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Point process limits of lattice processes

Published online by Cambridge University Press:  14 July 2016

Julian Besag ,
Robin Milne  and
Stan Zachary
Julian Besag*
Affiliation:
University of Durham
Robin Milne*
Affiliation:
University of Western Australia
Stan Zachary*
Affiliation:
Heriot-Watt University
*
∗Postal address: Department of Mathematics, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, U.K.
∗∗Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.
∗∗∗Postal address: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Riccarton, Currie, Edinburgh EH14 4AS, U.K.
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Abstract

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.

Keywords

SPATIAL POINT PROCESSAUTO-POISSON SCHEMEAUTO-LOGISTIC SCHEMESTRAUSS PROCESSPAIRWISE-INTERACTION PROCESSINHIBITORY POINT PROCESSHARD-CORE POINT PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 210 - 216
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Work done while this author was visiting the University of Durham.

References

Besag, J. E. (1974) Spatial interaction and the statistical analysis of lattice systems (with discussion). J. R. Statist. Soc. B 36, 192–236.Google Scholar
Besag, J. E. (1977) Some methods of statistical analysis for spatial data. Bull. Internat. Statist. Inst. 47, 77–92.Google Scholar
Carter, D. S. and Prenter, P. M. (1972) Exponential spaces and counting processes. Z. Wahrscheinlichkeitsth. 21, 1–19.Google Scholar
Kallenberg, O. (1978) On conditional intensities of point processes. Z. Wahrscheinlichkeitsth. 41, 205–220.Google Scholar
Kelly, F. P. and Ripley, B. D. (1976) A note on Strauss' model for clustering. Biometrika 63, 357–360.Google Scholar
Papangelou, F. (1974) The conditional intensity of general point processes and an application to line processes. Z. Wahrscheinlichkeitsth. 28, 207–226.Google Scholar
Preston, C. J. (1976) Random Fields. Lecture Notes in Mathematics 534, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Ripley, B. D. (1977) Modelling spatial patterns (with discussion). J. R. Statist. Soc. B 39, 172–212.Google Scholar
Ripley, B. D. and Kelly, F. P. (1977) Markov point processes. J. London Math. Soc. (2) 15, 188–192.CrossRefGoogle Scholar
Ruelle, D. (1969) Statistical Mechanics. Benjamin, New York.Google Scholar
Strauss, D. J. (1975) A model for clustering. Biometrika 62, 467–475.CrossRefGoogle Scholar