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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.

Let $w(\theta )=\pi (\theta )/\mu (\theta )$ be the importance weight. The two normalized densities giveSince $|F-Z|<C$, there is a finite constant $M$ such that $w(\theta )\le M$ everywhere. The Independence Metropolis–Hastings algorithm has an accepted transition density satisfyingConsequently the whole state space is a small set, with the one-step minorization condition $K(x,\cdot )\ge M^{-1}\pi (\cdot )$. Iterating this Doeblin condition gives uniform geometric convergence in total variation distance, so the chain is geometrically ergodic.