Sampling cluster point processes: a review

TL;DR

This paper reviews sampling methods for cluster and iterated cluster point processes, focusing on the Brix-Kendall exact sampling method and its extensions to non-Poissonian germ processes, including formal proofs and applications.

Contribution

It extends existing sampling techniques to non-Poissonian germ point processes and provides formal proofs for their validity, broadening the applicability of these methods.

Findings

Extended sampling methods to non-Poissonian germ processes.

Provided formal proofs using Laplace transforms.

Included exact sampling of Boolean models.

Abstract

The theme of this article is the sampling of cluster and iterated cluster point processes. It is partially a review, mainly of the Brix-Kendall exact sampling method for cluster point processes and its adaptation by M{\o}ller and Rasmussen to Hawkes branching point processes on the real line with light-tail fertility rate are reviewed. The main novel aspect of this review is the extension to non-Poissonian germ point processes. For this, a slight adaptation is needed. A formal proof via Laplace transforms of the validity of the method in terms of general clusters that are not necessarily point processes fits this purpose and allows to include the exact sampling of Boolean models.