Distance estimates for dependent thinnings of point processes with densities

Abstract

In [Schuhmacher, Electron. J. Probab. 10 (2005), 165--201] estimates of the Barbour-Brown distance d_2 between the distribution of a thinned point process and the distribution of a Poisson process were derived by combining discretization with a result based on Stein's method. In the present article we concentrate on point processes that have a density with respect to a Poisson process. For such processes we can apply a corresponding result directly without the detour of discretization and thus obtain better and more natural bounds not only in d_2 but also in the stronger total variation metric. We give applications for thinning by covering with an independent Boolean model and "Mat{\'e}rn type I"-thinning of fairly general point processes. These applications give new insight into the respective models, and either generalize or improve earlier results.