Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T05:23:34.194Z Has data issue: false hasContentIssue false
  • English
  • Français

Central Limit Theorem for Nonlinear Hawkes Processes

Part of: Stochastic processes Limit theorems

Published online by Cambridge University Press:  30 January 2018

Lingjiong Zhu
Lingjiong Zhu*
Affiliation:
New York University
*
∗ Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY-10012, USA. 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 Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance, and many other fields. In this paper we obtain a functional central limit theorem for the nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.

Keywords

Central limit theoremfunctional central limit theorempoint processHawkes processself-exciting process

MSC classification

Primary: 60G55: Point processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 760 - 771
Copyright
© Applied Probability Trust 

References

Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J. F. (2012). Scaling limits for Hawkes processes and application to financial statistics. Available at http://arxiv.org/abs/1202.0842v1.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.Google Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.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.CrossRefGoogle Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.Google Scholar
Dassios, A. and Zhao, H. (2011). A dynamic contagion process. Adv. Appl. Prob. 43, 814–846.CrossRefGoogle Scholar
Errais, E., Giesecke, K. and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financial Math. 1, 642–665.CrossRefGoogle 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
Heyde, C. C. and Scott, D. J. (1973). Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments. Ann. Prob. 1, 428–436.CrossRefGoogle Scholar
Karabash, D. and Zhu, L. (2013). Limit theorems for marked Hawkes processes with application to a risk model. Available at http://arxiv.org/abs/1211.4039v2.Google Scholar
Liniger, T. (2009). Multivariate Hawkes Processes. , ETH.Google Scholar
Zhu, L. (2012). Large deviations for Markovian nonlinear Hawkes Processes. Available at http://arxiv.org/abs/1108.2432v2.Google Scholar
Zhu, L. (2013). Process-level large deviations for nonlinear Hawkes point processes. To appear in Ann. Inst. H. Poincaré Prob. Statist. Google Scholar