## The Annals of Probability

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

### A Relation between Brownian Bridge and Brownian Excursion

#### 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: 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. The Annals of Probability 7 (1979), no. 1, 143--149. doi:10.1214/aop/1176995155. http://projecteuclid.org/euclid.aop/1176995155.