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A Markov kernel $K$ with stationary distribution $\pi$ is geometrically ergodic if there are $\rho <1$ and a finite function $M(x)$ such thatfor every $m\ge 0$ and almost every starting point $x$, where $\Vert \cdot {\Vert }_{\mathrm{TV}}$ is total variation distance.

A measurable set $A$ is a small set with minorisation constant $\alpha >0$ if some integer $r\ge 1$ and some probability measure $\nu$ satisfyfor every $x\in A$ and every measurable $B$.

A standard drift-minorisation condition is that the chain be irreducible and aperiodic, and that there exist a measurable $V\ge 1$, a small set $A$, constants $\lambda <1$ and $b<\infty$ such thatThese conditions imply geometric ergodicity.

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.