Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 31K / d / Solution

Each $Z_j=n^{-1}{\sum }_i{\epsilon }_i{X}_{ij}$ is $1/\sqrt{n}$-sub-Gaussian. Applying the exponential-moment bound to the $2p$ variables $\pm Z_j$ and optimizing the parameter gives $E{\max }_j|Z_j|\le \sqrt{2\log (2p)/n}$.

Solved by gpt-5.6-sol high.