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

Eric Foxall, Clément Soubrier

TL;DR

The paper analyzes the time until a large gap appears in a high-density dynamic Poisson point process on [0,1], where points arrive at rate λ and depart at rate 1. By developing a regeneration-based framework and a precise flush-pin regeneration construction for the process, the authors prove that the gap-hitting time, when properly scaled by its mean, converges to an exponential distribution, and they derive the asymptotic form of the mean hitting time in terms of λ and the gap size w_λ. Key contributions include explicit asymptotics for π_λ(A(w_λ)) and q_ex(A(w_λ)), the regeneration-based proof of asymptotic exponentiality, and detailed estimates for the stationary and renewal quantities that govern the rare-gap event. The results illuminate the stability of dense clusters in dynamic spatial systems and provide a robust approach for analyzing rare events in regenerative Markov processes with infinite state spaces.

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 $λ$ and are removed independently at rate $1$, as a function of the parameter $λ$ and the specified gap size function $w_λ$ as $λ\to\infty$. If $w_λ$ is a large enough multiple of the typical largest gap $(\log(λ)+O(1))/λ$ 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 $\limsup_λw_λ< 1$ then the expected time scales like $e^{λw_λ}/(λ^2 w_λ(1-w_λ))$.

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

TL;DR

The paper analyzes the time until a large gap appears in a high-density dynamic Poisson point process on [0,1], where points arrive at rate λ and depart at rate 1. By developing a regeneration-based framework and a precise flush-pin regeneration construction for the process, the authors prove that the gap-hitting time, when properly scaled by its mean, converges to an exponential distribution, and they derive the asymptotic form of the mean hitting time in terms of λ and the gap size w_λ. Key contributions include explicit asymptotics for π_λ(A(w_λ)) and q_ex(A(w_λ)), the regeneration-based proof of asymptotic exponentiality, and detailed estimates for the stationary and renewal quantities that govern the rare-gap event. The results illuminate the stability of dense clusters in dynamic spatial systems and provide a robust approach for analyzing rare events in regenerative Markov processes with infinite state spaces.

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 in which particles are added uniformly in space at rate and are removed independently at rate , as a function of the parameter and the specified gap size function as . If is a large enough multiple of the typical largest gap 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 . If in addition then the expected time scales like .

Paper Structure

This paper contains 10 sections, 24 theorems, 166 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Rare event theory
  3. Stopping sequence, Regeneration time
  4. Exponential limit distribution
  5. Scaling of expected hitting time
  6. Regeneration time for $(X_\lambda)$
  7. Proof of Theorem \ref{['thm:main']}
  8. Estimates of $\pi_\lambda$
  9. Estimates of $N$
  10. Acknowledgments:

Key Result

Theorem 1.1

With $X_\lambda$ and $A(w_\lambda)$ as above, let $\tau_{A(w)}=\inf\{t\colon X_\lambda(t) \in A(w)\}$ denote the $w_\lambda$-gap time. For each $\alpha>0$ there is $C>0$ so that if then

Figures (2)

  • Figure 1: Schematic representation of the dynamic Poisson point process, with a large gap event $A(w)$.
  • Figure 2: Schematic representation of the setup of Proposition \ref{['prop:exp']}. (A) We suppose that we can define regeneration cycles which are one-dependent. During each cycle, the process can hit $A_m$ with probability $p_m\to0$(i). (B) $N_m$ counts the number of cycles before the process hits $A_m$.

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.1
  • Definition 2.1: Stopping sequence, trajectory segments
  • Definition 2.2: Time shift operator
  • Lemma 2.1: Excursion principle
  • proof
  • Definition 2.3: Regeneration time
  • Lemma 2.2: One-dependence
  • Proposition 2.1
  • Lemma 2.3: Estimate from the proof of the Theorem, Section 3, gutmart, $p=1$ case.
  • ...and 41 more