Stein's method, Malliavin calculus, Dirichlet forms and the fourth moment theorem (Q2800233)

In this expository article, the authors review the connections of Stein's method, Malliavin calculus and the fourth moment theorem. After a sketch of Stein's method, including an illustration of the bound on the Wasserstein distance of the normal approximation, a brief survey of Malliavin calculus is presented. The next sections are devoted to Malliavin calculus and Stein's method following the celebrated Nualart-Peccati fourth moment theorem on normal approximation in a fixed Wiener chaos. In particular, the proof of Nourdin and Peccati for bounds on the total variation distance of this normal approximation is reviewed. Finally, the reformulation of the Nourdin-Peccati result in terms of Dirichlet structures due to Ledoux is presented. Special attention is given to the involved operators and the background connections via integration by parts formulae. Moreover, the paper contains some interesting historical notes on the development of these topics.NEWLINENEWLINEFor the entire collection see [Zbl 1318.31001].