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A phi-irreducible Markov chain is Harris recurrent when every set of positive irreducibility measure is hit almost surely from every state. With an invariant probability, it becomes a positive Harris recurrent Markov chain.

For the Metropolis–Hastings algorithm, start at a state $\theta$ with positive target density, propose ${\theta }^{\prime }$ from $q({\theta }^{\prime }\mid \theta )$, and generate an independent uniform draw. Accept the proposal with probabilityOn rejection retain $\theta$, including repeated states in the chain; removing repeats would generally change its stationary law. The accepted-flow density satisfieswhich is symmetric in its two states. This proves detailed balance and hence invariance of the target distribution, including the holding probabilities on the diagonal. Unknown common normalization constants cancel in the ratio. Invariance alone does not establish convergence from arbitrary initial states. Choosing proposals that yield a positive Harris recurrent Markov chain with aperiodicity supplies the usual convergence guarantee, and the resulting sample is dependent. Use burn-in and account for autocorrelation when assessing Monte Carlo precision.

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.