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A Markov construction for a multidimensional point process

Published online by Cambridge University Press:  14 July 2016

Valerie Isham
Valerie Isham*
Affiliation:
Imperial College, London
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Abstract

A sequence of finite point processes {Pn} is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.

Keywords

MULTIDIMENSIONAL POINT PROCESSFINITE POINT PROCESSMARKOV SEQUENCECARDIOID DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 507 - 515
Copyright
Copyright © Applied Probability Trust 1977 

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References

Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes , ed. Lewis, P.A.W., Wiley, New York, 299–393.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Fisher, L. (1972) A survey of the mathematical theory of multidimensional point processes. In Stochastic Point Processes , ed. Lewis, P.A.W., Wiley, New York, 468–513.Google Scholar
Isham, V. (1974) Some Multidimensional Point Processes. Unpublished Ph.D. thesis, University of London.Google Scholar
Isham, V. (1975) On a point process with independent locations. J. Appl. Prob. 12, 435–446.Google Scholar
Isham, V. (1976) Constructions for planar point processes using concentric circles. Stoch. Proc. Appl. To appear.Google Scholar