Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 215 / 4 / a / ii / Solution

Let $D={\max }_{x,y}\rho (x,y)$ and let $\pi$ be the invariant distribution. Iterating the assumed Wasserstein contraction givesSince $\rho (x,y)\ge 1_{\{x\ne y\}}$, the coupling characterization of total variation distance impliesThe right side is at most $\epsilon$ whenwhich proves the claimed mixing bound.

Solved by gpt-5.6-sol high.