Markov chain - recurrence and transience

Does a Markov chain with infinitely many recurrent states and infinitely many transient states exist?

I believe it doesn't exists but I'm not sure how can I prove it.

Thanks guys! :D

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Irreducible?  – Did Mar 18 '13 at 12:07

Consider the discrete-time Markov chain with infinitely many states $i=1,2,\ldots$, such that $P(X_t=i\mid X_{t-1}=i)=1$ when $i$ is odd, and $P(X_t=i+2\mid X_{t-1}=i)=1$ when $i$ is even. Then every odd-indexed state is a recurrent state and every even-indexed state is a transient state.

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Thanks :) What was your thought process about coming up with this example? Any tips on how to approach such questions? – Manko Mar 18 '13 at 18:25
@Mankochka The simplest example that every state is recurrent is that every state remains unchanged. The simplest example that every state is transcient is a "right-shift", i.e. every state only transits to the next. Now I merely glue the two examples together by interlacing those states from the two chains according to the parities of indices. – user1551 Mar 18 '13 at 19:01