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March 2024 Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields
Leonardo Maini, Ivan Nourdin
Author Affiliations +
Ann. Probab. 52(2): 737-763 (March 2024). DOI: 10.1214/23-AOP1669


Let B=(Bx)xRd be a collection of N(0,1) random variables forming a real-valued continuous stationary Gaussian field on Rd, and set C(xy)=E[BxBy]. Let φ:RR be such that E[φ(N)2]< with NN(0,1), let R be the Hermite rank of φ, and consider Yt=tDφ(Bx)dx, t>0 with DRd compact.

Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for Yt have been constantly refined, extended and applied to an increasing number of diverse situations, to such an extent that it has become a field of research in its own right.

The common belief, representing the intuition that specialists in the subject have developed over the last four decades, is that as t the fluctuations of Yt around its mean are, in general (i.e., except possibly in very special cases), Gaussian when B has short memory, and non-Gaussian when B has long memory and R2.

We show in this paper that this intuition forged over the last 40 years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where Yt admits Gaussian fluctuations in a long memory context.

To achieve this goal, we state and prove a spectral central limit theorem, which extends the conclusion of the celebrated Breuer–Major theorem to situations where CLR(Rd). Our main mathematical tools are the Malliavin–Stein method and Fourier analysis techniques.

Funding Statement

L. Maini was supported by the Luxembourg National Research Fund PRIDE17/1224660/GPS. I. Nourdin was supported by the Luxembourg National Research O22/17372844/FraMStA.


We would like to thank the referee for a careful reading, constructive remarks and useful suggestions.


Download Citation

Leonardo Maini. Ivan Nourdin. "Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields." Ann. Probab. 52 (2) 737 - 763, March 2024.


Received: 1 August 2022; Revised: 1 May 2023; Published: March 2024
First available in Project Euclid: 4 March 2024

MathSciNet: MR4718405
Digital Object Identifier: 10.1214/23-AOP1669

Primary: 60F05 , 60G15 , 60H07

Keywords: Fourier analysis , Hermite rank , isotropic Gaussian fields , long memory , Malliavin–Stein method , short memory , Spectral central limit theorem , stationary Gaussian fields

Rights: This research was funded, in whole or in part, by Fonds National de la recherche - FNR, 022/17372844/FraMSLA. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant’s open access conditions.

Vol.52 • No. 2 • March 2024
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