Let $X = \{X_n\}_{n \ge 0}$ be a Markov chain with transition kernel $P$ on a state space $\mathsf{X}$, Harris ergodic with invariant probability distribution $\pi$, and let $f : \mathsf{X} \to \mathbb{R}$ be measurable. Write $\bar f_n = n^{-1} \sum_{i=1}^{n} f(X_i)$ for the sample average and $E_\pi f = \int f \, d\pi$. Assume one of the following six conditions:

1. the chain is polynomially ergodic of order $m > 1$ with $E_\pi M < \infty$ for the rate constant $M$, and $|f| < B$ $\pi$-almost surely for some $B$;
2. the chain is polynomially ergodic of order $m$ with $E_\pi M < \infty$, and $E_\pi |f|^{2+\delta} < \infty$ for some $\delta > 0$ with $m\delta > 2 + \delta$;
3. the chain is geometrically ergodic and $E_\pi |f|^{2+\delta} < \infty$ for some $\delta > 0$;
4. the chain is geometrically ergodic and $E_\pi [f^2 \log^+ |f|] < \infty$;
5. the chain is geometrically ergodic, reversible with respect to $\pi$ (detailed balance), and $E_\pi f^2 < \infty$;
6. the chain is uniformly ergodic and $E_\pi f^2 < \infty$.

Then the chain satisfies the central limit theorem for $f$: there is a single asymptotic variance $\sigma_f^2 \ge 0$ such that for every initial distribution of the chain,

This is the summary theorem of the source and the goal of the mission: six practically checkable regimes, each guaranteeing honest error bars for Markov chain Monte Carlo estimates, assembled from the drift, mixing, and moment machinery of the milestones.

Formalization Note "Harris ergodic" is encoded by its total-variation characterization: $\pi$ is invariant for $P$ and $\|P^n(x, \cdot) - \pi\| \to 0$ for every starting point $x$ (equivalent to the classical aperiodic, $\psi$-irreducible, positive Harris recurrent definition; the "every $x$" quantifier is exactly the Harris property). Convergence in distribution is weak convergence of laws, and $N(0, 0)$ is read as the point mass at $0$, which absorbs the source's "$\sigma_f^2 > 0$" caveat.