An Improved Second Order Poincar\'e Inequality for Functionals of Gaussian Fields

TL;DR

This paper introduces an improved second order Gaussian Poincaré inequality that enhances bounds on distributional distances for functionals of Gaussian fields, leading to more precise quantitative CLTs and optimal convergence rates.

Contribution

The paper develops a refined second order Poincaré inequality for Gaussian functionals, improving existing bounds and applications in quantitative CLTs for stationary Gaussian fields.

Findings

Enhanced bounds for distributional distances to normality.

Improved rates of convergence in quantitative CLTs.

Application to non-linear functionals of Gaussian fields.

Abstract

We present an improved version of the second order Gaussian Poincar\'e inequality, firstly introduced in Chatterjee (2009) and Nourdin, Peccati and Reinert (2009). These novel estimates are used in order to bound distributional distances between functionals of Gaussian fields and normal random variables. Several applications are developed, including quantitative CLTs for non-linear functionals of stationary Gaussian fields related to the Breuer-Major theorem, improving previous findings in the literature and obtaining presumably optimal rates of convergence.