On hitting times and fastest strong stationary times for skip-free chains
On hitting times and fastest strong stationary times for skip-free chains
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关于无跳链的击球时间和最快强静止时间
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发表时间：
2007
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通讯作者：
J. A. Fill
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作者：
J. A. Fill
以一组非负整数作为状态空间的（向上）无跳过马尔可夫链是向上跳跃可能仅具有单位大小的链；向下跳跃没有限制。在 1987 年的一篇论文中，Brown 和 Shao 确定了对于不可约连续时间无跳跃链和任意 d，从状态 0 到状态 d 的传递时间分布。当生成器的非零特征值 νj 均为实数时，其结果表明通过时间分布为 d 个独立指数随机变量的总和，速率为 νj。我们给出他们定理的另一个证明。在生死链的情况下，我们的证明将流逝时间明确表示为独立指数随机变量的总和。 Diaconis 和 Miclo 最近获得了第一个这样的表示，但我们的构造要简单得多。对于从状态 0 开始的随机单调时间反转的遍历连续时间无跳跃链的最快强平稳时间 T，我们获得了类似的（新的）结果，并且我们还获得了所有结果的离散时间模拟。 AMS 2000 主题分类。初级60J25；次级 60J35、60J10、60G40。
An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues νj of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates νj. We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. AMS 2000 subject classifications. Primary 60J25; secondary 60J35, 60J10, 60G40.