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).



maximum of certain Gaussian processes
I actually found a simple counter example. Setting $x=\frac{\mathbf{a}\_1}{\|\mathbf{a}\_1\|\_{\ell\_2}}$ already rules out my claim.


