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

Fierro, Raúl; Leiva, Víctor; Møller, Jesper

J. Appl. Probab. 52, No. 1, 37-54 (2015).

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.

MSC:

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

Keywords:

Hawkes process; central limit theorem; clustering effect; law of large numbers; unpredictable marks

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