Solution
= Solution
Each $Z_j=n^{-1}\sum_i\varepsilon_iX_{ij}$ is $1/\sqrt n$-sub-Gaussian. Applying the exponential-moment bound to the $2p$ variables $\pm Z_j$ and optimizing the parameter gives $E\max_j|Z_j|\le\sqrt{2\log(2p)/n}$.
= Solution
Each $Z_j=n^{-1}\sum_i\varepsilon_iX_{ij}$ is $1/\sqrt n$-sub-Gaussian. Applying the exponential-moment bound to the $2p$ variables $\pm Z_j$ and optimizing the parameter gives $E\max_j|Z_j|\le\sqrt{2\log(2p)/n}$.