An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process

Eran Assaf, Jeremiah Buckley, Naomi Feldheim

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)999-1036
Number of pages38
JournalProbability Theory and Related Fields
Volume187
Issue number3-4
DOIs
StatePublished - 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.

FundersFunder number
Simons Collaboration Grant550029
Engineering and Physical Sciences Research CouncilEP/L025302/1, EP/V002449/1
Israel Science Foundation1327/19

    Keywords

    • Fluctuations of zeroes
    • Gaussian process
    • Stationary process
    • Wiener Chaos

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