Abstract
We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth.
Original language | English |
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Pages (from-to) | 999-1036 |
Number of pages | 38 |
Journal | Probability Theory and Related Fields |
Volume | 187 |
Issue number | 3-4 |
DOIs | |
State | Published - Dec 2023 |
Bibliographical note
Publisher Copyright:© 2023, The Author(s).
Funding
Mikhail Sodin first suggested the project to us. We had a number of interesting and fruitful discussions with Yan Fyodorov, Marie Kratz and Igor Wigman on various topics related to this article. Eugene Shargorodsky contributed the main idea in the proof of Claim . We are thankful to Ohad Feldheim and the anonymous referees for suggesting improvements to the presentation of this paper. The research of E.A. is supported by a Simons Collaboration Grant (550029, to John Voight), his research was partly conducted while hosted in King’s College London and supported by EPSRC grant EP/L025302/1. The research of J.B. is supported in part by EPSRC New Investigator Award EP/V002449/1. The research of N.F. is partially supported by Israel Science Foundation grant 1327/19.
Funders | Funder number |
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Simons Collaboration Grant | 550029 |
Engineering and Physical Sciences Research Council | EP/L025302/1, EP/V002449/1 |
Israel Science Foundation | 1327/19 |
Keywords
- Fluctuations of zeroes
- Gaussian process
- Stationary process
- Wiener Chaos