A Relation between Brownian Bridge and Brownian Excursion

Wim Vervaat
Source: Ann. Probab. Volume 7, Number 1 (1979), 143-149.

Abstract

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.

First Page:
Primary Subjects: 60J65
Secondary Subjects: 60B10, 60C05
Full-text: Open access