Is this sub-Gaussian random variable?

Question

Say $X_1, X_2$ are i.i.d. sub-Gaussian variables following the definition in High-Dimensional Probability. Denote $X = (X_1, X_2)^T$, and $\eta$ is some positive constant. Is the following random variable still sub-Gaussian? How to prove it rigorously? $$\sup_{t\in\mathbb{R}^2, ||t||_2\leq \eta} X^Tt$$ I guess the proof may need to use techniques in empirical processes?

Answer 1

You don't have to use any empirical process techniques here. Note that

$$ \sup_{t \in \mathbb{R}^2, \| t \|_2 \leq \eta} X^T t = \frac{1}{\eta} \sup_{t \in \mathbb{R}^2, \| t \|_2 \leq 1} X^t t = \frac{1}{\eta} \| (X_1, X_2) \|_2, $$

which is the Euclidean norm of a random vector with subgaussian coordinates.

Chapter 3 in the same book covers random vectors with subgaussian coordinates.