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Quantitative Limit Theorems for Cox-Poisson and Cox-Binomial Point Processes

Hamza Adrat, Laurent Decreusefond

TL;DR

The paper addresses the problem of quantifying the convergence of two Cox point processes to Poisson limits, using Stein's method with a generator (Glauber dynamics) framework and the Stein-Dirichlet representation. The authors derive explicit, non-asymptotic bounds for a Cox-Poisson process on ${\mathbb R}^2$ built from a Poisson line process and for a Cox-Binomial process on ${\mathbb S^2}$ modeling satellites along great-circle orbits, achieving a $O(1/\lambda_n)$ rate in the former and an $O(1/n)$ rate in the latter. The results provide concrete error controls in total-variation and Wasserstein metrics, with clear applicability to stochastic geometry and spatial statistics. Overall, the work showcases the versatility of the Stein-generator method for handling complex, transformed point-process models and offers practical guidance for approximation accuracy in high-dimensional spatial settings.

Abstract

This paper establishes quantitative limit theorems for two classes of Cox point processes, quantifying their convergence to a Poisson point process (PPP). We employ Stein's method for PPP aproximation, leveraging the generator approach and the Stein-Dirichlet representation formula associated with the Glauber dynamics. First, we investigate a Cox-Poisson process constructed by placing one-dimensional PPPs on the lines of a Poisson line process in $\mathbb{R}^2. We derive an explicit bound on the convergence rate to a homogeneous PPP as the line intensity grows and the point intensity on each line diminishes. Second, we analyze a Cox-Binomial process on the unit sphere $\mathbb{S}^2$, modeling a system of satellites. This process is generated by placing PPPs on great-circle orbits, whose positions are determined by a Binomial point process. For this model, we establish a convergence rate of order $O(1/n)$ to a uniform PPP on the sphere, where n is the number of orbits. The derived bounds provide precise control over the approximation error in both models, with applications in stochastic geometry and spatial statistics.

Quantitative Limit Theorems for Cox-Poisson and Cox-Binomial Point Processes

TL;DR

The paper addresses the problem of quantifying the convergence of two Cox point processes to Poisson limits, using Stein's method with a generator (Glauber dynamics) framework and the Stein-Dirichlet representation. The authors derive explicit, non-asymptotic bounds for a Cox-Poisson process on built from a Poisson line process and for a Cox-Binomial process on modeling satellites along great-circle orbits, achieving a rate in the former and an rate in the latter. The results provide concrete error controls in total-variation and Wasserstein metrics, with clear applicability to stochastic geometry and spatial statistics. Overall, the work showcases the versatility of the Stein-generator method for handling complex, transformed point-process models and offers practical guidance for approximation accuracy in high-dimensional spatial settings.

Abstract

This paper establishes quantitative limit theorems for two classes of Cox point processes, quantifying their convergence to a Poisson point process (PPP). We employ Stein's method for PPP aproximation, leveraging the generator approach and the Stein-Dirichlet representation formula associated with the Glauber dynamics. First, we investigate a Cox-Poisson process constructed by placing one-dimensional PPPs on the lines of a Poisson line process in \mathbb{S}^2O(1/n)$ to a uniform PPP on the sphere, where n is the number of orbits. The derived bounds provide precise control over the approximation error in both models, with applications in stochastic geometry and spatial statistics.

Paper Structure

This paper contains 12 sections, 10 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Poisson and Binomial point processes
  5. Stein's method
  6. Quantitative Limit Theorem for Cox-Poisson Point Processes
  7. Quantitative Limit Theorem for Cox-Binomial Point Processes
  8. Conclusion
  9. Proofs
  10. Proof of Lemma \ref{['lem:coarea_formula']}
  11. Proof of Theorem \ref{['th:convergence_cox_process']}
  12. Proof of Theorem \ref{['th:convergence_satellites_BPP']}

Key Result

Theorem 3.3

Let $\Phi$ be a PPP on $\mathbb X$ with intensity measure $\sigma$. Then, for any measurable function $F \colon \mathbb X \times {\mathfrak N}_{\mathbb X} \to {\mathbb R}$ such that we have

Figures (2)

  • Figure 1: Cox-Poisson point process.
  • Figure 2: Reduction to $K$ of the Cox-Poisson point process.

Theorems & Definitions (16)

  • Definition 3.1: Poisson point process
  • Definition 3.2: Binomial point process
  • Theorem 3.3: Campbell-Mecke formula for a PPP
  • Theorem 3.4: Campbell-Mecke formula for a BPP
  • Theorem 3.5: Invariance property
  • Definition 3.6: Glauber semigroup
  • Theorem 3.7: Semigroup of a PPP
  • Theorem 3.8: Infinitesimal generator of a PPP
  • Definition 3.9: $1$-Lipschitz function
  • Definition 3.10: Total variation distance
  • ...and 6 more