Abstract
Let be a collection of random variables forming a real-valued continuous stationary Gaussian field on , and set . Let be such that with , let R be the Hermite rank of φ, and consider , with compact.
Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for 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 the fluctuations of 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 .
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 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 . 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.
Acknowledgments
We would like to thank the referee for a careful reading, constructive remarks and useful suggestions.
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. https://doi.org/10.1214/23-AOP1669
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