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

   
@article{EJP20,
	author = {Jean Bertoin},
	title = {Cauchy's Principal Value of Local Times of Lévy Processes with no Negative Jumps via Continuous Branching Processes},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {2},
	year = {1997},
	keywords = {Cauchy's principal value, Lévy process with no negative jumps, branching process.},
	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.},
	pages = {no. 6, 1-12},
	issn = {1083-6489},
	doi = {10.1214/EJP.v2-20},    
        url = {http://ejp.ejpecp.org/article/view/20}}