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1997 Cauchy's Principal Value of Local Times of Lévy Processes with no Negative Jumps via Continuous Branching Processes
Jean Bertoin
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Electron. J. Probab. 2: 1-12 (1997). DOI: 10.1214/EJP.v2-20

Abstract

Let $X$ be a recurrent Lévy process with no negative jumps and $n$ the measure of its excursions away from $0$. Using Lamperti's connection that links $X$ to a continuous state branching process, we determine the joint distribution under $n$ of the variables $C^+_T=\int_{0}^{T}{\bf 1}_{{X_s>0}}X_s^{-1}ds$ and $C^-_T=\int_{0}^{T}{\bf 1}_{{X_s<0}}|X_s|^{-1}ds$, where $T$ denotes the duration of the excursion. This provides a new insight on an identity of Fitzsimmons and Getoor on the Hilbert transform of the local times of $X$. Further results in the same vein are also discussed.

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Jean Bertoin. "Cauchy's Principal Value of Local Times of Lévy Processes with no Negative Jumps via Continuous Branching Processes." Electron. J. Probab. 2 1 - 12, 1997. https://doi.org/10.1214/EJP.v2-20

Information

Accepted: 1 September 1997; Published: 1997
First available in Project Euclid: 26 January 2016

zbMATH: 0890.60069
MathSciNet: MR1475864
Digital Object Identifier: 10.1214/EJP.v2-20

Subjects:
Primary: 60J30
Secondary: 60F05, 60G10

JOURNAL ARTICLE
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Vol.2 • 1997
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