# 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

-
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.