Pair correlation functions and limiting distributions of iterated cluster point processes

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1 Citation (Scopus)

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.
Original language English Journal of Applied Probability 55 3 789-809 21 0021-9002 https://doi.org/10.1017/jpr.2018.50 Published - 1 Sep 2018

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Pair Correlation Function
Point Process
Limiting Distribution
Markov chain
Equilibrium Distribution
Superposition
Closed-form
Moment
First-order
Converge
Limiting distribution
Point process
Equilibrium distribution
Model

Keywords

• Coupling
• Equilibrium
• Independent clustering
• Markov chain
• Pair correlation function
• Reproducing population weighted determinantal and permanental point processes

Cite this

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title = "Pair correlation functions and limiting distributions of iterated cluster point processes",
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.",
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In: Journal of Applied Probability, Vol. 55, No. 3, 01.09.2018, p. 789-809.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Pair correlation functions and limiting distributions of iterated cluster point processes

AU - Møller, Jesper

AU - Christoffersen, Andreas Dyreborg

PY - 2018/9/1

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N2 - 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.

AB - 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.

KW - Coupling

KW - Equilibrium

KW - Independent clustering

KW - Markov chain

KW - Pair correlation function

KW - Reproducing population weighted determinantal and permanental point processes

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U2 - 10.1017/jpr.2018.50

DO - 10.1017/jpr.2018.50

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JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

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