Functional Gaussian approximations on Hilbert-Poisson spaces (Q6564544)

Given a Gaussian process \(Z\), and a square integrable and measurable transformation \(X\) of a Poisson process, each defined on the same separable Hilbert space, the authors give explicit bounds on the distance between \(X\) and \(Z\) in terms of moments and contractions. These bounds are established by a combination of Stein's method of exchangeable pairs and Malliavin calculus in an infinite-dimensional and non-diffusive setting, and provide an analogue of the fourth moment theorem in this setting. These bounds are illustrated with two applications: the first is to the Brownian approximation of Poisson processes with growing intensity in Besov--Liouville spaces, and the second is to an edge-counting statistic in a random geometric graph.