Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 2 / c

Put $s=\Vert \hat{\beta }{\Vert }_0+\Vert {\beta }^0{\Vert }_0$. Since $\delta$ has at most $s$ nonzero coordinates, $\Vert \delta {\Vert }_1^2\le s\Vert \delta {\Vert }_2^2$. On ${\Omega }_2$,where $\hat{\Sigma }=X^{T}X/n$. Write $R={\delta }^T\hat{\Sigma }\delta$. On ${\Omega }_1$,Squaring after division by $\sqrt{R}$ proves