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A convenient general-state-space ergodic theorem is the following. For a positive Harris recurrent Markov chain with invariant probability measure $\pi$, and a measurable function $h$ with $\pi |h|<\infty$,Here $\pi h=\int hd\pi$ is the stationary expected value. Harris recurrence means that every set of positive irreducibility measure is visited almost surely from every state; positive recurrence supplies an invariant probability rather than only an infinite invariant measure. The ergodic theorem for a positive Harris recurrent Markov chain holds from any starting state under these Harris hypotheses. aperiodicity is not necessary just for averages.

A useful sufficient form of the central limit theorem for a geometrically ergodic Markov chain adds aperiodicity, geometric ergodicity, and $\pi (|h{|}^{2+\delta })<\infty$ for some $\delta >0$. It givesThese are sufficient hypotheses, not a claim that irreducibility alone ensures a central limit theorem. The Markov chain Monte Carlo asymptotic variance uses stationary covariances ${\gamma }_h(k)={\mathrm{Cov}}_{\pi }(h(X_0),h(X_k))$, with ${\gamma }_h(0)={\mathrm{Var}}_{\pi }h$. The series is absolutely convergent under the stated sufficient assumptions. If ${\gamma }_h(0)>0$, write ${\rho }_h(k)={\gamma }_h(k)/{\gamma }_h(0)$ and $v_h={\gamma }_h(0){\tau }_{\mathrm{int}}$, where the integrated autocorrelation time is ${\tau }_{\mathrm{int}}=1+2{\sum }_{k\ge 1}{\rho }_h(k)$. When $v_h>0$, the effective sample size of a Markov chain is approximately $n/{\tau }_{\mathrm{int}}$. Negative correlations can reduce the asymptotic variance; a zero asymptotic variance gives a degenerate normal limit. For constant $h$, variance is zero and the autocorrelation normalization is undefined.

First require a proper target: $0<Z={\int }_{{\mathbb{R}}^d}\pi (x)dx<\infty$ and $\bar{\pi }=\pi /Z$. The Metropolis–Hastings algorithm uses the Metropolis–Hastings acceptance probabilitySufficient general conditions are phi-irreducibility, aperiodicity, and a drift-minorisation condition establishing geometric ergodicity of the resulting chain, together with a stationary $(2+\delta )$th moment for the observable being averaged. These imply the central limit theorem for a geometrically ergodic Markov chain. The observable's moment condition must be included: conditions on the sampler alone cannot give the theorem for every arbitrary function.

A concrete stronger condition, directly in terms of the proposal, isTogether with a bounded observable, this is an especially simple sufficient answer. The accepted proposal density is $\min \{q(y\mid x),\bar{\pi }(y)q(x\mid y)/\bar{\pi }(x)\}$, so it is at least $ϵ\bar{\pi }(y)$. The kernel satisfies a global minorization condition with the target, hence uniform geometric ergodicity, positive Harris recurrence, and aperiodicity. An independent proposal whose importance weight $\bar{\pi }(y)/q(y)$ is uniformly bounded is one example. No claim of these properties follows merely from writing down a positive proposal.