Distributions of the Longest Excursions in a Tied Down Simple Random Walk and in a Brownian Bridge

2007/12/01 by Andreas Lindell, Lars Holst
Computer Science · Mathematics · Decision Sciences · #Bayesian Methods and Mixture Models #Stochastic processes and statistical mechanics #Probability and Risk Models

paper · doi:10.1017/s0021900200003739

Abstract

Expressions for the joint distribution of the longest and second longest excursions as well as the marginal distributions of the three longest excursions in the Brownian bridge are obtained. The method, which primarily makes use of the weak convergence of the random walk to the Brownian motion, principally gives the possibility to obtain any desired joint or marginal distribution. Numerical illustrations of the results are also given.

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