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Introduction to the Theory of Gibbs Point Processes

  • David DereudreEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2237)

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 with 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 \(\mathbb {R}^d\). In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space \(\mathbb {R}^d\). 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 we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takacs-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.

Notes

Acknowledgements

The author thanks P. Houdebert, A. Zass and the anonymous referees for the careful reading and the interesting comments. This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the CNRS GdR 3477 GeoSto and the ANR project PPP (ANR-16-CE40-0016).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University LilleVilleneuve-d’AscqFrance

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