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A Fourier Approach for the Level Crossings of Shot Noise Processes with Jumps

Part of: Stochastic processes Distribution theory - Probability Real functions Limit theorems

Published online by Cambridge University Press:  04 February 2016

Hermine Biermé  and
Agnès Desolneux
Hermine Biermé*
Affiliation:
Université Paris Descartes
Agnès Desolneux*
Affiliation:
Université Paris Descartes
*
∗ Postal address: MAP5 (UMR CNRS 8145), Université Paris Descartes, 45 rue des Saints-Pères, 75006 Paris, France.
∗∗ Email address: agnes.desolneux@mi.parisdescartes.fr
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Abstract

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We use a change-of-variable formula in the framework of functions of bounded variation to derive an explicit formula for the Fourier transform of the level crossing function of shot noise processes with jumps. We illustrate the result in some examples and give some applications. In particular, it allows us to study the asymptotic behavior of the mean number of level crossings as the intensity of the Poisson point process of the shot noise process goes to infinity.

Keywords

Shot noise processlevel crossingstationary processPoisson point processcharacteristic functionfunctions of bounded variationchange-of-variable formula

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60E10: Characteristic functions; other transforms 26A45: Functions of bounded variation, generalizations 60G10: Stationary processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 100 - 113
Copyright
© Applied Probability Trust 

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