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I am wondering if there is a general formula for doing this.

Consider my favorite simple example: Consider stochastic matrix $P$ with entities $p_{11}=1/3$, $p_{22}=5/6$, $p_{12}=2/3$, $p_{21}=1/6$.

Is there a way to find out what are the recurrent and transient states without having to find a general formula for $P^n$?

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    Either from first principles, showing the hitting time of i starting from 3-i is almost surely finite by computing its distribution. Or from the elementary theory of Markov chains, which ensures that every irreducible finite Markov chain is recurrent. So, it all depends on what you know and what you don't--a subject on which you are totally silent.2012-01-26

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