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

Jean Bertoin (Université Paris VI)


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-12

Publication Date: September 1, 1997

DOI: 10.1214/EJP.v2-20


  1. M. T. Barlow. Continuity of local times for L'evy processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 69 (1985), 23-35. Math Review link
  2. J. Bertoin. Sur la d'ecomposition de la trajectoire d'un processus de L'evy spectralement positif en son infimum. Ann. Inst. Henri Poincar'e 27 (1991), 537-547. Math Review link
  3. J. Bertoin. An extension of Pitman's theorem for spectrally positive L'evy processes. Ann. Probab. 20 (1992), 1464-1483. Math Review link
  4. J. Bertoin. On the Hilbert transform of the local times of a L'evy process. Bull. Sci. Math. 119 (1995), 147-156. Math Review link
  5. Ph. Biane and M. Yor. Valeurs principales associ'ees aux temps locaux browniens. Bull. sc. Math. 111 (1987), 23-101. Math Review link
  6. N. H. Bingham. Continuous branching processes and spectral positivity, Stochastic Process. Appl. 4 (1976), 217-242. Math Review link
  7. P. J. Fitzsimmons and R. K. Getoor. On the distribution of the Hilbert transform of the local time of a symmetric L'evy process. Ann. Probab. 20 (1992), 1484-1497. Math Review link
  8. D. R. Grey. Asymptotic behaviour of continuous-time continuous state-space branching processes. J. Appl. Prob. 11 (1974), 669-677. Math Review link
  9. M. Jirina. Stochastic branching processes with continuous state-space. Czech. Math. J. 8 (1958), 292-313. Math Review link
  10. K. Kawazu and S. Watanabe. Branching processes with immigration and related limit theorems. Th. Probab. Appl. 16 (1971), 36-54. Math Review link
  11. J. Lamperti. Continuous-state branching processes. Bull. Amer. Math. Soc. 73 (1967), 382-386. Math Review link
  12. J. Lamperti. The limit of a sequence of branching processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 7 (1967), 271-288. Math Review link
  13. M. L. Silverstein. A new approach to local time. J. Math. Mech. 17 (1968), 1023-1054. Math Review link

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.