Geometric ergodicity

A Markov kernel is geometrically ergodic when its iterates converge to stationarity in total variation at a geometric rate, with a finite state-dependent prefactor.

A minorization condition gives a common component of transition laws: $K^m(x,\cdot )\ge \alpha \nu (\cdot )$ for every $x$ in a specified set, some $m\ge 1$, $\alpha >0$, and a probability measure $\nu$.

A Doeblin condition is a minorization of the whole state space. It implies uniform geometric convergence in total variation distance.

A set $A$ is $\alpha$-small when some iterate satisfies $K^m(x,\cdot )\ge \alpha \nu (\cdot )$ for every $x\in A$ and some probability measure $\nu$.

A geometric drift $KV\le \lambda V+b{1}_A$ with $\lambda <1$, together with a small-set minorisation on $A$ and irreducibility and aperiodicity, implies geometric ergodicity.