We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting,
thus obtaining a) quantitative central limit theorems for approximation of arbitrary nondegenerate
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.First author draf
