Electronic Journal of Probability

Distance estimates for dependent thinnings of point processes with densities

Dominic Schuhmacher

Full-text: Open access

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

Permanent link to this document
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

Subjects
Primary: 60G55: Point processes
Secondary: 60E99: None of the above, but in this section 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
point process Poisson process approximation Stein's method point process density random field thinning total variation distance Barbour-Brown distance

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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


Export citation

References

  • A. Baddeley. Spatial point processes and their applications. Stochastic geometry. Vol. 1892 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2007. pp. 1–75.
  • A.D. Barbour and T.C. Brown. Stein's method and point process approximation. Stochastic Process. Appl. 43 (1992), 9–31.
  • A.D. Barbour and L.H.Y. Chen (eds). An introduction to Stein's method. Vol. 4 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. Singapore University Press, Singapore, 2005.
  • A.D. Barbour, L. Holst and S. Janson. Poisson approximation. Vol. 2 of Oxford Studies in Probability. Oxford University Press, Oxford, 1992.
  • T. Brown. Position dependent and stochastic thinning of point processes. Stochastic Process. Appl. 9 (1979), 189–193.
  • T.C. Brown and A. Xia. On metrics in point process approximation. Stochastics Stochastics Rep. 52 (1995), 247–263.
  • L.H.Y. Chen and A. Xia. Stein's method, Palm theory and Poisson process approximation. Ann. Probab. 32 (2004), 2545–2569.
  • D.J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988.
  • P.J. Diggle. Statistical analysis of spatial point patterns. Second edition. Arnold, London, 2003.
  • P. Doukhan. Mixing. Vol. 85 of Lecture Notes in Statistics. Springer-Verlag, New York, 1994.
  • P. Hall. Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrsch. Verw. Gebiete 70 (1985), 237–261.
  • O. Kallenberg. Limits of compound and thinned point processes. J. Appl. Probability 12 (1975), 269–278.
  • O. Kallenberg. Random measures. Fourth edition. Akademie-Verlag, Berlin, 1986.
  • M. Månsson and M. Rudemo. Random patterns of nonoverlapping convex grains. Adv. in Appl. Probab. 34 (2002), 718–738.
  • J. Mecke. Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete 9 (1967), 36–58.
  • J. Mecke. Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrsch. Verw. Gebiete 11 (1968), 74–81.
  • J. Møller and R.P. Waagepetersen. Statistical inference and simulation for spatial point processes. Vol. 100 of Monographs on Statistics and Applied Probability. Chapman & Hall/CRC, Boca Raton, FL, 2004.
  • J. Neveu. Mathematical foundations of the calculus of probability. Translated by Amiel Feinstein. Holden-Day Inc., San Francisco, Calif., 1965.
  • X.X. Nguyen and H. Zessin. Integral and differential characterizations of the Gibbs process. Math. Nachr. 88 (1979), 105–115.
  • C. Preston. Random fields. Vol. 534 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1976.
  • R.-D. Reiss. A course on point processes. Springer Series in Statistics. Springer-Verlag, New York, 1993.
  • D. Schuhmacher. Distance estimates for Poisson process approximations of dependent thinnings. Electron. J. Probab. 10 (2005), 165–201 (electronic).
  • D. Schuhmacher. Estimation of distances between point process distributions. PhD thesis, University of Zurich, 2005. http://www.dissertationen.unizh.ch/2006/schuhmacher/diss.pdf (
  • C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971). Vol. II: Probability theory. Univ. California Press, Berkeley, Calif., 1972, pp. 583–602.
  • D. Stoyan, W.S. Kendall and J. Mecke. Stochastic geometry and its applications. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Ltd., Chichester, 1987.
  • A. Xia. Stein's method and Poisson process approximation. An introduction to Stein's method. Vol. 4 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. Singapore Univ. Press, Singapore, 2005, pp. 115–181.