(also asked on math.se, with no answers)

Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:

$$E\|\Sigma_m-\Sigma\| \le \epsilon \|\Sigma\|$$

Vershynin (High-Dimensional Probability Remark 5.6.3) gives the following sample requirement for arbitrary distribution in terms of intrinsic dimension $r=\text{tr}\ \Sigma/\|\Sigma\|$ $$m \approx \epsilon^{-2} r \log n$$

Is there a tighter bound for the Gaussian case? In particular, I'm wondering if a dimension-free bound is possible.