Stein's method and Papangelou intensity for Poisson or Cox process approximation

Functional inequalities for marked point processes

In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincaré inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the …

The Pólya sum kernel and Bayes estimation

We consider a particular Cox process from a Bayesian viewpoint and show that the Bayes estimator of the intensity measure is the so-called Pólya sum kernel, which occurred recently in the context of the construction of the so-called Papangelou processes. More precisely, if the prior, the directing measure of the Cox process, is a Poisson-Gamma random measure, then the posterior is again a Poisson-Gamma random measure and the Bayes estimator of the intensity is the Pólya sum kernel. Moreover, we extend this result to doubly stochastic Poisson-Gamma priors and give conditions under which one can…

Cox-Poisson与Cox-二项点过程的定量极限定理

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

本文采用Stein方法，分别建立了二维平面上Cox-Poisson过程和单位球面上Cox-二项过程向Poisson点过程收敛的显式速率界，为相关逼近误差提供精确控制，可应用于随机几何与空间统计领域。

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and $U$-statistics

This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_π$, stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the …