Uniform geometric ergodicity
= Uniform geometric ergodicity
Uniform geometric ergodicity strengthens <geometric ergodicity> by bounding the prefactor uniformly over the starting state. If $P(x,\cdot)\geq\varepsilon\mu(\cdot)$ globally and $\mu$ is invariant, then
$$
\sup_x\|P^n(x,\cdot)-\mu\|_{\mathrm{TV}}\leq(1-\varepsilon)^n.
$$
Write $P=\varepsilon\Pi+(1-\varepsilon)R$, where $\Pi$ draws independently from $\mu$, to obtain the bound.