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

Eran Assaf; Jeremiah Buckley; Naomi Feldheim

doi:https://doi.org/10.1007/s00440-023-01218-4

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

Eran Assaf, Jeremiah Buckley, Naomi Feldheim

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-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 language	English
Pages (from-to)	999-1036
Number of pages	38
Journal	Probability Theory and Related Fields
Volume	187
Issue number	3-4
DOIs	https://doi.org/10.1007/s00440-023-01218-4
State	Published - Dec 2023

Keywords

Fluctuations of zeroes
Gaussian process
Stationary process
Wiener Chaos

All Science Journal Classification (ASJC) codes

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

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https://doi.org/10.1007/s00440-023-01218-4

Cite this

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N2 - 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.

AB - 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.

KW - Fluctuations of zeroes

KW - Gaussian process

KW - Stationary process

KW - Wiener Chaos

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

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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