Question about irreducibility ,recurrence of a Finite M. Chain

Jan 2010
150
29
Mexico City
hi!

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!
 
Nov 2009
517
130
Big Red, NY
Theorem. If C is a finite closed and irreducible set, then all states in C are recurrent.

-Essentials of Stochastic Processes, Rick Durrett.
 
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Jan 2010
150
29
Mexico City
thank you, but what about positive recurrence?