Functional Gaussian approximations on Hilbert-Poisson spaces

Solesne Bourguin; Simon Campese; Thanh Dang

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arXiv · 2110.04877

Functional Gaussian approximations on Hilbert-Poisson spaces

Solesne Bourguin·Simon Campese·Thanh Dang

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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.

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Solesne Bourguin, Simon Campese, Thanh Dang. 2021-10-10. Functional Gaussian approximations on Hilbert-Poisson spaces. https://arxiv.org/abs/2110.04877

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