Gaussian approximation of functionals: Malliavin calculus and Stein’s method

Abstract

Combining Malliavin calculus and Stein’s method has recently lead to a new framework for normal and for chi-square approximation, and for second-order Poincaré inequalities. Applications include functionals of Gaussian random fields as well as functionals of infinite Poisson and Rademacher sequences. Here we present the framework and an extension which leads to invariance principles for multilinear homogeneous sums of general centered independent random variables. In the spirit of Stein’s method, the key is weak dependence, which can be quantified by the influence function of a variable. In particular we see that Wiener chaos is universal, in the sense that the normal approximations of any homogeneous sum can be completely characterised by first switching to its Wiener chaos counterpart. Then the Malliavin–Stein framework yields simple upper bounds and convergence criteria to the normal (and to the chi-square) distribution. These results are joint work with Ivan Nourdin and Giovanni Peccati.