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February, 1979 A Relation between Brownian Bridge and Brownian Excursion
Wim Vervaat
Ann. Probab. 7(1): 143-149 (February, 1979). DOI: 10.1214/aop/1176995155


It is shown that Brownian excursion is equal in distribution to Brownian bridge with the origin placed at its absolute minimum. This explains why the maximum of Brownian excursion and the range of Brownian bridge have the same distribution, a fact which was discovered by Chung and Kennedy. The result is proved by establishing similar relations for "Bernoulli excursions" and "Bernoulli bridges" constructed from symmetric Bernoulli walks, and exploiting known weak convergence results. Some technical complications arise from the fact that Bernoulli bridges assume their minimum value with positive probability more than once.


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Wim Vervaat. "A Relation between Brownian Bridge and Brownian Excursion." Ann. Probab. 7 (1) 143 - 149, February, 1979.


Published: February, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60058
MathSciNet: MR515820
Digital Object Identifier: 10.1214/aop/1176995155

Primary: 60J65
Secondary: 60B10 , 60C05

Keywords: Brownian bridge , Brownian excursion , invariance principles under conditioning , symmetric Bernoulli walk

Rights: Copyright © 1979 Institute of Mathematical Statistics

Vol.7 • No. 1 • February, 1979
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