Ladder heights, Gaussian random walks and the Riemann zeta function
Ladder heights, Gaussian random walks and the Riemann zeta function
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阶梯高度、高斯随机游走和黎曼 zeta 函数
DOI：
10.1214/aop/1024404419
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发表时间：
1997
影响因子：
2.3
通讯作者：
Y. Peres
中科院分区：
文献类型：
--
作者：
Joseph T. Chang;Y. Peres
令{S n：n ≥ 0}是具有正态分布增量的随机游动，平均值为θ，方差为1，设τ是随机游动首次取正值的时间，因此S τ是第一阶梯高度。然后，期望值E θ S τ，最初定义为正θ，可以扩展为在整个复平面上复变量θ的解析函数，除了某些分支点奇点。特别地，关于θ = 0的泰勒展开式中的系数可以被明确地写为涉及黎曼zeta函数的简单表达式。以前只有第一个系数的系列开发在这里是已知的;这个术语已被广泛用于开发近似的边界交叉问题的高斯随机游动。知识的完整系列，使更精细的结果成为可能，我们应用它来获得渐近的边界交叉概率和限制预期的超调。
Let {S n : n ≥ 0} be a random walk having normally distributed increments with mean θ and variance 1, and let τ be the time at which the random walk first takes a positive value, so that S τ is the first ladder height. Then the expected value E θ S τ , originally defined for positive θ, may be extended to be an analytic function of the complex variable θ throughout the entire complex plane, with the exception of certain branch point singularities. In particular, the coefficients in a Taylor expansion about θ = 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.