## Electronic Journal of Probability

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

Jean Bertoin

#### 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

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

Rights

#### 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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