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Let $(p_t)_{t\in\mathbb{N}}$ be an stochastic process on a countable (probability measure) space. Supose it has the Markov and the Martingale properties. It converges almost surely to a random variable $p_\infty$ and the support of this random variable lies entirely in a measurable set $C$. This set is absorbing, in the sense that once $p_t\in C$, it is stopped and $p_{t+1}=p_t$.

Question: are there some general condition to ensure that the process converges almost surely in finite time?

Insights: I think this has to do to with the first hitting time in set $C$. Since the process converges almost surely to a random variable with support in $C$, either the first hitting time is infinite with positive change or its expected value must be finite. I could not find any general conditions to characterize when it will have finite expectation.

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  • $\begingroup$ What does “converge in finite time” mean? $\endgroup$ – Nap D. Lover May 4 at 14:31
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    $\begingroup$ Maybe that the expected value of the first hitting time is almost surely finite? I am trying to understand results from an article that does explicitly state some definitions. $\endgroup$ – Caio Lorecchio May 5 at 15:37

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