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

Eran Assaf, Jeremiah Buckley, Naomi Feldheim

Published 2021-01-11Version 1

We study the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description of the variance of this random variable, under mild mixing conditions. In particular, we give a linear lower bound for any non-degenerate process. 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. We state some conjectures which generalise our results.