Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 205 / 6 / b / Solution

Put $\hat{\Sigma }=X^{T}X/n$. Direct substitution of $Y=X{\beta }^0+\epsilon$ into the Debiased Lasso givesConditionally on the deterministic design,The assumed approximate inverse of a Gram matrix property and Holder inequality implyThus $\rho (n,p)=\sqrt{\log p}$.