2025/10/06 by Adrat, Hamza, Decreusefond, Laurent
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2510.05004
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 ℝ2. 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 \mathbbS2, 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.