I just need to set the next implications straight since literature can be very confusing when talking about finite state space Markov chains.

If \(\displaystyle X_n\) is an irreducible chain with finite state space then

is \(\displaystyle X_n\) recurrent?

even more, is \(\displaystyle X_n\) positive recurrent?

I have found out the answer to both questions is positive but i would like to confirm it, is anybody sure about it? Can you think of any text where it explicitly shows the above results?

Thank you!