# Positive and Null recurrence of Markov Chains on a General State Space

Suppose $X_n$ is an irreducible, aperiodic and Harris recurrent Markov chain. It is well known that in this case, $X_n$ has a stationary distribution $\pi$.

Are there any conditions that are sufficient to guarantee that $\pi$ is finite? In other words, what criteria are there to show that the chain is positive recurrent and not null-recurrent?

Let's call your state space $\mathcal{X}$. Then, you want to be able to find a function $V \colon \mathcal{X} \to \mathbf{R}_+$ and a small set $C$ such that $$PV \le V - 1 + K \mathbf{1}_C\;,$$ where $P$ is your Markov operator, $K$ is some constant, and $\mathbf{1}_C$ denotes the indicator function of $C$. Since $V$ bounds the expectation of the return time to $C$, this is what you want. This condition is actually both sufficient and necessary since the expected return time to $C$ does satisfy that bound. In some cases, such a Lyapunov function $V$ can easily be guessed, in other cases it is more of an artform to engineer a good one. (See https://arxiv.org/abs/0810.5431 for a class of very interesting examples in continuous time.)