19 research outputs found
Lectures on Gaussian approximations with Malliavin calculus
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 Coll\`ege de
France between January and March 2012, within the framework of the annual prize
of the Fondation des Sciences Math\'ematiques 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.Comment: 72 pages. To be published in the S\'eminaire de Probabilit\'es. Mild
update: typos, referee comment