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Honest bernoulli excursions

Published online by Cambridge University Press:  14 July 2016

Laurel Smith  and
Persi Diaconis
Laurel Smith
Affiliation:
Texas A&M University
Persi Diaconis*
Affiliation:
Stanford University
*
∗ Postal address: Department of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305, USA.
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Abstract

For simple random walk on the integers, consider the chance that the walk has traveled distance k from its start given that its first return is at time 2n. We derive a limiting approximation accurate to order 1/n. We give a combinatorial explanation for a functional equation satisfied by the limit and show this yields the functional equation of Riemann's zeta function.

Keywords

EXCURSION DISTANCERANDOM WALK LIMITING DISTRIBUTIONZETA FUNCTIONFIRST RETURN DISTANCE
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 3 , September 1988 , pp. 464 - 477
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported by NSF Grant DMS 86-00235.

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