= Solution
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 probability>
$$
a(x,y)=\min\!\left\{1,\frac{\pi(y)q(x\mid y)}{\pi(x)q(y\mid x)}\right\}.
$$
Sufficient 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, is
$$
\boxed{q(y\mid x)\geq\epsilon\bar\pi(y)\quad\text{for all }x,y\text{ in the target support},\qquad\epsilon>0.}
$$
Together 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 $\epsilon\bar\pi(y)$. 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.
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