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Markov properties of cluster processes

  • A. J. Baddeley (a1), M. N. M. Van Lieshout (a2) and J. Møller (a3)

Abstract

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

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Corresponding author

Postal address: Department of Mathematics, University of Western Australia, Nedlands WA 6009, Australia.
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, U.K.
∗∗∗ Postal address: Department of Mathematics and Computer Science, University of Aalborg, F. Bajers Vej 7E, DK-9220 Aalborg Ø, Denmark.

References

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[1] Baddeley, A. J. and Van Lieshout, M. N. M. (1996) A nonparametric measure of spatial interaction in point patterns. Statist. Neerland. To appear.
[2] Baddeley, A. J. and Møller, J. (1989) Nearest-neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89121.
[3] Carter, D. S. and Prenter, P. M. (1972) Exponential spaces and counting processes. Z. Wahrscheinlichkeitsth. 21, 119.
[4] Diggle, P. J., Fiksel, T., Ogata, Y., Stoyan, D. and Tanemura, M. (1994) On parameter estimation for pairwise interaction processes. Int. Statist. Rev. 62, 99117.
[5] Kendall, W. S. (1990) A spatial Markov property for nearest-neighbour Markov point processes. J. Appl. Prob. 28, 767778.
[6] Kingman, J. F. C. (1977) Remarks on the spatial distribution of a reproducing population. J. Appl. Prob. 14, 577583.
[7] Møller, J. (1994) Discussion contribution. Scand. J. Statist. 21, 346349.
[8] Møller, J. (1994) Markov chain Monte Carlo and spatial point processes. Research Report 293. Department of Theoretical Statistics, University of Aarhus. To appear in Proc. Seminaire Européen de Statistique Toulouse 1996 ‘Stochastic Geometry: Theory and Applications’. ed. Barndorff-Nielsen, O., Kendall, W. S., Letac, G. and van Lieshout, M. N. M. Chapman and Hall, London.
[9] Preston, C. J. (1976) Random Fields. Springer, Berlin.
[10] Ripley, B. D. (1988) Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge.
[11] Ripley, B. D. (1989). Gibbsian interaction models. In Spatial Statistics: Past, Present and Future. pp. 119. ed. Griffiths, D. A. Image, New York.
[12] Ripley, B. D. and Kelly, F. P. (1977) Markov point processes. J. London Math. Soc. 15, 188192.
[13] Ruelle, D. (1969) Statistical Mechanics. Wiley, New York.

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Markov properties of cluster processes

  • A. J. Baddeley (a1), M. N. M. Van Lieshout (a2) and J. Møller (a3)

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