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

Fix $\delta >0$ and chooseUnder $P$, the weak law of large numbers makes the normalized log likelihood ratio converge in probability to $D(P\Vert Q)$, so $P^{\otimes n}(B_n({\tau }_n))\to 1$. On this Neyman-Pearson decision region,and henceLetting $\delta \downarrow 0$ proves the direct bound.