2007/03/20 by A. J. E. M. Janssen, Johan S. H. van Leeuwaarden, J. S. H. van Leeuwaarden
Mathematics · #Analytic Number Theory Research #Mathematical functions and polynomials #Random Matrices and Applications #math.PR #msc:11M06 #msc:30B40 #msc:60G50 #msc:60G51 #msc:65B15
paper · pdf · doi:10.1214/105051606000000781
published as Annals of Applied Probability 2007, Vol. 17, No. 2, 421-439 · 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)
openalex publication_date 2007/03/20 · arxiv created 2007/03/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.