On Lerch's transcendent and the Gaussian random walk

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dc.creatorJanssen, A. J. E. M.
dc.creatorvan Leeuwaarden, J. S. H.
dc.date2007-03-30
dc.date.accessioned2026-07-07T07:55:06Z
dc.date.available2026-07-07T07:55:06Z
dc.descriptionLet $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-β<0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max\{S_n:n\geq0\}.$ 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 $β\downarrow0$. 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.
dc.descriptionPublished 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)
dc.identifierhttps://arxiv.org/abs/math/0703908
dc.identifierhttp://arxiv.org/abs/math/0703908
dc.identifierAnnals of Applied Probability 2007, Vol. 17, No. 2, 421-439
dc.identifierdoi:10.1214/105051606000000781
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/126852
dc.subjectProbability
dc.subject11M06, 30B40, 60G50, 60G51, 65B15 (Primary)
dc.titleOn Lerch's transcendent and the Gaussian random walk
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