Journal of Applied Probability

The Hawkes process with different exciting functions and its asymptotic behavior

Raúl Fierro, Víctor Leiva, and Jesper Møller

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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.

Article information

J. Appl. Probab. Volume 52, Number 1 (2015), 37-54.

First available in Project Euclid: 17 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems

Central limit theorem clustering effect law of large numbers unpredictable marks


Fierro, Raúl; Leiva, Víctor; Møller, Jesper. The Hawkes process with different exciting functions and its asymptotic behavior. J. Appl. Probab. 52 (2015), no. 1, 37--54. doi:10.1239/jap/1429282605.

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