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.