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Ladder heights, Gaussian random walks and the Riemann zeta function. (English) Zbl 0880.60070
Summary: 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 $$\theta$$, may be extended to be an analytic function of the complex variable $$\theta$$ throughout the entire complex plane, with the exception of certain branch point singularities. 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.

##### MSC:
 60G50 Sums of independent random variables; random walks 30B40 Analytic continuation of functions of one complex variable 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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