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

Part of: Markov processes Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Andreas Lindell  and
Lars Holst
Andreas Lindell*
Affiliation:
Royal Institute of Technology, Stockholm
Lars Holst*
Affiliation:
Royal Institute of Technology, Stockholm
*
∗Current address: Lokevägen 9, Täby, SE 18776, Sweden. Email address: andreas.lindell@swedbank.se
∗∗Postal address: Department of Mathematics, Royal Institute of Technology, Stockholm, SE 10044, Sweden. Email address: lholst@math.kth.se
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Abstract

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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.

Keywords

Tied down random walkBrownian bridgeranked excursion lengthweak convergencegenerating functionKummer function

MSC classification

Secondary: 60J65: Brownian motion 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 1056 - 1067
Copyright
Copyright © Applied Probability Trust 2007 

References

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