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Distance estimates for dependent thinnings of point processes with densities. (English) Zbl 1196.60089

Summary: In [Schuhmacher, Electron. J. Probab. 10, 165–2010 (2005; Zbl 1071.60034)] 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 which we can apply a corresponding result directly without the detour of discretization. This enables us to obtain better and more natural bounds in the \(d_2\)-metric, and for the first time also bounds in the stronger total variation metric. We give applications for thinning by covering with an independent Boolean model and “Matern type I” thinning of fairly general point processes. These applications give new insight into the respective models, and either generalize or improve earlier results.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry

Citations:

Zbl 1071.60034
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