Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 224 / 1 / d / Solution

For any region $B_n$ with $P^{\otimes n}(B_n)\ge 1-\epsilon$, letThe weak law of large numbers gives $P^{\otimes n}(C_n)\to 1$, so $P^{\otimes n}(B_n\cap C_n)\ge 1-\epsilon -o(1)$. ThereforeThus $\mathrm{limsup}-n^{-1}\log {\beta }_n\le D(P\Vert Q)+\delta$; let $\delta \downarrow 0$.