Approximation of Hilbert-valued Gaussians on Dirichlet structures
Abstract
We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments bounds for Hilbert-valued random variables with possibly infinite chaos expansion, which include, as special cases, all finite-dimensional four moments results for Gaussian approximation in a diffusive context proved earlier by various authors. Our main ingredient is a combination of an infinite-dimensional version of Stein's method as developed by Shih and the so-called Gamma calculus. As an application, rates of convergence for the functional Breuer-Major theorem are established.
Cite
@article{arxiv.1905.05127,
title = {Approximation of Hilbert-valued Gaussians on Dirichlet structures},
author = {Solesne Bourguin and Simon Campese},
journal= {arXiv preprint arXiv:1905.05127},
year = {2020}
}