Let $X$ be a $\sigma$-subGaussian random vector on $\mathbb R^n$, meaning that the random variable $X^Tv$ is $\sigma$-subGaussian for every unit vector $v \in \mathbb R^n$. Consider the $d$-by-$d$ psd matrix $\Sigma := \mathbb E[\tilde{X}\tilde{X}^T]$, where $\tilde{X} := X/\|X\|_2$. It is clear that every eigenvalue of $\Sigma$ lies in the interval $[0, 1]$. In fact, $\text{tr}\Sigma \le 1$.

Question 1. What is a good estimate for the largest eigenvalue of $\Sigma $ ?

Question 2. Same question without the sub-Gaussianity assumption.