Superconvergence phenomenon in Wiener chaoses

Abstract

We establish, in full generality, an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. Namely, for every sequence of chaotic random variables, convergence in law to the Gaussian distribution is automatically upgraded to superconvergence: the regularity of the densities increases along the convergence, and all the derivatives converge uniformly on the real line. Our findings strikingly strengthen known results regarding modes of convergence for normal approximation on Wiener chaoses. Without additional assumptions, quantitative convergence in total variation is established by Nourdin and Peccati (Probab. Theory Related Fields 145 (2009) 75–118), and later on amplified to convergence in relative entropy by Nourdin, Peccati and Swan (J. Funct. Anal. 266 (2014) 3170–3207).

Our result is then extended to the multivariate setting and for polynomial mappings of a Gaussian field, provided the projection on the Wiener chaos of maximal degree admits a nondegenerate Gaussian limit. While our findings potentially apply to any context involving polynomial functionals of a Gaussian field, we emphasize, in this work, applications regarding: improved Carbery–Wright estimates near Gaussianity, normal convergence in entropy and in Fisher information, superconvergence for the spectral moments of Gaussian orthogonal ensembles, moments bounds for the inverse of strongly correlated Wishart-type matrices, and superconvergence in the Breuer–Major Theorem.

Our proofs leverage Malliavin’s historical idea to establish smoothness of the density via the existence of negative moments of the Malliavin gradient, and we further develop a new paradigm to study this problem. Namely, we relate the existence of negative moments to some explicit spectral quantities associated with the Malliavin Hessian. This link relies on an adequate choice of the Malliavin gradient, which provides a novel decoupling procedure of independent interest. Previous attempts to establish convergence beyond entropy have imposed restrictive assumptions ensuring finiteness of negative moments for the Malliavin derivatives Our analysis renders these assumptions superfluous.

The terminology superconvergence was introduced by Bercovici and Voiculescu (Probab. Theory Related Fields 103 (1995) 215–222) for the central limit theorem in free probability.