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A Relation between Brownian Bridge and Brownian Excursion
The Annals of Probability
Vol. 7, No. 1 (Feb., 1979), pp. 143-149
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2242845
Page Count: 7
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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.
The Annals of Probability © 1979 Institute of Mathematical Statistics