An error implies that some $\theta\ne\theta^*$ has likelihood at least that of $\theta^*$. A finite <union bound> and part (b) therefore give
$$
\limsup_{n\to\infty}\frac1n\log
\mathbb P(\widehat\theta_n\ne\theta^*)
\leq-C^*(\Theta),
$$
where
$$
C^*(\Theta)=
\min_{\theta\ne\theta^*}
\inf_{R:\,D(R\Vert P_{\theta^*})\geq D(R\Vert P_\theta)}
D_e(R\Vert P_{\theta^*}).
$$
Every inner infimum is strictly positive by the argument in part (b), and the minimum of finitely many positive numbers is positive.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-224/2/c/solution