The Hawkes process with different exciting functions and its asymptotic behavior. (English) Zbl 1315.60055

Summary: 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.


60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
Full Text: DOI Euclid


[1] 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. · Zbl 1292.60032
[2] Brémaud, P. (1981). Point Processes and Queues. Martingale Dynamics . Springer, New York.
[3] Brémaud, P. Nappo, G. and Torrisi, G. L. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123-136. · Zbl 1005.60062
[4] 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.
[5] 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. · Zbl 1026.60061
[6] 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. · Zbl 1242.62093
[7] 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. · Zbl 1266.60013
[8] Gänssler, P. and Haeusler, E. (1986). On martingale central limit theory. In Dependence in Probability and Statistics , Birkhäuser, Boston, MA, pp. 303-334.
[9] 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. · Zbl 1095.62126
[10] Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. Ser. B 33, 438-443. · Zbl 0238.60094
[11] Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83-90. · Zbl 0219.60029
[12] Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493-503. · Zbl 0305.60021
[13] Jacod, J. (1974). Multivariate point processes: predictable projection, Radon-Nikodým derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235-253. · Zbl 0302.60032
[14] Møller, J. and Rasmussen, J. G. (2005). Perfect simulation of Hawkes processes. J. Appl. Prob. 37, 629-646. · Zbl 1074.60057
[15] Møller, J. and Rasmussen, J. G. (2006). Approximate simulation of Hawkes processes. Methodology Comput. Appl. Prob. 8, 53-64. · Zbl 1101.60032
[16] Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1044.62101
[17] Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9-27.
[18] Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Ann. Inst. Statist. Math. 50, 379-402. · Zbl 0947.62061
[19] Pernice, V., Staude, B., Cardanobile, S. and Rotter, S. (2012). Recurrent interactions in spiking networks with arbitrary topology. Phys. Rev. E 85, 031916.
[20] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, New York. · Zbl 0544.60045
[21] Zhu, L. (2013). Central limit theorem for nonlinear Hawkes processes. J. Appl. Prob. 50, 760-771. · Zbl 1306.60015
[22] Zhu, L. (2013). Nonlinear Hawkes processes. Doctoral Thesis, New York University. · Zbl 1306.60015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.