Let $x\_1,...,x\_n\in\mathbb{R}^p$ be i.i.d. random vectors with mean 0 and covariance $\Sigma\_p$. Let $S\_{n,p}=\sum\_{i=1}^nx\_ix\_i^T/n$ be the sample covariance. We consider the asymptotics of the empirical distribution of eigenvalues of $S\_{n,p}$. We let $p/n\to c\in(0,1)$ where $c$ is a constant. 


It is well-known that when $\Sigma\_p$ is identity, the limiting distribution of the spectrum of $S\_{n,p}$ follows the [Marchenko-Pastur distribution](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution)


What about for general $\Sigma\_p$? What kind of asymptotic regime do we need on $\Sigma\_p$ in order to have a meaningful limiting distribution of the spectrum of $S\_{n,p}$? 


For example, we can assume the empirical distribution of eigenvalues of $\Sigma\_p$ converges to a deterministic measure $\mu$. Is this condition enough? I suppose this is not enough and the eigenvectors of $\Sigma\_p$ also seems matter. If this is indeed enough, what is the relationship between $\mu$ and the limiting spectrum distribution of $S\_{n,p}$?



Marchenko-Pastur Law under general covariance structure
In the Gaussian case, you can rewrite $x\_i=R^{1/2}y\_i$ where $y\_i$ now possess iid entries.
This leads you to computing the eigenvalues of $Y^\*RY$, this is actually the problem solved by Pastur and Marchenko, see Math. USSR. Sbornik vol 1 (1967), with an explicit equation satisfied by the Stieltjes transform of $\mu$. You can find also a discussion in the book of Bai and Silverstein, Chapter 4. All that matters is the asymptotic limit of empirical measure of eigenvalues of $R$. 


For a large class of non-Gaussian entries, you can apply <https://arxiv.org/pdf/1312.0037.pdf> to go back to the Gaussian case.


