# Irreducible, finite Markov chains are positive recurrent

I am under the impression that an irreducible, finite Markov chain is necessarily positive recurrent. How might I show this?

Regards, Jon

Here is one way to look at it. If $x$ is a null state, then the chain spends very little time in $x$, more precisely, $${1\over n}\sum_{j=1}^n 1_{[X_j=x]}\to 0 \text{ almost surely.}$$ Therefore, for any finite set $F$ of null states we also have $${1\over n}\sum_{j=1}^n 1_{[X_j\in F]}\to 0 \text{ almost surely.}$$
• To turn the definition of null recurrent into the statement above that the fraction of visits to $x$ tends to zero almost surely, one needs to use the ergodic theorem for Markov chains. – gj255 Sep 9 '17 at 9:48