Stein's method and exact Berry--Esseen asymptotics for functionals of Gaussian fields

TL;DR

This paper refines the understanding of normal approximation for Gaussian field functionals by establishing optimal Berry--Esseen bounds using advanced probabilistic techniques, with applications to stationary processes, Brownian sheets, and fractional Brownian motion.

Contribution

It introduces a new approach combining Malliavin calculus, Stein's method, and cumulants to achieve precise Berry--Esseen bounds and local Edgeworth expansions for Gaussian functionals.

Findings

Established optimal Berry--Esseen bounds for Gaussian functionals.

Extended results to quadratic functionals of stationary processes.

Applied methods to Brownian sheets and fractional Brownian motion.

Abstract

We show how to detect optimal Berry--Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein's method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75--118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395--406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551--579]); (ii) to ``exploding'' quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer--Major CLT for functionals of a fractional Brownian motion.