Journal of Theoretical Probability

, Volume 13, Issue 1, pp 259–277

# An Extension of Vervaat's Transformation and Its Consequences

• L. Chaumont
Article

## 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 ms such that when we exchange the pre-ms-part and the post-ms-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 ms 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.

Brownian bridge Brownian excursion uniform law path transformation occupation time quantile

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