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An Extension of Vervaat's Transformation and Its Consequences

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  • L. Chaumont

Abstract

Vervaat(18) proved that by exchanging the pre-minimum and post-minimum parts of a Brownian bridge one obtains a normalized Brownian excursion. Let s ∈ (0, 1), then we extend this result by determining a random time m s such that when we exchange the pre-m s-part and the post-m s-part of a Brownian bridge, one gets a Brownian bridge conditioned to spend a time equal to s under 0. This transformation leads to some independence relations between some functionals of the Brownian bridge and the time it spends under 0. By splitting the Brownian motion at time m s in another manner, we get a new path transformation which explains an identity in law on quantiles due to Port. It also yields a pathwise construction of a Brownian bridge conditioned to spend a time equal to s under 0.

Suggested Citation

  • L. Chaumont, 2000. "An Extension of Vervaat's Transformation and Its Consequences," Journal of Theoretical Probability, Springer, vol. 13(1), pages 259-277, January.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:1:d:10.1023_a:1007795228700
    DOI: 10.1023/A:1007795228700
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    References listed on IDEAS

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    1. 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.
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