Stein's method meets Malliavin calculus: a short survey with new estimates

TL;DR

This paper surveys recent techniques combining Stein's method and Malliavin calculus for Gaussian approximation, introduces new moment estimates for Wiener chaos, and applies these to central limit theorems for fractional Brownian motion.

Contribution

It establishes explicit links between Stein's method and the method of moments, providing new estimates for moments in Wiener chaos and applying them to fractional Brownian motion CLTs.

Findings

New moment estimates for Wiener chaos variables

Explicit connections between Stein's method and Malliavin calculus

Central limit theorems for fractional Brownian motion quadratic variation

Abstract

We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ``Stein's method'' for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.