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Given a finite-state Markov chain $M$ and assume there is a terminate state $f$ which is reached with prob. 1, I am interested in the distribution of hitting time $T$ of $f$, namely, $T$ is defined as a RV which is the number of steps before reaching $f$, and $P(T\geq N)$ for natural number $N$ is the probability of reaching $f$ by at least $N$ steps. In particular, I want to obtain a nontrivial bound of $N$ such that $P(T\geq N)\leq \epsilon$, in terms of the transition probabilities of $M$ and $\epsilon$.

Is there any existing research on this problem, as it seems to be a very natural problem to consider.

Thanks.

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You can take a look at the book: Markov Chains and mixing times by Y.Peres, D.evin and E.Wilmer (you can get this from the webpage of Yuval Peres). See chapter 10 for example.

Different techniques are useful for different models, It would be helpful to know the model.

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