Malliavin-Stein

NOTE: If you are interested in the Malliavin-Stein approach, you can take a look at my dedicated webpage HERE

• In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the ``fourth moment theorem'' nowadays; alternative proofs can be found here, here and here) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution towards the standard normal law is in fact equivalent to convergence of the fourth moment only. Shortly after, Peccati and Tudor gave a multidimensional version of this characterization.

• Since the publication of these two pathbreaking papers, many improvements and developments on this theme have been considered. Among them is the work of 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 paper "Stein's method on Wiener chaos" written jointly with Peccati in which, by bringing together Stein's method with Malliavin calculus, we were able (among other things) to associate quantitative bounds with the fourth moment theorem.

• It turns out that Stein's method and Malliavin calculus fit together admirably well, and that their interaction has led to remarkable new results. The webpage on Stein's method and Malliavin calculus aims to bring together all the research papers linked, in a way or another, to the fourth moment theorem.

• With Giovanni Peccati, we wrote in 2012 a book entitled "Normal Approximations with Malliavin Calculus: from Stein's Method to Universality". It provides an introduction both to Stein's method and Malliavin calculus, from the point of view of normal approximations on a Gaussian space. It is published by Cambridge University Press in the series "Cambridge Tracts in Mathematics''.

• Between January and March 2012, I gave a series of lectures of three hours each (with the exception of the first lecture, which was `only' two hours and was followed by a cocktail) at the prestigious Collège de France. I mainly talked about my recent research on Stein's method, Malliavin calculus and free probability. Here is the dedicated poster. The associated lecture notes are published in the Séminaire de Probabilités XLV, pp. 3-89. And here is another link, where you can watch the videos of the lectures.