Central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic Markov chain, say $S_n = g(X_1)+ \cdots + g(X_n)$ where $E[g(X_1)]= 0$ and $E[g(X_1)^2]<\infty$. The conditions imposed restrict the moments of $g$ and the growth of the conditional means $E(S_n|X_1)$. No other restrictions on the dependence structure of the chain are required. When specialized to shift processes,the conditions are implied by simple integral tests involving $g$.
"Central limit theorems for additive functionals of Markov chains." Ann. Probab. 28 (2) 713 - 724, April 2000. https://doi.org/10.1214/aop/1019160258