Open Access
Translator Disclaimer
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.


Download Citation

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

Keywords: analytic continuation , boundary crossing probability , ladder height , Random walk , Riemann zeta function

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 2 • April 1997
Back to Top