Time of appearance of a large gap in a dynamic Poisson point process

TL;DR

This paper analyzes the timing of large gaps in a dynamic spatial Poisson process, showing how the distribution and expected time depend on the process parameters and initial conditions.

Contribution

It provides a detailed asymptotic analysis of the distribution of the first large gap time in a spatial birth-death process as the intensity parameter grows.

Findings

Gap time scaled by its mean converges to exponential distribution.

Expected gap time scales exponentially with the process intensity and gap size.

Results depend on initial particle density and the size of the considered gap.

Abstract

We study the distribution of the 'gap time', the first time that a large gap appears, in the spatial birth and death point process on $[0,1]$ in which particles are added uniformly in space at rate $\lambda$ and are removed independently at rate $1$, as a function of the parameter $\lambda$ and the specified gap size function $w_\lambda$ as $\lambda\to\infty$. If $w_\lambda$ is a large enough multiple of the typical largest gap $(\log(\lambda)+O(1))/\lambda$ and the initial distribution has a high enough local density of particles and not too many particles in total, then the gap time, scaled by its expected value, converges in distribution to exponential with mean $1$. If in addition $\limsup_\lambda w_\lambda < 1$ then the expected time scales like $e^{\lambda w_\lambda}/(\lambda^2 w_\lambda(1-w_\lambda))$.