Stein meets Malliavin in normal approximation

TL;DR

This paper explores the connection between Stein's method and Malliavin calculus for normal approximation, highlighting the fourth moment theorem and providing error bounds in Gaussian process functionals.

Contribution

It offers an exposition of Nourdin and Peccati's work linking Stein's method with Malliavin calculus, including the fourth moment theorem and error bounds in CLTs.

Findings

Established a fundamental connection between Stein's method and Malliavin calculus.

Provided error bounds in total variation for Gaussian functionals.

Highlighted the significance of the fourth moment theorem in normal approximation.

Abstract

Stein's method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation the functional equation is a first order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (2009) established a fundamental connection between Stein's method for normal approximation and Malliavin calculus through integration by parts. This connection is exploited to obtain error bounds in total variation in central limit theorems for functionals of general Gaussian processes. Of particular interest is the fourth moment theorem which provides error bounds of the order $\sqrt{\mathbb{E}(F_n^4)-3}$ in the central limit theorem for elements $\{F_n\}_{n\ge 1}$ of Wiener chaos of any fixed order such that $\mathbb{E}(F_n^2) = 1$. This paper is an exposition of the work of Nourdin and Peccati with a brief introduction to Stein's method and Malliavin calculus. It is based on a lecture delivered at the Annual Meeting of the Vietnam Institute for Advanced Study in Mathematics in July 2014.