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

Abstract

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.

Variance linearity for real Gaussian zeros

“…It remains open to find a necessary and sufficient condition. During the revision of the present work, Assaf, Buckley and Feldheim [2] have given an upper bound for the variance, and a mixing condition under which the condition (C, C ∈ L 2 ) is necessary and sufficient. Regarding point (iii), they also investigated the behaviour of the variance when there is an atom at ±x 0 , that they call a special atom.…”

“…In the rest of this section we take X as a stationary Gaussian process whose reduced covariance function C satisfies Geman's condition (2). In particular, C and C (0) exist, hence Fatou's lemma yields that the spectral measure µ has a finite second moment, and…”

Diophantine Gaussian excursions and random walks

“…Elementary considerations yield that the average number of crossings on an interval is proportional to the length of the interval. Furthermore, if µ contains more than one (symmetrised) atom, the variance of the number of crossings is quadratic [24,2]. We focus here on the Lebesgue measure of the nodal excursions {X > 0} = {t ∈ R d : X(t) > 0}.…”