## Electronic Journal of Probability

### Distance estimates for dependent thinnings of point processes with densities

Dominic Schuhmacher

#### 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 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.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 38, 1080-1116.

Dates
Accepted: 26 May 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819499

Digital Object Identifier
doi:10.1214/EJP.v14-643

Mathematical Reviews number (MathSciNet)
MR2506126

Zentralblatt MATH identifier
1196.60089

Rights

#### Citation

Schuhmacher, Dominic. Distance estimates for dependent thinnings of point processes with densities. Electron. J. Probab. 14 (2009), paper no. 38, 1080--1116. doi:10.1214/EJP.v14-643. https://projecteuclid.org/euclid.ejp/1464819499

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