May 2024 Superconvergence phenomenon in Wiener chaoses
Ronan Herry, Dominique Malicet, Guillaume Poly
Author Affiliations +
Ann. Probab. 52(3): 1162-1200 (May 2024). DOI: 10.1214/24-AOP1689


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.

Funding Statement

R.H. gratefully acknowledges funding from Centre Henri Lebesgue (ANR-11-LABX-0020-01) through a research fellowship in the framework of the France 2030 program.
This work was supported by the ANR Grant UNIRANDOM (ANR-17-CE40-0008).


The authors are grateful to G. Cébron for suggesting the reference [5] regarding superconvergence in the free central limit theorem. The authors are indebted to the anonymous referee. Their thorough review and thoughtful comments have greatly helped us in the preparation of the final version of this paper.


Download Citation

Ronan Herry. Dominique Malicet. Guillaume Poly. "Superconvergence phenomenon in Wiener chaoses." Ann. Probab. 52 (3) 1162 - 1200, May 2024.


Received: 1 May 2023; Revised: 1 January 2024; Published: May 2024
First available in Project Euclid: 23 April 2024

Digital Object Identifier: 10.1214/24-AOP1689

Primary: 28C20 , 60B12 , 60H07

Keywords: Malliavin calculus , Malliavin–Stein approach , superconvergence , Wiener Chaos

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 3 • May 2024
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