Electronic Journal of Probability

Cauchy's Principal Value of Local Times of Lévy Processes with no Negative Jumps via Continuous Branching Processes

Jean Bertoin

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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.

Article information

Source
Electron. J. Probab., Volume 2 (1997), paper no. 6, 12 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453839982

Digital Object Identifier
doi:10.1214/EJP.v2-20

Mathematical Reviews number (MathSciNet)
MR1475864

Zentralblatt MATH identifier
0890.60069

Subjects
Primary: 60J30
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes

Keywords
Cauchy's principal value Lévy process with no negative jumps branching process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bertoin, Jean. Cauchy's Principal Value of Local Times of Lévy Processes with no Negative Jumps via Continuous Branching Processes. Electron. J. Probab. 2 (1997), paper no. 6, 12 pp. doi:10.1214/EJP.v2-20. https://projecteuclid.org/euclid.ejp/1453839982


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