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Power spectra of general shot noises and Hawkes point processes with a random excitation

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

P. Brémaud  and
L. Massoulié
P. Brémaud*
Affiliation:
ENS, Paris, and EPFL, Lausanne
L. Massoulié*
Affiliation:
Microsoft Research
*
∗ Postal address: École Normale Supérieure, Departement d'Informatique, 45 rue d'Ulm, F75230 Paris Cedex 05, France. Email address: pierre.bremaud@ens.fr
∗∗ Postal address: Microsoft Research, 7 J. J. Thomson Avenue, Cambridge CB3 0FB, UK.
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Abstract

We give (i) the Cramér power spectral measure of the general shot noise process with random excitation and non-Poisson stationary driving point processes and (ii) the Bartlett power spectral measure of the self-exciting Hawkes point process with random excitation, also called the Hawkes branching point process with random fertility rate. The latter is obtained via the isometry formula for integrals with respect to the canonical martingale measure associated with a marked point process.

Keywords

Point processshot noiseHawkes processBartlett spectra

MSC classification

Primary: 60G12: General second-order processes 60G55: Point processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 1 , March 2002 , pp. 205 - 222
Copyright
Copyright © Applied Probability Trust 2002 

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