Persistence of Gaussian stationary processes: a spectral perspective

TL;DR

This paper investigates how the likelihood of a Gaussian stationary process staying positive over time relates to its spectral measure, revealing intricate dependencies near zero and infinity.

Contribution

It provides a spectral analysis framework connecting persistence probabilities with spectral measure behavior for Gaussian stationary processes.

Findings

Persistence probability linked to spectral measure near zero and infinity

Spectral perspective offers new insights into process longevity

Analytical tools for spectral and persistence relationship

Abstract

We study the persistence probability of a centered stationary Gaussian process on $\mathbb{Z}$ or $\mathbb{R}$, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the behavior of the spectral measure of the process near zero and infinity.