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Markov properties of cluster processes

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


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.


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.


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[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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