Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 2 / a / Solution

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.