Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T18:37:19.354Z Has data issue: false hasContentIssue false
  • English
  • Français

Hawkes branching point processes without ancestors

Part of: Stochastic processes Inference from 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.
Get access Rights & Permissions [Opens in a new window]

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 

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

Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, New York.Google Scholar
Billingsley, P. (1968). Weak Convergence of Probability Measures. John Wiley, New York.Google Scholar
Brandt, A., and Last, G. (1995). Marked Point Processes on the Real Line. The Dynamic Approach. Springer, New York.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer, New York.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.Google Scholar
Cox, D. R. (1984). Long range dependence: a review. In Statistics: an Appraisal (Proc. 50th Anniv. Conf. Iowa State Statist. Lab.), eds David, H. A. and David, H. T. Iowa State University Press.Google Scholar
Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Daley, D. J., and Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stoch. Proc. Appl. 70, 265–282.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
Hawkes, A. G., and Oakes, D. (1974). A cluster representation of a self-exciting point process. J. Appl. Prob. 11, 493–503.Google Scholar
Jacod, J. (1975). Multivariate point processes: predictable projections, Radon–Nikodým derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Springer, Berlin.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Massoulié, L. (1995). Stabilité, simulation et optimisation de systèmes à événements discrets. Doctoral Thesis, Université Paris-Sud.Google Scholar
Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Proc. Appl. 75, 1–30.Google Scholar
Rozanov, Yu. A. (1967). Stationary Random Processes. Holden-Day, San Francisco.Google Scholar