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Hawkes branching point processes without ancestors

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Pierre Brémaud  and
Laurent Massoulié
Pierre Brémaud*
Affiliation:
CNRS, France, and École Polytechnique Fédérale de Lausanne
Laurent Massoulié*
Affiliation:
Microsoft Research
*
∗ Postal address: École Supérieure d'Électricité, Laboratoire des Signaux et Systèmes, Plateau du Moulon, 91192 Gif sur Yvette, France. Email address: bremaud@lss.supelec.fr
∗∗ Postal address: Microsoft Research, St George House, 1 Guildhall Street, Cambridge CB2 3NH, UK.
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Abstract

In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the fertility rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usual critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancestors. We also prove an ergodic decomposition result for stationary critical Hawkes point processes as a mixture of critical Hawkes point processes, and we give conditions for weak convergence to stationarity of critical Hawkes point processes.

Keywords

stochastic processespoint processesHawkes processesspectral analysislong-rang dependencecoupling

MSC classification

Primary: 60G55: Point processes 62M15: Spectral analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 122 - 135
Copyright
Copyright © by the Applied Probability Trust 2001 

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