For a $p$-dimensional random vector $X \sim N(0,\Sigma)$, it is known that with high probability $$ \| X \|_{\max} \le C \sqrt{\log p} $$ where $C$ is some constant possibly depending on $\Sigma$. I would like to know if there is a simple dimension free bound that generalizes this result.
For context on a related problem, consider the $\| \cdot \|_2$ setting, then example Exercise 6.3.5. in Vershynin's High Dimensional Probability states that for a $m \times n$ constant matrix $B$, $Z \sim N(0,I)$, that $$ P( \| BZ\|_{2} \ge C \| B \|_F + t) \le \exp \left ( -\frac{ct^2}{\| B \|^2_\text{op}}\right) $$ where the subscript $F$ is the frobenius norm. Choosing $B = \sqrt{\Sigma}$ yields $$ P( \| X \|_{2} \ge C \sqrt{\text{Tr}(\Sigma)} + t) \le \exp \left ( -\frac{ct^2}{\|\Sigma\|^2_{\text{op}}}\right), $$ so that with high probability $\| X \|_2 \le C\sqrt{\text{Tr}(\Sigma)}$. In the worst case, $\Sigma = I$ and we get back the standard bound $\| X \|_2 \le C\sqrt{p}$, but for many problems of interest $\Sigma$ has a nice structure that makes the first bound more useful.
Is there a similar bound that generalizes the $\sqrt{\log p}$ bound? Ideally it would involve the trace, $\|X\|_{\max} \stackrel{?}{\le} C \sqrt{\log \text{Tr}(\Sigma)}$.
Note also that all results mentioned hold more generally for subGaussian random variables (with different universal constants), but for simplicity I am just looking at the Gaussian case.
An update: I have found a related dimension free result, Lemma 2.3 of this paper by Van Handel states that for $X_1,\dots, X_n$ be not necessarily independent subGaussian random variables with subGaussian norm $\sigma_i$, then $$ E [\max_{i \le n} X_i] \le C \max_{i \le n} \sigma_{(i)} \sqrt{\log(i+1)} $$ where $\sigma_{(1)} \ge \sigma_{(2)} \ge \dots \sigma_{(n)}$ are the ordered subGaussian norms. If the $X$'s are independent Gaussian there is a matching lower bound.