arXiv 1408.08222014-08-04math.PR

马尔可夫链中的惊奇概率

Surprise probabilities in Markov chains

James Norris, Yuval Peres, Alex Zhai

摘要：

在状态$x$开始的马尔可夫链中，命中时间$τ(y)$是链第一次到达另一个状态$y$。我们研究概率$\mathbf{P}_x(τ(y) = t)$，第一次访问$y$恰好发生在一个给定的时间$t$。非正式地说，在大时间$t$访问新状态的事件可以被认为是"惊喜"。我们证明了以下三个界：1）在任何具有$n$状态的马尔可夫链中，$\mathbf{P}_x(τ(y) = t) \le \frac{n}{t}$。2)在具有$n$状态的可逆链中，$\mathbf{P}_x(τ(y) = t) \le \frac{\sqrt{2n}}{t}$为$t \ge 4n + 4$。3)对于具有$n \ge 2$个顶点的简单图上的随机游动，得到了$\mathbf{P}_x(τ(y) = t) \le \frac{4e \log n}{t}$.我们构造的例子表明，这些界限接近最优。我们的边界的主要特点是，他们需要很少的知识的马尔可夫链的结构。为了证明图上随机游动的界，我们建立了Aldous，Ding和Oveis-Gharan（private communication）的估计：对于$n$-顶点图上的随机游动，对每个初始顶点$x$，\[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \]

英文摘要：

In a Markov chain started at a state $x$, the hitting time $τ(y)$ is the first time that the chain reaches another state $y$. We study the probability $\mathbf{P}_x(τ(y) = t)$ that the first visit to $y$ occurs precisely at a given time $t$. Informally speaking, the event that a new state is visited at a large time $t$ may be considered a "surprise". We prove the following three bounds: 1) In any Markov chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{n}{t}$. 2) In a reversible chain with $n$ states, $\mathbf{P}_x(τ(y) = t) \le \frac{\sqrt{2n}}{t}$ for $t \ge 4n + 4$. 3) For random walk on a simple graph with $n \ge 2$ vertices, $\mathbf{P}_x(τ(y) = t) \le \frac{4e \log n}{t}$. We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain. To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): For random walk on an $n$-vertex graph, for every initial vertex $x$, \[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \]

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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