The Hawkes Process with Different Exciting Functions and its Asymptotic Behavior. (March 2015)

• Record Type: Journal Article
• Authors: Fierro, Raúl; Leiva, Víctor; Møller, Jesper
• Is Part Of: Journal of applied probability. Volume 52:Number 1(2015)
• Page Start: 37
• Page End: 54
• Publication Date: 2015-03
• Subjects: Central limit theorem, -- clustering effect, -- law of large numbers, -- unpredictable marks; 60G55, -- 60F05; 519.2
• DOI: 10.1017/S0021900200012183
• Languages: English
• ISSNs: 0021-9002

Abstract

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.