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Rate of convergence to equilibrium of marked Hawkes processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

P. Brémaud ,
G. Nappo  and
G. L. Torrisi
P. Brémaud*
Affiliation:
ENS, Paris, and EPFL, Lausanne
G. Nappo*
Affiliation:
Università di Roma ‘La Sapienza’
G. L. Torrisi*
Affiliation:
Istituto per le Applicazioni del Calcolo, Roma
*
∗ Postal address: Département d’Informatique, École Normale Supérieure, 45 rue d’Ulm, F-75230 Paris Cedex 05, France.
∗∗ Postal address: Dipartimento di Matematica, Università di Roma ‘La Sapienza’, Piazzale A. Moro 1, 00185-Roma, Italy. Email address: [email protected]
∗∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘M. Picone’ (IAC-CNR), Viale del Policlinico 137, 00161 Roma, Italy.
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Abstract

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

Keywords

Point processesstochastic intensityHawkes processesrenewal theorybranching processesconvergence in variationstochastic simulation

MSC classification

Primary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 123 - 136
Copyright
Copyright © Applied Probability Trust 2002 

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