Exponential concentration for zeroes of stationary Gaussian processes

TL;DR

This paper proves that the number of zeroes of certain stationary Gaussian processes concentrates sharply around its mean, with deviations decreasing exponentially fast as the interval length increases.

Contribution

It establishes exponential concentration results for zero counts of stationary Gaussian processes with specific spectral properties, extending understanding of their zero distribution.

Findings

Zero counts are within ηT of the mean with high probability.

Deviations from the mean decay exponentially with T.

Results apply to processes with integrable covariance and spectral measures with finite exponential moments.

Abstract

We show that for any centered stationary Gaussian process of integrable covariance, whose spectral measure has compact support, or finite exponential moments (and some additional regularity), the number of zeroes of the process in $[0,T]$ is within $\eta T$ of its mean value, up to an exponentially small in $T$ probability.