The Fourth-Moment Theorem on Hilbert Spaces
Marie-Christine Düker, Pavlos Zoubouloglou
TL;DR
This work develops an infinite-dimensional Fourth-Moment Theorem for Hilbert-space-valued Wiener chaos, establishing conditions under which chaos sequences converge in distribution to a nondegenerate Gaussian measure. A sharp Stein–Malliavin bound in the trace-class norm for the covariance operators together with a d2-distance framework enables precise control of convergence, including fixed, finite, and infinite chaos expansions. The results recover classical finite-dimensional theorems and provide quantitative bounds, while addressing identifiability issues present in prior Hilbert-space formulations. The paper also demonstrates practical impact through applications to a quantitative Cremers–Kadelka CLT for L^2-valued functionals, a functional CLT for Kernel Ridge Regression in RKHS, and weak-error/invariant-measure analysis for the stochastic heat equation. Overall, it advances Gaussian approximation in infinite dimensions with explicit rates and broad applicability in SPDEs and statistical learning in Hilbert spaces.
Abstract
In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-Itô chaos representation to a nondegenerate Gaussian measure on a separable Hilbert space. Our first main result shows that, assuming convergence of the associated covariance operators in the trace-class norm, a sequence lying in a fixed Wiener-Itô chaos converges in distribution if and only if its fourth weak moments converge to the corresponding Gaussian moments. For general sequences with infinite chaos expansions, we derive analogous sufficient conditions for convergence in distribution. A key ingredient in our approach is a Stein-Malliavin bound formulated with respect to a distance that metrizes weak convergence of probability measures on separable Hilbert spaces. The results are infinite-dimensional extensions of the classical real-valued Fourth-Moment Theorem of Nualart and Peccati [Ann. Probab. 33, 177-193 (2005)]. Our work builds upon the work by Bourguin and Campese [Electron. J. Probab. 25, 1-30 (2020)] who claimed a Fourth-Moment Theorem in separable Hilbert spaces. However, a recent work by Bassetti, Bourguin, Campese, and Peccati [arXiv:2509.13427 (2025)] showed that the distance employed in the former article does not metrize weak convergence of probability measures on separable Hilbert spaces. Consequently, the conditions stated in Bourguin and Campese are not sufficient to recover a valid Fourth-Moment Theorem in the Hilbert-space setting.