On Lerch's transcendent and the Gaussian random walk

Abstract

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta <0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\ge 0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max \{S_n:n\ge 0\}.$ These expressions are in terms of Taylor series about $\beta =0$ with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for $\beta \downarrow 0$. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802], and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin summation as key ingredients.

Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Functionals of the Brownian motion, localization and metric graphs
• Complete corrected diffusion approximations for the maximum of a random walk

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