Reference : Approximation of Hilbert-valued Gaussians on Dirichlet structures
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Approximation of Hilbert-valued Gaussians on Dirichlet structures
Bourguin, Solesne [Boston University > Mathematics and Statistics]
Campese, Simon mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
Electronic Journal of Probability
Institute of Mathematical Statistics
Yes (verified by ORBilu)
[en] Stein's method ; Malliavin calculus ; Gaussian approximation ; Gamma calculus ; functional central limit theorem ; quantitative central limit theorem ; fourtth moment theorem
[en] 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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