Long gaps between sign-changes of Gaussian Stationary Processes

TL;DR

This paper investigates the probability that a Gaussian stationary process remains positive over a large interval, establishing bounds on its decay rate based on spectral measure properties.

Contribution

It generalizes existing bounds on positivity probability decay for Gaussian stationary processes, linking spectral measure density conditions to exponential bounds.

Findings

Probability decays exponentially with interval length N

Bounds depend on spectral measure density bounds

Extends previous specific case results

Abstract

We study the probability of a real-valued stationary process to be positive on a large interval $[0,N]$. We show that if in some neighborhood of the origin the spectral measure of the process has density which is bounded away from zero and infinity, then the decay of this probability is bounded between two exponential functions in $N$. This generalizes similar bounds obtained for particular cases, such as a recent result by Artezana, Buckley, Marzo, Olsen.