Introduction to the theory of Gibbs point processes
Abstract
The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated to the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in Rd. In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space Rd. Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally, in a last part, we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takac-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.
Keywords
Cite
@article{arxiv.1701.08105,
title = {Introduction to the theory of Gibbs point processes},
author = {David Dereudre},
journal= {arXiv preprint arXiv:1701.08105},
year = {2018}
}
Comments
The manuscript is based on a mini course given during the conference of GDR 3477 g\'eom\'etrie stochastique, at university of Nantes in April 2016