaction has led to some remarkable new results involving central and non-central limit theorems for functionals of infinite-dimensional Gaussian fields. The current survey aims to introduce the main features of this recent theory. It originates from a series of lectures I delivered at the Collège de France between January and March 2012, within the framework of the annual prize of the Fondation des Sciences Mathématiques de Paris. It may be seen as a teaser for the book "Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality" (jointly written with Peccati), in which the interested reader will find much more than in this short survey.
72 pages. To be published in the Séminaire de Probabilités. Mild update: typos, referee comments
References in corpus (9)
- Central limit theorems for sequences of multiple stochastic integrals
- Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos
- Stein's method and exact Berry--Esseen asymptotics for functionals of Gaussian fields
- Noncentral convergence of multiple integrals
- Hitting times for Gaussian processes
- The Chen-Stein method for Poisson functionals
- Density estimates and concentration inequalities with Malliavin calculus
- Fourth Moment Theorem and q-Brownian Chaos
- Invariance principles for homogeneous sums of free random variables