On the probability that a stationary Gaussian process with spectral gap remains non-negative on a long interval

TL;DR

This paper establishes an exponential upper bound on the probability that a stationary Gaussian process with a spectral gap remains non-negative over a long interval, highlighting the influence of spectral properties on process behavior.

Contribution

It provides a sharp, explicit bound on non-negativity probability for Gaussian processes with spectral gaps, extending understanding of their long-term behavior.

Findings

Probability decays exponentially with interval length and spectral gap size

Bound is sharp without extra assumptions

Spectral gap critically influences process positivity

Abstract

Let $f$ be a zero-mean continuous stationary Gaussian process on ${\mathbb R}$ whose spectral measure vanishes in a $\delta$-neighborhood of the origin. Then the probability that $f$ stays non-negative on an interval of length $L$ is at most $e^{-c\delta^2 L^2}$ with some absolute $c>0$ and the result is sharp without additional assumptions.