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

TL;DR

This paper derives an asymptotic formula for the variance of zero counts in stationary Gaussian processes, revealing how spectral measure atoms influence growth rates under mild mixing conditions.

Contribution

It provides a simple asymptotic description of variance growth and characterizes the impact of spectral measure atoms on this growth.

Findings

Small atoms at a special frequency do not affect asymptotic growth

Atoms at other frequencies lead to maximal variance growth

Results apply to various examples of stationary Gaussian processes

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. Our results allow us to analyse a large number of interesting examples.