Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 224 / 2 / b / Solution

The test accepts $P$ when $D({\hat{P}}_n\Vert P)\le \delta$. Under $P$, the method of types givessoUnder $Q$, accepting $P$ requires a type in the closed set $\{R:D(R\Vert P)\le \delta \}$. ThereforewhereConsequently ${\mathrm{limsup}}_n{n}^{-1}{\log }_2{e}_i^{(n)}\le -D_i(\delta )$ for $i=1,2$. The first exponent is positive when $\delta >0$. Because relative entropy vanishes only when its arguments agree, the second is positive exactly while $Q$ lies outside the constraint set. Thus both are strictly positive for