In this chapter, we show how Malliavin calculus and Stein's method may be combined into a powerful and flexible tool for studying probabilistic approximations. In particular, our aim is to use these two techniques to assess the distance between the laws of regular functionals of an isonormal Gaussian process and a one-dimensional normal distribution.

The highlight of the chapter is arguably Section 5.2, where we deduce a complete characterization of Gaussian approximations inside a fixed Wiener chaos. As discussed below, the approach developed in this chapter yields results that are systematically stronger than the so-called ‘method of moments and cumulants’, which is the most popular tool used in the proof of central limit theorems for functional of Gaussian fields.

Note that, in view of the chaos representation (2.7.8), any general result involving random variables in a fixed chaos is a key for studying probabilistic approximations of more general functionals of Gaussian fields. This last point is indeed one of the staples of the entire book, and will be abundantly illustrated in Section 5.3 as well as in Chapter 7.

Throughout the following, we fix an isonormal Gaussian process X = {X(h): h ∈ h}, defined on a suitable probability space (Ω, ℱ, P) such that ℱ = σ {X}. We will also adopt the language and notation of Malliavin calculus introduced in Chapter 2.

This is a text about probabilistic approximations, which are mathematical statements providing estimates of the distance between the laws of two random objects. As the title suggests, we will be mainly interested in approximations involving one or more normal (equivalently called Gaussian) random elements. Normal approximations are naturally connected with central limit theorems (CLTs), i.e. convergence results displaying a Gaussian limit, and are one of the leading themes of the whole theory of probability.

The main thread of our text concerns the normal approximations, as well as the corresponding CLTs, associated with random variables that are functionals of a given Gaussian field, such as a (fractional) Brownian motion on the real line. In particular, a pivotal role will be played by the elements of the socalled Gaussian Wiener chaos. The concept of Wiener chaos generalizes to an infinite-dimensional setting the properties of the Hermite polynomials (which are the orthogonal polynomials associated with the one-dimensional Gaussian distribution), and is now a crucial object in several branches of theoretical and applied Gaussian analysis.

The cornerstone of our book is the combination of two probabilistic techniques, namely the Malliavin calculus of variations and Stein's method for probabilistic approximations.

The Malliavin calculus of variations is an infinite-dimensional differential calculus, whose operators act on functionals of general Gaussian processes. Initiated by Paul Malliavin (starting from the seminal paper [69], which focused on a probabilistic proof of Hörmander's ‘sum of squares’ theorem), this theory is based on a powerful use of infinite-dimensional integration by parts formulae.