Given i.i.d. observations from an unknown $d$-dimensional Gaussian distribution $\mathcal{N}(\mu,\Sigma)$, it is "well-known" that $n=\Omega(d^2/\varepsilon^2)$ are necessary to produce an estimate $\widehat{\Sigma}$ of $\Sigma$ close in relative Frobenius norm: $$ \lVert \Sigma^{-1/2}\widehat{\Sigma}\Sigma^{1/2}-I_d\rVert_F \leq \varepsilon $$ with probability at least, say $9/10$. I have seen this claim (or the equivalent assertion that "learning a high-dimensional Gaussian to total variation distance $\varepsilon$ requires $n=\Omega(d^2/\varepsilon^2)$") stated as "folklore" since at least 2012 in various papers.
However, I cannot find a reference. Rather, the only references I can find for this convergence rate lower bound (again, in Frobenius norm, not operator norm) are papers from 2017 and 2018:
- https://arxiv.org/abs/1710.05209 (Theorem 6.3)
- https://arxiv.org/abs/1806.06887 (Corollary 1.2)
where they are claimed as new.
Is there a published reference, from before 2012, which provides the lower bound on the rate of covariance estimation, for $d$-dimensional Gaussian distributions, under the Frobenius norm?
