Let $\mathbf{a}_k\in\mathbb{C}^n$ for $k=1,2,\ldots,m$ be i.i.d. standard complex normal random vectors with distribution $c\mathcal{N}(0,\mathbf{I})$. I am interested in a tight upper bound on the following quantities with high probabilities (say with probability at least $1-\frac{1}{n}$ or something similar): \begin{align} \underset{\mathbf{x}\in \mathbb{C}^ns.t.\|\mathbf{x}\|_{\ell_2}=1}{\text{max}}\frac{1}{m}\sum_{k=1}^m |\mathbf{a}_k^*\mathbf{x}|^4\le ? \end{align} \begin{align} \underset{\mathbf{x},\mathbf{y}\in \mathbb{C}^ns.t.\|\mathbf{x}\|_{\ell_2}=\|\mathbf{y}\|_{\ell_2}=1} {\text{max}}\frac{1}{m}\sum_{k=1}^m |\mathbf{a}_k^*\mathbf{x}|^2Re(\mathbf{y}^*\mathbf{a}_k\mathbf{a}_k^*\mathbf{x})\le? \end{align}
Of course I know how to do this for fixed quantities of $\mathbf{x}$ and $\mathbf{y}$. However, the corresponding probabilities do not allow for a covering argument. I am hoping that the upper bound for all $\mathbf{x},\mathbf{y}$ is comparable to the result for fixed $\mathbf{x},\mathbf{y}$ up to constant/log factors. If anybody knows of a counter argument which shows that this is not the case that would also be very helpful. Assume that $m \ge c n$ for a sufficiently large numerical constant $c$. I would also be ok with an argument which assumes $m \ge c n (\log n)^\alpha$ for some small $\alpha$ like $\alpha=1,2,3,4$ (the smaller of course the better).
