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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Abstract

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

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

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1429282605

Digital Object Identifier
doi:10.1239/jap/1429282605

Mathematical Reviews number (MathSciNet)
MR3336845

Zentralblatt MATH identifier
1315.60055

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

Keywords
Central limit theorem clustering effect law of large numbers unpredictable marks

Citation

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. https://projecteuclid.org/euclid.jap/1429282605.


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