In the topic of Principal component analysis in high dimensions I have given the following task:
Let $X\in\mathbb R^{n\times d}$ and $w\sim\mathcal N(0,\sigma^2I_n)$.
Show that for any $\lambda>0$ we have: $$\mathbb P\left(\bigg\|\frac{X^T w}n\bigg\|_\infty\geq\lambda\right)\leq2e^{-\frac{n\lambda^2}{2\sigma^2}+\log(d)}$$
can someone give me a hint?