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

Part of: Combinatorial probability Markov processes

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: [email protected]
∗∗Postal address: Department of Mathematics, Royal Institute of Technology, Stockholm, SE 10044, Sweden. Email address: [email protected]
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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

[1] Abramovitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Csáki, E. and Hu, Y. (2004). Invariance principles for ranked excursion lengths and heights. Electron. Commun. Prob. 9, 14–21.CrossRefGoogle Scholar
[3] Gourdon, X. (1998). Largest component in random combinatorial structures. Disc. Math. 180, 185–209.CrossRefGoogle Scholar
[4] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
[5] Lindell, A. (2000). Numerical investigations of the distributions of the longest excursions in tied down simple random walks and Brownian bridges. . Royal Institute of Technology, Stockholm.Google Scholar
[6] Pitman, J. and Yor, M. (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25, 855–900.Google Scholar
[7] Wendel, J. G. (1964). Zero-free intervals in semi-stable Markov processes. Math. Scand. 14, 21–34.Google Scholar