Pair correlation functions and limiting distributions of iterated cluster point processes

TL;DR

This paper studies a Markov chain of point processes, focusing on their pair correlation functions and equilibrium distributions, extending previous models of reproducing populations with a new coupling construction method.

Contribution

It introduces a method to construct equilibrium distributions for a class of iterated cluster point processes using coupling arguments.

Findings

Pair correlation functions converge in certain models.

Equilibrium distributions can be explicitly constructed.

The approach extends previous population models.

Abstract

We consider a Markov chain of point processes such that each state is a super position of an independent cluster process with the previous state as its centre process together with some independent noise process. The model extends earlier work by Felsenstein and Shimatani describing a reproducing population. We discuss when closed term 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.