Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 205 / 2 / c

Because $\Vert X_j{\Vert }_2=\sqrt{n}$, each coordinate of $X^T\epsilon /n$ is $N(0,{\sigma }^2/n)$. The Gaussian tail bound and a union bound giveOn the complementary score event, the standard Basic inequality for the Lasso, cone argument, and compatibility oracle inequality give, for $S=\mathrm{supp}({\beta }^0)$,Using part b and ${\lambda }^2=A^2{\sigma }^2\log p/n$ proveswith the required probability.