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2021 Zeros of smooth stationary Gaussian processes
Michele Ancona, Thomas Letendre
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Electron. J. Probab. 26: 1-81 (2021). DOI: 10.1214/21-EJP637

Abstract

Let f:RR be a stationary centered Gaussian process. For any R>0, let νR denote the counting measure of {xRf(Rx)=0}. Under suitable assumptions on the regularity of f and the decay of its correlation function at infinity, we derive the asymptotics as R+ of the central moments of the linear statistics of νR. In particular, we derive an asymptotics of order Rp2 for the p-th central moment of the number of zeros of f in [0,R]. As an application, we prove a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures νR. More precisely, after a proper rescaling, νR converges almost surely towards the Lebesgue measure in weak-∗ sense. Moreover, the fluctuation of νR around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the k-point function of the zero point process of f, for any k2. Our analysis yields two results of independent interest. First, we derive an equivalent of this k-point function near any point of the large diagonal in Rk, thus quantifying the short-range repulsion between zeros of f. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of f.

Funding Statement

This work was supported by the French National Research Agency grants UniRaNDom (ANR-17-CE40-0008) and SpInQS (ANR-17-CE40-0011) and by the Israeli Science Foundation Grants 382/15 and 501/18.

Acknowledgments

Thomas Letendre thanks Julien Fageot for useful discussions about Fernique’s Theorem, Benoit Laslier for his help in the proof of Lemma C.2 and Hugo Vanneuville for pointing out the relation between ergodicity and decay of correlations. The authors are grateful to Damien Gayet for suggesting they write this paper in the first place, and to Misha Sodin for bringing to their attention the intrinsic interest of clustering properties for k-point functions. They also thank Jean-Yves Welschinger for his support and Louis Gass for spotting an error in an earlier version of Lemma 7.8. Finally, the authors thank the anonymous referee for their careful reading of the paper, and their comments that helped to improve the exposition of the main results.

Citation

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Michele Ancona. Thomas Letendre. "Zeros of smooth stationary Gaussian processes." Electron. J. Probab. 26 1 - 81, 2021. https://doi.org/10.1214/21-EJP637

Information

Received: 1 September 2020; Accepted: 27 April 2021; Published: 2021
First available in Project Euclid: 12 May 2021

Digital Object Identifier: 10.1214/21-EJP637

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