Lectures on Gaussian approximations with Malliavin calculus

Overview. In a seminal paper of 2005, Nualart and Peccati [40] 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 [46] 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 [39], 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 [27] (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 [32], in which the interested reader will find much more than in this short survey.