Functional Gaussian approximations on Hilbert-Poisson spaces

TL;DR

This paper introduces a new functional Stein-Malliavin method for Hilbert-Poisson spaces, enabling quantitative CLTs and moment bounds for Gaussian approximations in infinite-dimensional settings.

Contribution

It develops an infinite-dimensional Stein's method using Gamma calculus for Gaussian approximation in Hilbert spaces, with applications to Poisson processes and geometric graph statistics.

Findings

Quantitative CLTs for Gaussian elements in Hilbert spaces

Fourth moment bounds for chaos expansion sequences

Applications to Poisson process approximation and geometric graph analysis

Abstract

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary non-degenerate Gaussian random elements taking values in a separable Hilbert space and b) fourth moment bounds for approximating sequences with finite chaos expansion. Our results rely on an infinite-dimensional version of Stein's method of exchangeable pairs combined with the so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic of a random geometric graph.