Poisson approximation of the largest gaps between zeros of a stationary Gaussian process

TL;DR

This paper proves that the largest gaps between zeros of certain stationary Gaussian processes, after rescaling, follow a Poisson process, even with slow polynomial decay of correlations.

Contribution

It introduces a novel approximate splitting property for gap events, enabling Poisson approximation under minimal correlation decay conditions.

Findings

Largest gaps, after rescaling, converge to a Poisson process

Approximate splitting property holds for processes with slow polynomial decay

Method applies to a broad class of stationary Gaussian processes

Abstract

We study the largest gaps between successive zeros of a smooth stationary Gaussian process. Our main result is that, if correlations decay at least polynomially, then after suitable rescaling of the locations and sizes of the largest gaps in a growing interval, the resulting joint process converges to a Poisson point process. The main novel step in the proof is to establish an approximate splitting property, with multiplicative error, for gap events in well-separated intervals; notably we achieve this for processes with arbitrarily slow polynomial decay of correlations.