The Annals of Applied Probability

On Lerch’s transcendent and the Gaussian random walk

A. J. E. M. Janssen and J. S. H. van Leeuwaarden

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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 {Sn: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.

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Ann. Appl. Probab., Volume 17, Number 2 (2007), 421-439.

First available in Project Euclid: 19 March 2007

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Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 30B40: Analytic continuation 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; Lévy processes 65B15: Euler-Maclaurin formula

Gaussian random walk all-time maximum Lerch’s transcendent Riemann zeta function Spitzer’s identity Euler–Maclaurin summation


Janssen, A. J. E. M.; van Leeuwaarden, J. S. H. On Lerch’s transcendent and the Gaussian random walk. Ann. Appl. Probab. 17 (2007), no. 2, 421--439. doi:10.1214/105051606000000781.

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