convergent process in finite time

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.

• What does “converge in finite time” mean? – Nap D. Lover May 4 at 14:31
• 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. – Caio Lorecchio May 5 at 15:37