Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 210 / 3

Write $K=K_n$, $ϵ={ϵ}_n$, and chooseTake a $\delta$-net $({\theta }^1,\dots ,{\theta }^N)$ of $H_1$ in ${\ell }^2$, with centers in $H_1$. Since every alternative lies in the $K$-dimensional Euclidean ball of radius $n$, the volumetric covering bound givesfor all large $n$. Put $v_j={\theta }^j-{\theta }_0$ and define the Gaussian net testThe inner products are well-defined Gaussian linear functionals because $v_j\in {\ell }^2$. Under $H_0$,and $\Vert v_j{\Vert }_2\ge Mϵ$. A Gaussian tail bound and a union bound give

For any $\theta \in H_1$, choose $j$ with $\Vert \theta -{\theta }^j{\Vert }_2\le \delta$. ThenAnother Gaussian tail bound givesBecause $n{ϵ}_n^2=K_n\log n$, the entropy term in the type-I bound is dominated by the signal exponent when $M$ is sufficiently large. Also ${ϵ}_n^2\to 0$ implies $K_n<n$ eventually, so $\log n\ge \log K_n$. Given $c_1>0$, choose $M$ large enough to obtainEnlarging $M$ if necessary handles the finitely many initial $n$.