The cover time of a Markov chain on a finite state space is the expected time until all states are visited. We show that if the cover time of a discrete-time Markov chain with rational transitions probabilities is bounded, then it is a rational number. The result is proved by relating the cover time of the original chain to the hitting time of a set in another higher dimensional chain. We prove this result in a more general setting where $k\ge 1$ independent copies of a Markov chain are run simultaneously on the same state space.

中文翻译：

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具有合理转移概率的（多）马尔可夫链的覆盖时间是合理的

有限状态空间上马尔可夫链的覆盖时间是直到所有状态都被访问的预期时间。我们证明，如果具有有理转移概率的离散时间马尔可夫链的覆盖时间是有界的，那么它是一个有理数。通过将原始链条的覆盖时间与另一个更高维度链条中的一组撞击时间相关联来证明该结果。我们在更一般的环境中证明了这一结果，其中$ķ\ge 1$马尔可夫链的独立副本在同一状态空间上同时运行。