Open Access
2020 Approximation of Hilbert-Valued Gaussians on Dirichlet structures
Solesne Bourguin, Simon Campese
Electron. J. Probab. 25: 1-30 (2020). DOI: 10.1214/20-EJP551


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.


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Solesne Bourguin. Simon Campese. "Approximation of Hilbert-Valued Gaussians on Dirichlet structures." Electron. J. Probab. 25 1 - 30, 2020.


Received: 18 February 2020; Accepted: 11 November 2020; Published: 2020
First available in Project Euclid: 23 December 2020

Digital Object Identifier: 10.1214/20-EJP551

Primary: ‎46G12 , 46N30 , 60B12 , 60F17

Keywords: Dirichlet structures , fourth moment conditions , Functional limit theorems , Gaussian approximation , Gaussian measures on Hilbert spaces , probabilistic metrics , Stein’s method on Banach spaces

Vol.25 • 2020
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