I have an irreducable discrete time Markov chain (DTMC) $(X_n)_{n\geq0}$ with finite state space $\mathcal{X}$. The DTMC has a stationary distribution $\pi$, such that $\pi = \pi P$, where $P$ is the transition probability matrix of the Markov chain. I need to prove that the Markov chain is positive recurrent.
I have that, if the Markov chain is a closed, finite, communicating class then it is positive recurrent. How does having an invariant distribution imply positive recurrence (or that the Markov chain is a finite, closed communicating class)? Thanks.