Normal Approximations with Malliavin Calculus

TLDR

Stein's method uses differential operators to measure distances between probability distributions. The book introduces the combination of Stein's method and Malliavin calculus to derive quantitative central limit theorems for functionals of Gaussian fields. It surveys recent developments such as fourth moment theorems, density estimates, Breuer–Major theorems, recursive cumulant computations, optimal rates, and universality results for homogeneous sums. The book is self‑contained, suitable for self‑study, and appeals to researchers and graduate students interested in the interplay between Stein's method and Malliavin calculus.

Abstract

Stein's method is a collection of probabilistic techniques that allow one to assess the distance between two probability distributions by means of differential operators. In 2007, the authors discovered that one can combine Stein's method with the powerful Malliavin calculus of variations, in order to deduce quantitative central limit theorems involving functionals of general Gaussian fields. This book provides an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space. Many recent developments and applications are studied in detail, for instance: fourth moment theorems on the Wiener chaos, density estimates, Breuer–Major theorems for fractional processes, recursive cumulant computations, optimal rates and universality results for homogeneous sums. Largely self-contained, the book is perfect for self-study. It will appeal to researchers and graduate students in probability and statistics, especially those who wish to understand the connections between Stein's method and Malliavin calculus.