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Occupation time distributions for Lévy bridges and excursions

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  • Fitzsimmons, P. J.
  • Getoor, R. K.

Abstract

Let X be a one-dimensional Lévy process. It is shown that under the bridge law for X starting from 0 and ending at 0 at time t, the amount of time X spends positive has a uniform distribution on [0, t]. When 0 is a regular point, this uniform distribution result leads to an explicit expression for the Laplace transform of the joint distribution of the pair (R, AR), where R is the length of an excursion of X from 0, and AR is the total time X spends positive during the excursion. More concrete expressions are obtained for stable processes by specialization. In particular, a formula determining the distribution of AR/R is given in the stable case.

Suggested Citation

  • Fitzsimmons, P. J. & Getoor, R. K., 1995. "Occupation time distributions for Lévy bridges and excursions," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 73-89, July.
    • Handle: RePEc:eee:spapps:v:58:y:1995:i:1:p:73-89
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    References listed on IDEAS

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    1. Bertoin, Jean, 1993. "Splitting at the infimum and excursions in half-lines for random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 17-35, August.
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    Cited by:

    1. Marchal, Philippe, 1998. "Distribution of the occupation time for a Lévy process at passage times at 0," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 123-131, May.
    2. Yano, Kouji & Yano, Yuko, 2008. "Remarks on the density of the law of the occupation time for Bessel bridges and stable excursions," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2175-2180, October.
    3. Borovkov, Konstantin & McKinlay, Shaun, 2012. "The uniform law for sojourn measures of random fields," Statistics & Probability Letters, Elsevier, vol. 82(9), pages 1745-1749.
    4. L. Chaumont, 2000. "An Extension of Vervaat's Transformation and Its Consequences," Journal of Theoretical Probability, Springer, vol. 13(1), pages 259-277, January.
    5. Giovanni Conforti & Tetiana Kosenkova & Sylvie Rœlly, 2019. "Conditioned Point Processes with Application to Lévy Bridges," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2111-2134, December.

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