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 $\Sigma := \mathbb E[UU^T]$, where $U := X/\|X\|_2$. 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.


My Current approach
---
Let $\lambda$ be an eigenvalue of $\Sigma$ and $z$ be a unit vector in the corresponding eigenspace.

For $\delta \in [0,1]$, let $G_\delta := \{x \in \mathbb R^n \mid |x^Tz| > \delta\}$

$$
\begin{split}
\lambda &= \lambda\|z\|^2 = z^T(\lambda z) = z^T\Sigma z = z^T E[UU^T]z = E[z^TUU^Tz] = E|U^Tz|^2\\
&= E[|U^Tz|^2 \mid U \in G_\delta]P(U \in G_\delta) + E[|U^Tz|^2 \mid U \in G^c_\delta)P(U \in G^c_\delta)\\
&\le P(U \in G_\delta)  + \delta^2P(U \in G_\delta^c) = (1 - \delta^2)P(U \in G_\delta) + \delta^2.
\end{split}
$$

That is,

$$
\lambda \le (1-\delta^2)P(|U^Tz| > \delta) + \delta^2,\; \forall \delta \in [0, 1].
\tag{1}
$$

>Thus, if I had a bound on $P(|U^Tz| > \delta)$, I could plug it in (1) and then minimize over $\delta \in [0, 1]$ to get (a perhaps good) upper bound on $\lambda$.