Quantitative Limit Theorems for Cox-Poisson and Cox-Binomial Point Processes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916991433965568 |
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| author | Adrat, Hamza Decreusefond, Laurent |
| author_facet | Adrat, Hamza Decreusefond, Laurent |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Limit Theorems for Cox-Poisson and Cox-Binomial Point Processes Adrat, Hamza Decreusefond, Laurent Probability 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. |
| title | Quantitative Limit Theorems for Cox-Poisson and Cox-Binomial Point Processes |
| topic | Probability |
| url | https://arxiv.org/abs/2510.05004 |