Steins method, Malliavin calculus and infinite-dimensional Gaussian analysis

This expository paper is a companion of the four one-hour tutorial lectures given in the occasion of the special month Progress in Stein’s Method, held at the University of Singapore in January 2009. We will explain how one can combine Stein’s method with Malliavin calculus, in order to obtain explicit bounds in the normal and Gamma approximation of functionals of in…nite-dimensional Gaussian …elds. The core of our discussion is based on a series of papers jointly written with I. Nourdin, as well as with I. Nourdin and A. Réveillac. Key Words: Central limit theorem; Gamma approximation; Gaussian approximation; Gaussian processes; Malliavin calculus; Stein’s method; Wiener chaos. Mathematics Subject Classi…cation: 60F05 60G15 60H05 60H07 Contents 1 Overview and motivation 2 2 Preliminary example: exploding quadratic Brownian functionals without Stein’s method 4 2.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The method of cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Random time-changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Gaussian measures 9 4 Wiener-Itô integrals 11 4.1 Single integrals and the …rst Wiener chaos . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Multiplication formulae 16 5.1 Contractions and multiplications . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Multiple stochastic integrals as Hermite polynomials . . . . . . . . . . . . . . . . 17 5.3 Chaotic decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Équipe Modal’X, Université Paris Ouest Naterre La Défense, 200 Avenue de la République, 92000 Nanterre and LSTA, Université Paris VI, France. E-mail: [email protected]. yThese notes will be posted and updated on my website www.geocities.com/giovannipeccati. Please, send me corrections/comments!