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 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.

Modelling columnarity of pyramidal cells in the human cerebral cortex

“…When Φ is a DPP, we call P xy ( Y S ) a determinantal Thomas point process (DTPP). The DTPP is discussed to some extent in M⊘ller & Christoffersen (2018), where a closed form expression of its PCF is found. Thus, the DLCPP can be fitted by fitting a DTPP to the projected data using a minimum contrast procedure (see Section 2.3).…”

Modelling columnarity of pyramidal cells in the human cerebral cortex

“…When Φ is a DPP, we call P xy (Y S ) a determinantal Thomas point process (DTPP). The DTPP is discussed to some extent in Møller and Christoffersen (2018), where a closed form expression of its PCF is found. Thus, the DLCPP can be fitted by fitting a DTPP to the projected data using a minimum contrast procedure (see Section 2.3).…”