$\newcommand{\E}{\mathbb{E}}$$\newcommand{\P}{\mathbb{P}}$The Central Limit Theorem (CLT) states that for $X_1,X_2,\dots$ independent and identically distributed (iid) with $\E[X_i]=0$ and $\operatorname{ Var} (X_i)<\infty$, the sum converges to a normal distribution as $n\to\infty$: $$ \sum_{i=1}^n X_i \to N\left(0, \sqrt{n}\right). $$
Assume instead that $X_1,X_2,\dots$ form a finite-state Markov chain with a stationary distribution $\P_\infty$ with expectation 0 and bounded variance. Is there a simple extension of CLT for this case?
The papers I've found on CLT for Markov Chains generally treat much more general cases. I would be very grateful for a pointer to the relevant general result and an explanation of how it applies.