Functional second-order Gaussian Poincar\'e inequalities

TL;DR

This paper develops a functional second-order Gaussian Poincaré inequality within Hilbert-valued Wiener structures, providing versatile bounds for Gaussian process approximation applicable to diverse fields like neural networks and spatial statistics.

Contribution

It introduces a new functional second-order Gaussian Poincaré inequality that generalizes existing bounds for Gaussian process approximation.

Findings

Provides abstract bounds for Gaussian process approximation in $d_2$ distance.

Applicable to functional Breuer-Major theorems, neural networks, and SPDE spatial statistics.

Abstract

In this paper, we work in the framework of Hilbert-valued Wiener structures and derive a functional version of the second-order Gaussian Poincar\'e inequality that leads to abstract bounds for Gaussian process approximation in $d_2$ distance. Our abstract bounds are flexible and can be applied in various examples including functional Breuer-Major central limit theorems, shallow neural networks, and spatial statistics of SPDEs solutions.