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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.

References

Biane, Ph. and Yor, M. (1987) Valeurs principales associées aux temps locaux Browniens. Bull. Soc. Math. 2e série 111, 23–101.Google Scholar
Chandrasekharan, K. (1970) Arithmetical Functions. Springer-Verlag, Berlin.Google Scholar
Chung, K. L. (1974) A Course in Probability Theory, Academic Press, London.Google Scholar
Chung, K. L. (1976) Excursions in Brownian motion. Arch. Math. 14, 157–179.Google Scholar
Csáki, E. and Mohanty, S. (1981) Excursion and meander in random walk. Canad. J. Statist. 9, 57–70.Google Scholar
Csáki, E. and Mohanty, S. (1986) Some joint distributions for conditional random walks. Canad. J. Statist. 14, 19–28.Google Scholar
Diaconis, P. (1980) Average running times of the fast Fourier transform. J. Algorithms 1, 187–208.Google Scholar
Durrett, R. and Iglehart, D. (1977) Functionals of Brownian meander and Brownian excursion. Ann. Prob. 5, 130–135.Google Scholar
Durrett, R., Iglehart, D. and Miller, D. (1977) Weak convergence to Brownian meander and Brownian excursion. Ann. Prob. 5, 117–129.Google Scholar
Edwards, H. M. (1974) Riemann's Zeta-Function. Academic Press, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Gnedenko, B. V. (1954) Proof of the limiting probability distribution in two independent samples. Math. Nachr. 12, 29–66. (In Russian.)Google Scholar
Golomb, S. (1970) A class of probability distributions on the integers. J. Number Theory 2, 189–192.Google Scholar
Good, I. J. (1986) Statistical applications of Poisson's work. Statist. Sci. 2, 157–180.Google Scholar
Ito, K. and Mckean, H. P. (1965) Diffusion Processes and Their Sample Paths. Springer-Verlag.Google Scholar
Ivic, I. A. (1985) The Riemann Zeta-Function. Wiley, New York.Google Scholar
Kaigh, W. D. (1976) An invariance principle for random walk conditioned by a late return to zero. Ann. Prob. 4, 115–121.Google Scholar
Kaigh, W. D. (1978) An elementary derivation of the distribution of the maxima of Brownian meander and Brownian excursion. Rocky Mountain J. Math. 8, 641–645.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Kemperman, J. H. B. (1959) Asymptotic expansions for the Smirnov test and for the range of cumulative sums. Ann. Math. Statist. 30, 448–462.Google Scholar
Kiefer, J. (1959) K-sample analogues of the Kolmogorov–Smirnov and Cramer–von Mises tests. Ann. Math. Statist. 30, 420–447.Google Scholar
Knight, F. B. (1980) On the excursion process of Brownian motion. Trans. Amer. Math. Soc. 258, 77–86.Google Scholar
Lloyd, S. P. (1984) Ordered prime divisors of a random integer. Ann. Prob. 12, 1205–1212.Google Scholar
Newman, C. M. (1976) Fourier transforms with only real zeros. Proc. Amer. Math. Soc. 61, 245–251.Google Scholar
Polya, G. (1921) Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann. 84, 149–160.Google Scholar
Riemann, B. (1859) Über die Anzahl der Primzahlen unter eine gegebener Grosse. Monatsfer. Akad. Berlin, 671–680.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton.Google Scholar
Takács, L. (1957) Remarks on random walk problems. Publ. Math. Inst. Hungar. Acad. Sci. 11, 175–181.Google Scholar
Titchmarsh, E. C. (1951) The Theory of the Riemann Zeta-Function. Clarendon Press, Oxford.Google Scholar
Vervaat, V. (1979) A relation between Brownian bridge and Brownian excursion. Ann. Prob. 7, 143–149.Google Scholar