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Pair correlation functions and limiting distributions of iterated cluster point processes

Part of: Stochastic processes Geometric probability and stochastic geometry Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  16 November 2018

Jesper Møller  and
Andreas D. Christoffersen
Jesper Møller*
Affiliation:
Aalborg University
Andreas D. Christoffersen*
Affiliation:
Aalborg University
*
* Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark.
* Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark.
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Abstract

We consider a Markov chain of point processes such that each state is a superposition of an independent cluster process with the previous state as its centre process together with some independent noise process and a thinned version of the previous state. The model extends earlier work by Felsenstein (1975) and Shimatani (2010) describing a reproducing population. We discuss when closed-form expressions of the first- and second-order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.

Keywords

Couplingequilibriumindependent clusteringMarkov chainpair correlation functionreproducing populationweighted determinantal and permanental point processes

MSC classification

Primary: 60G55: Point processes 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry 62M30: Spatial processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 789 - 809
Copyright
Copyright © Applied Probability Trust 2018 

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