The Annals of Probability

Ladder heights, Gaussian random walks and the Riemann zeta function

Joseph T. Chang and Yuval Peres

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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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Ann. Probab. Volume 25, Number 2 (1997), 787-802.

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

Primary: 60J15 30B40: Analytic continuation

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


Chang, Joseph T.; Peres, Yuval. Ladder heights, Gaussian random walks and the Riemann zeta function. Ann. Probab. 25 (1997), no. 2, 787--802. doi:10.1214/aop/1024404419.

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