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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 38 open journal systems 


Distance estimates for dependent thinnings of point processes with densities

Dominic Schuhmacher, University of Western Australia, Australia


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.


Full text: PDF

Pages: 1080-1116

Published on: May 26, 2009
Bibliography
  1. A. Baddeley. Spatial point processes and their applications. Stochastic geometry. Vol. 1892 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2007. pp. 1--75. Math. Review 2008c:60045
  2. A.D. Barbour and T.C. Brown. Stein's method and point process approximation. Stochastic Process. Appl. 43 (1992), 9--31. Math. Review 93k:60120
  3. 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. Math. Review 2007j:60001
  4. A.D. Barbour, L. Holst and S. Janson. Poisson approximation. Vol. 2 of Oxford Studies in Probability. Oxford University Press, Oxford, 1992. Math. Review 93g:60043
  5. T. Brown. Position dependent and stochastic thinning of point processes. Stochastic Process. Appl. 9 (1979), 189--193. Math. Review 80i:60077
  6. T.C. Brown and A. Xia. On metrics in point process approximation. Stochastics Stochastics Rep. 52 (1995), 247--263. Math. Review 97j:60081
  7. L.H.Y. Chen and A. Xia. Stein's method, Palm theory and Poisson process approximation. Ann. Probab. 32 (2004), 2545--2569. Math. Review 2005d:60080
  8. D.J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988. Math. Review 90e:60060
  9. P.J. Diggle. Statistical analysis of spatial point patterns. Second edition. Arnold, London, 2003. Math. Review 85m:62205 (first edition)
  10. P. Doukhan. Mixing. Vol. 85 of Lecture Notes in Statistics. Springer-Verlag, New York, 1994. Math. Review 96b:60090
  11. P. Hall. Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrsch. Verw. Gebiete 70 (1985), 237--261. Math. Review 86k:60020
  12. O. Kallenberg. Limits of compound and thinned point processes. J. Appl. Probability 12 (1975), 269--278. Math. Review 52 #12072
  13. O. Kallenberg. Random measures. Fourth edition. Akademie-Verlag, Berlin, 1986. Math. Review 87k:60137
  14. M. Månsson and M. Rudemo. Random patterns of nonoverlapping convex grains. Adv. in Appl. Probab. 34 (2002), 718--738. Math. Review 2003j:60014
  15. J. Mecke. Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete 9 (1967), 36--58. Math. Review 37 #3611
  16. J. Mecke. Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrsch. Verw. Gebiete 11 (1968), 74--81. Math. Review 39 #3614
  17. 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. Math. Review 2004h:62003
  18. J. Neveu. Mathematical foundations of the calculus of probability. Translated by Amiel Feinstein. Holden-Day Inc., San Francisco, Calif., 1965. Math. Review 33 #6660
  19. X.X. Nguyen and H. Zessin. Integral and differential characterizations of the Gibbs process. Math. Nachr. 88 (1979), 105--115. Math. Review 80i:60081a
  20. C. Preston. Random fields. Vol. 534 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1976. Math. Review 56 #6936
  21. R.-D. Reiss. A course on point processes. Springer Series in Statistics. Springer-Verlag, New York, 1993. Math. Review 94b:60058
  22. D. Schuhmacher. Distance estimates for Poisson process approximations of dependent thinnings. Electron. J. Probab. 10 (2005), 165--201 (electronic). Math. Review 2005j:60099
  23. D. Schuhmacher. Estimation of distances between point process distributions. PhD thesis, University of Zurich, 2005. http://www.dissertationen.unizh.ch/2006/schuhmacher/diss.pdf (Math. Review number not available.)
  24. 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. Math. Review 53 #6687
  25. 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. Math. Review 88j:60034a
  26. 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. Math. Review MR2235450
















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Electronic Journal of Probability. ISSN: 1083-6489