Random Walk Conditioned to Stay Positive
Random Walk Conditioned to Stay Positive
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DOI：
10.1112/s0024610702003708
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发表时间：
2003-02
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影响因子：
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通讯作者：
J. Biggins
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文献类型：
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作者：
J. Biggins
一个确定访问（0，∞）的随机游走通过一个适当的h-变换与它相关联，一个马尔可夫链被称为“条件为正的随机游走”，它在下面被正确定义。在连续时间中，如果随机游动被布朗运动代替，则类似的相关过程是贝塞尔3。令φ（x）= log log x。本文得到的主要结果（在定理1中正式陈述）是，当随机游动具有零均值和有限方差时，对于适当的（非随机）正L和有限U，当x趋于无穷大时，随机游动保持为正的总时间最终位于Lx 2/φ（x）和Ux 2 φ（x）之间。对于Bessel-3，确定了最佳L和U。
A random walk that is certain to visit (0, ∞) has associated with it, via a suitable h‐transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel‐3. Let φ(x) = log log x. The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below x ultimately lies between Lx2/φ(x) and Ux2φ(x), for suitable (non‐random) positive L and finite U, as x goes to infinity. For Bessel‐3, the best L and U are identified.