Ladder heights, gaussian random walks and the Riemann zeta function
成果类型：
Article
署名作者：
Chang, JT; Peres, Y
署名单位：
Yale University; University of California System; University of California Berkeley; Hebrew University of Jerusalem
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1997
页码：
787-802
关键词：
摘要：
Let {S-n: n greater than or equal to 0} be a random walk having normally distributed increments with mean theta and variance 1, and let tau be the time at which the random walk first takes a positive value, so that S-tau is the first ladder height. Then the expected value EthetaStau, originally defined for positive theta, may be extended to be an analytic function of the complex variable theta throughout the entire complex plane, with the exception of certain branch point singularities. In particular, the coefficients in a Taylor expansion about theta = 0 may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot.