Hostname: page-component-7479d7b7d-68ccn Total loading time: 0 Render date: 2024-07-11T12:21:18.266Z Has data issue: false hasContentIssue false
  • English
  • Français

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: nappo@mat.uniroma1.it
∗∗∗ Postal address: Istituto per le Applicazioni del Calcolo ‘M. Picone’ (IAC-CNR), Viale del Policlinico 137, 00161 Roma, Italy.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]. Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
[2]. Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
[3]. Brémaud, P. (1981). Point Processes and Queues. Springer, New York.Google Scholar
[4]. Brémaud, P. and Massoulié, L. (1996). Stability of non-linear Hawkes processes. Ann. Prob. 24, 1563–1588.Google Scholar
[5]. Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[6]. Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438–443.Google Scholar
[7]. Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
[8]. Hawkes, A. G., and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.Google Scholar
[9]. Kerstan, J. (1964). Teilprozesse Poissonscher Prozesse. In Trans. 3rd Prague Conf. Inf. Theory, Statist. Decision Functions, Random Process. (Liblice, 1962), Czechoslovak Academy of Sciences, Prague, pp. 377–403.Google Scholar
[10]. Last, G., and Brandt, A. (1995). Marked Point Processes on the Real Line. Springer, New York.Google Scholar
[11]. Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[12]. Massoulié, L. (1995). Stabilité, simulation et optimisation de systèmes à evénements discrets. Doctoral Thesis, Université de Paris-Sud.Google Scholar
[13]. Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Proc. Appl. 75, 1–30.Google Scholar
[14]. Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Trans. Inf. Theory 27, 23–31.Google Scholar
[15]. Torrisi, G. L. (2000). Rate of convergence to stationarity of non-linear Hawkes processes. Doctoral Thesis, University of Milan.Google Scholar
[16]. Torrisi, G. L. (2002). A class of interacting marked point processes: rate of convergence to equilibrium. J. Appl. Prob. 39, 137–160.Google Scholar