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For a homogeneous discrete time Markov chain with transition matrix $p$, a state $i$ is recurrent, if and only there exists $n \geq 1$ s.t. $p_{ii}^{(n)} =1$?

I have it copied from somewhere in my notes, but I forgot about the source. I didn't write the chain is finite-state, so can I assume the above statement, if it is true, can apply to countable-state case?

Thanks and regards!

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up vote 1 down vote accepted

Not at all. Counterexample: simple random walk on any connected finite graph (with at least 3 vertices, say).

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Thanks, Did! Is it possible that there is a correct statement that looks similar to the wrong statement in my post? – Tim Apr 14 '13 at 15:18
No. $ $ $ $ $ $ – Did Apr 14 '13 at 15:19

As a matter of definition, state $i$ is recurrent for the Markov chain $(X_n)_{n\ge 0}$ if and only if $$ P(X_n = i \hbox{ for some }n\ge 1 \,|\, X_0=i) =1. $$ It is well known that $i$ is recurrent if and only if $\sum_{n\ge 1} p^{(n)}_{ii} =\infty$.

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