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 S_n=X₁+⋯+X_n, with S₀=0, and refer to the process {S_n:n≥0} as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum M=max {S_n: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.