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April 1997 Ladder heights, Gaussian random walks and the Riemann zeta function
Joseph T. Chang, Yuval Peres
Ann. Probab. 25(2): 787-802 (April 1997). DOI: 10.1214/aop/1024404419


Let $\{S_n: n \geq 0\}$ be a random walk having normally distributed increments with mean $\theta$ and variance 1, and let $\tau$ be the time at which the random walk first takes a positive value, so that $S_{\tau}$ is the first ladder height. Then the expected value $E_{\theta} S_{\tau}$, originally defined for positive $\theta$, maybe extended to be an analytic function of the complex variable $\theta$ throughout the entire complex plane, with the exception of certain branch point sin-gularities. In particular, the coefficients in a Taylor expansion about $\theta = 0$ may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot.


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Joseph T. Chang. Yuval Peres. "Ladder heights, Gaussian random walks and the Riemann zeta function." Ann. Probab. 25 (2) 787 - 802, April 1997.


Published: April 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0880.60070
MathSciNet: MR1434126
Digital Object Identifier: 10.1214/aop/1024404419

Primary: 30B40, 60J15

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 1997
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