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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: [email protected]
∗∗ 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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References

Bartlett, M. S. (1963). The spectral analysis of point processes. J. R. Statist. Soc. B 29, 264–296.Google Scholar
Bondesson, L. (1988). Shot noise processes and distributions. In Encyclopedia of Statistical Science, Vol. 8, John Wiley, New York, pp. 448–452.Google Scholar
Brémaud, P., (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.Google Scholar
Brémaud, P., (2000). An insensitivity property of Lundberg's estimate for delayed claims. J. Appl. Prob. 37, 914–917.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Prob. 38, 122–135.Google Scholar
Brémaud, P., Nappo, G. and Torrisi, G. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123–136.Google Scholar
Cioczek-Georges, R. and Mandelbrot, B. (1995). A class of micropulses and antipersistent fractional Brownian motion. Stoch. Process. Appl. 60, 1–18.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438–443.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
Hawkes, A. G. (1972). Spectra of some mutually exciting point processes with associated random variables. In Stochastic Point Processes, ed. Lewis, P. A. W., John Wiley, New York, pp. 261–271.Google Scholar
Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.Google Scholar
Last, G. and Brandt, A. (1995). Marked Point Process on the Real Line. Springer, New York.Google Scholar
Lewis, P. A. W. (1964). A branching point process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 26, 398–456.Google Scholar
Lowen, S. B. and Teich, M. C. (1990). Power law shot noise. IEEE Trans. Inf. Theory 36, 1302–1318.Google Scholar
Massoulié, L., (1995). Simulation et optimisation de systèmes à événements discrets. , Université Paris-Sud.Google Scholar
Massoulié, L., (1998). Stability results for a general class of interacting point process dynamics, and applications. Stoch. Process. Appl. 75, 1–30.Google Scholar
Parulekar, M. and Makowski, A. M. (1997). M/G/∞: a versatile class of models for network traffic. In Proc. IEEE INFOCOM'97 (Kobe, Japan), pp. 1452–1459.Google Scholar
Rice, S. O. (1944). Mathematical analysis of random noise. Bell Systems Tech. J. 23, 1–51.Google Scholar
Rice, S. O. (1977). On general shot noise. Adv. Appl. Prob. 9, 553–565.Google Scholar
Schottky, W. (1918). Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern. Ann. Phys. 57, 541–567.Google Scholar
Vere-Jones, D. (1970). Stochastic models for earthquake occurrences. J. R. Statist. Soc. B 32, 1–42.Google Scholar