Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T11:37:23.644Z Has data issue: false hasContentIssue false
  • English
  • Français

The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Raúl Fierro ,
Víctor Leiva  and
Jesper Møller
Raúl Fierro*
Affiliation:
Pontificia Universidad Católica de Valparaíso and Universidad de Valparaíso
Víctor Leiva*
Affiliation:
Universidad de Valparaíso and Universidad Adolfo Ibáñez
Jesper Møller*
Affiliation:
Aalborg University
*
∗ Postal address: Pontificia Universidad Católica de Valparaíso, Brasil 2950, Casilla 4059, Valparaíso, chile. Email address: [email protected]
∗∗ Postal address: Universidad de Valparaíso, Gran Bretaña 1111, Casilla 5030, Valparaíso, Chile. Email address: [email protected]
∗∗∗ Postal address: Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Ø, Denmark. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

Keywords

Central limit theoremclustering effectlaw of large numbersunpredictable marks

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 37 - 54
Copyright
© Applied Probability Trust 

References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123, 2475–2499.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Martingale Dynamics. Springer, New York.Google Scholar
Brémaud, P. Nappo, G. and Torrisi, G. L. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123–136.Google Scholar
Carstensen, L., Sandelin, A., Winther, O. and Hansen, N. R. (2010). Multivariate Hawkes process models of the occurrence of regulatory elements. BMC Bioinformatics 11, 456.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Embrechts, P., Liniger, T. and Lu, L. (2011). Multivariate Hawkes processes: an application to financial data. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A), Applied Probability Trust, Sheffield, pp. 367–378.Google Scholar
Fierro, R., Leiva, V., Ruggeri, F. and Sanhueza, A. (2013). On a Birnbaum–Saunders distribution arising from a non-homogeneous Poisson process. Statist. Prob. Lett. 83, 1233–1239.CrossRefGoogle Scholar
Gänssler, P. and Haeusler, E. (1986). On martingale central limit theory. In Dependence in Probability and Statistics, Birkhäuser, Boston, MA, pp. 303–334.Google Scholar
Gusto, G. and Schbath, S. (2005). FADO: a statistical method to detect favored or avoided distances between occurrences of motifs using the Hawkes' model. Statist. Appl. Genet. Molec. Biol. 4, 24.CrossRefGoogle ScholarPubMed
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. Ser. B 33, 438–443.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 process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.Google Scholar
Jacod, J. (1974). Multivariate point processes: predictable projection, Radon–Nikodým derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
Møller, J. and Rasmussen, J. G. (2005). Perfect simulation of Hawkes processes. J. Appl. Prob. 37, 629–646.Google Scholar
Møller, J. and Rasmussen, J. G. (2006). Approximate simulation of Hawkes processes. Methodology Comput. Appl. Prob. 8, 53–64.Google Scholar
Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9–27.Google Scholar
Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50, 379–402.Google Scholar
Pernice, V., Staude, B., Cardanobile, S. and Rotter, S. (2012). Recurrent interactions in spiking networks with arbitrary topology. Phys. Rev. E 85, 031916.Google Scholar
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.CrossRefGoogle Scholar
Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Prob. 50, 760–771.Google Scholar
Zhu, L. (2013). Nonlinear Hawkes processes. , New York University.Google Scholar