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

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 ${\mathbb{R}}^2.WederiveanexplicitboundontheconvergenceratetoahomogeneousPPPasthelineintensitygrowsandthepointintensityoneachlinediminishes.Second,weanalyzeaCox-Binomialprocessontheunitsphere$\mathbb{S}^2$,modelingasystemofsatellites.ThisprocessisgeneratedbyplacingPPPsongreat-circleorbits,whosepositionsaredeterminedbyaBinomialpointprocess.Forthismodel,weestablishaconvergencerateoforder$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.