Geometric and uniform ergodicity for MCMC CLTs

Let $P$ be a probability kernel with invariant probability $\pi$ on a measurable space $(\mathsf{X},\mathcal{X})$. Write $P^n(x,\cdot )$ for its $n$-step transition probability and use $\mathrm{\parallel }\mu -\nu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}={\sup }_{A\in \mathcal{X}}\mathrm{\mid }\mu (A)-\nu (A)\mathrm{\mid }$ for probability measures.

The chain is geometrically ergodic if there are a measurable finite function $M:\mathsf{X}\to [0,\mathrm{\infty })$ and $r\in (0,1)$ such that $\mathrm{\parallel }{P}^n(x,\cdot )-\pi {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le M(x)r^n(x\in \mathsf{X},n\ge 1)\mathrm{.}$

It is uniformly ergodic if this bound holds with $M$ bounded on $\mathsf{X}$. These are convergence-rate properties, not conclusions about a particular function's CLT or long-run variance. If a proof needs $\int Md\pi <\mathrm{\infty }$, that integrability is an additional stated hypothesis unless a cited theorem establishes it.