On Lerch's transcendent and the Gaussian random walk

• 成果类型: Article
• 署名作者: Janssen, A. J. E. M.; van Leeuwaarden, J. S. H.
• 署名单位: Philips; Philips Research
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051606000000781
• 发表日期: 2007
• 页码: 421-439
• 关键词: corrected diffusion approximations; heavy-traffic limits; Servers; queues

摘要

Let X-1, X-2.... be independent variables, each having a normal distribution with negative mean -beta < 0 and variance 1. We consider the partial sums Sn = X-1 + - - - + X-n, with S-0 = 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 beta = 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 beta down arrow 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.