(also [asked on math.se][2], 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][1] 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 $\log n$ term can be dropped


  [1]: https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf
  [2]: https://math.stackexchange.com/questions/4169322/estimate-nearly-singular-gaussian-covariance-matrix