The Annals of Probability

A Relation between Brownian Bridge and Brownian Excursion

Wim Vervaat

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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.

Article information

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176995155

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176995155

Mathematical Reviews number (MathSciNet)
MR515820

Zentralblatt MATH identifier
0392.60058

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60B10: Convergence of probability measures 60C05: Combinatorial probability

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

Citation

Vervaat, Wim. A Relation between Brownian Bridge and Brownian Excursion. Ann. Probab. 7 (1979), no. 1, 143--149. doi:10.1214/aop/1176995155. http://projecteuclid.org/euclid.aop/1176995155.


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