马尔可夫链的条件是永远不要在原点等待太久
Markov chains conditioned never to wait too long at the origin
摘要：
受Feller抛硬币问题的启发，我们考虑了一个不可约的马尔可夫链在0点上等待时间过长的条件。用$τ$表示链$X$在原点等待至少一个时间单位的第一次，我们考虑以事件$(τ>T)$为链的条件。我们表明，有一个弱极限的情况下，无论是有限的状态空间或$X$是瞬态的$T\to \infty$。我们给出了在其他情况下存在弱极限的充分条件，并表明如果达到零的时间比$τ$具有更轻的尾部并且$τ$是亚指数的，则我们淡收敛到有缺陷的极限。
英文摘要：
Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by $τ$ the first time that the chain, $X$, waits for at least one unit of time at the origin, we consider conditioning the chain on the event $(τ>T)$. We show there is a weak limit as $T\to \infty$ in the cases where either the statespace is finite or $X$ is transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than $τ$ and $τ$ is subexponential.
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