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Convergence of spatial birth-and-death processes

Published online by Cambridge University Press:  24 October 2008

H. W. Lotwick  and
B. W. Silverman
H. W. Lotwick
Affiliation:
University of Bath
B. W. Silverman
Affiliation:
University of Bath
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Abstract

Some models for spatial point processes are difficult to simulate directly and are most easily realized as the equilibrium distribution of certain spatial-temporal Markov processes. This paper examines the convergence of such processes, concentrating mainly on the ‘hard core’ case when the points represent the centres of non-overlapping discs. Coupling methods from the theory of Markov chains are used to establish sufficient conditions for the processes to converge to the required equilibrium, and to give a lower bound on the rate of convergence. One technique used is to couple processes when they become close in a suitable metric.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 1 , July 1981 , pp. 155 - 165
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Billingsley, P.Convergence of probability measures (John Wiley, New York, 1968).Google Scholar
(2)Chung, K. L.A course in probability theory (Academic Press, New York, 1974).Google Scholar
(3)Crane, M. A. and Iglehart, D. L.Simulating stable stochastic systems. I. General multiserver queues. J. Assoc. Comput. Mach. 21 (1974), 103–113.CrossRefGoogle Scholar
(4)Doob, J. L.Stochastic processes (John Wiley, New York, 1953).Google Scholar
(5)Kelly, F. P. and Ripley, B. D.A note on Strauss's model for clustering. Biometrika 63 (1976), 357–360.CrossRefGoogle Scholar
(6)Lindvall, T.A probabilistic proof of Blackwell's Renewal Theorem. Ann. Probability 5 (1977), 482–485.CrossRefGoogle Scholar
(7)Pitman, J. W.Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrsch. Verw. Gebiete 29 (1974), 193–227.CrossRefGoogle Scholar
(8)Preston, C. J.Spatial birth-and-death processes. Bull. Inst. Internat. Statist. 46 (1976), 371–391.Google Scholar
(9)Ripley, B. D.Modelling spatial patterns (with discussion). J. Roy. Statist. Soc., Ser. B 39 (1977), 172–212.Google Scholar
(10)Ripley, B. D.Simulating spatial patterns: dependent samples from a multivariate density. J. Roy. Statisti. Soc., Ser. C 28 (1979), 109–112.Google Scholar
(11)Strauss, D. J.A model for clustering. Biometrika 62 (1975), 467–475.CrossRefGoogle Scholar