# In finite-state Markov chain state $i$ is transient

Can you help me please with proof of this question:

Prove, that in finite-state Markov chain state $i$ is transient if and only if is exist state $k$ such that $i\rightarrow k$ but k $\nrightarrow i$.

Give counterexamples at case of infinite chain.

Thanks!

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Try showing that in a communicating class, all states are the same. That is, either they are all transient or recurrent. Thus, if there is a transient state $i$ which communicates with every single state, that means all the other states are transitive. But then there must be a "sink" somewhere, because not every single state can be transitive in a Markov chain! The other way of saying this is that at least one state must be recurrent in a finite Markov chain. The last sentence becomes false in the case of infinite chains (why!?) –  Alex R. Dec 22 '12 at 7:05
Try showing what you did, the current version of the question has 0% of personal input. –  Did Dec 28 '12 at 12:05