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Critical cluster cascades

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  13 September 2022

Matthias Kirchner
Matthias Kirchner*
Affiliation:
Institute of Teacher Education NMS Bern
*
*Postal address: Waisenhausplatz 27, 3000 Bern, Switzerland. Email address: matthias.kirchner@phnmsbern.ch
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Abstract

We consider a sequence of Poisson cluster point processes on $\mathbb{R}^d$ : at step $n\in\mathbb{N}_0$ of the construction, the cluster centers have intensity $c/(n+1)$ for some $c>0$ , and each cluster consists of the particles of a branching random walk up to generation n—generated by a point process with mean 1. We show that this ‘critical cluster cascade’ converges weakly, and that either the limit point process equals the void process (extinction), or it has the same intensity c as the critical cluster cascade (persistence). We obtain persistence if and only if the Palm version of the outgrown critical branching random walk is locally almost surely finite. This result allows us to give numerous examples for persistent critical cluster cascades.

Keywords

Critical branching random walkcluster point process

MSC classification

Primary: 60G55: Point processes
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 2 , June 2023 , pp. 357 - 381
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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