On lerch's transcendent and the Gaussian random walk

Abstract

Let X1, X2, ... be independent variables, each having a normal distribution with negative mean -β < 0 and variance 1. We consider the partial sums Sn = X1 +⋯+ Xn, with S0 = 0, and refer to the process {Sn : n ≥ 0} as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum M = max{5n : n ≥ 0}. These expressions are in terms of Taylor series about β = 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 β ↓ 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. © Institute of Mathematical Statistics, 2007.