Central limit theorem for a geometrically ergodic Markov chain ($\sqrt{n}({\overline{h}}_n-\pi h)\Rightarrow N(0,v_h)$)

For an aperiodic positive Harris recurrent Markov chain with geometric ergodicity, an observable with stationary moment $\pi (|h{|}^{2+\delta })<\infty$ for some $\delta >0$ satisfies a central limit theorem. Its asymptotic variance is ${\gamma }_h(0)+2{\sum }_{k\ge 1}{\gamma }_h(k)$, with stationary covariances. Ergodicity alone does not guarantee this theorem.