 Solution

The likelihood-ratio event depends only on the type $R={\hat{P}}_n$ and isequivalently $D(R\Vert Q)\ge D(R\Vert P)$. This constraint defines a closed subset of the finite probability simplex, so Sanov's upper bound gives the exponentIt is strictly positive: the only distribution with zero divergence from $Q$ is $R=Q$, but $Q$ violates the constraint because $0=D(Q\Vert Q)<D(Q\Vert P)$. Compactness and continuity under full support keep the infimum away from zero. This exponent is the Chernoff information between $P$ and $Q$.