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The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Raúl Fierro ,
Víctor Leiva  and
Jesper Møller
Raúl Fierro*
Affiliation:
Pontificia Universidad Católica de Valparaíso and Universidad de Valparaíso
Víctor Leiva*
Affiliation:
Universidad de Valparaíso and Universidad Adolfo Ibáñez
Jesper Møller*
Affiliation:
Aalborg University
*
∗ Postal address: Pontificia Universidad Católica de Valparaíso, Brasil 2950, Casilla 4059, Valparaíso, chile. Email address: rfierro@ucv.cl
∗∗ Postal address: Universidad de Valparaíso, Gran Bretaña 1111, Casilla 5030, Valparaíso, Chile. Email address: victorleivasanchez@gmail.com
∗∗∗ Postal address: Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg Ø, Denmark. Email address: jm@math.aau.dk
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Abstract

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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.

Keywords

Central limit theoremclustering effectlaw of large numbersunpredictable marks

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 37 - 54
Copyright
© Applied Probability Trust 

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