Distance estimates for dependent thinnings of point processes with densities

In [Schuhmacher, Electron. J. Probab. 10 (2005), 165–201] estimates of the Barbour-Browndistance d2 between the distribution of a thinned point process and the distribution of a Poissonprocess were derived by combining discretization with a result based on Stein’s method. Inthe present article we concentrate on point processes that have a density with respect to aPoisson process, for which we can apply a corresponding result directly without the detour ofdiscretization. This enables us to obtain better and more natural bounds in the d2-metric, andfor the first time also bounds in the stronger total variation metric.We give applications for thinning by covering with an independent Boolean model and “Matérntype I”-thinning of fairly general point processes. These applications give new insight into therespective models, and either generalize or improve earlier results.