Can a birth death chain having a finite space contain any recurrent state?

Question

I am studying Birth-Death Chains from Introduction to Stochastic Processes by Paul G. Hoel. From the definition of birth death chain it is immediate that it does not contain any absorbing state. We know that absorbing states are recurrent but not conversely. So there may exist recurrent states in a birth death chain inspite of the non existence of absorbing states.

So my question is "Can a birth death chain contain a recurrent state when we consider the underlying state space to be finite?" What happens if the state space is infinite?

Please help me in this regard.

Thank you very much.

Edit $:$ If one state is recurrent then so are all the other states since birth death chains are irreducible.

Answer 1

If the state space is finite then no state can be transient. If it is infinite there are plenty of examples where all states are transient. Ref.: books of Karlin and Taylor, KL Chung etc.