The Fourth-Moment Theorem on Hilbert Spaces

Abstract: In a work by Bourguin and Campese [Electron. J. Probab. 25, 1-30 (2020)], a fourth-moment theorem for weak convergence to a Gaussian measure on separable Hilbert spaces was proposed. However, a very 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 on such spaces; consequently, one of the main results therein does not hold as stated. In this paper, we characterize convergence in distribution to a non-degenerate Gaussian measure on a separable Hilbert space for sequences of multiple Wiener-It^o integrals of fixed order. Assuming convergence of the associated covariance operators in the trace class norm, we prove that weak convergence holds if and only if the fourth weak moments converge to their Gaussian counterparts. A key tool in our approach is a Stein-Malliavin bound for a distance metrizing weak convergence on Hilbert spaces, hence extending the classical real-valued result of Nualart and Peccati [Ann. Probab. 33, 177-193 (2005)].