Assume I have i.i.d samples $X_i\sim\mathcal{N}(0,\Sigma)$, $1\leq i\leq n$, with an arbitrary positive definite covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$.
What does the spectral distribution of the sample covariance $\frac{1}{n}\sum_{i}X_iX_i^T$ look like?
The literature on the topic seems to be too progressed for me to find an answer to this sample question. I've found papers that allow way too general dependence structures but don't even talk about simple cases like mine, and I've found lecture notes or short-proof-of papers that discuss the $\Sigma=I$ (Marchenko--Pastur) case at lengths, but not the in between case I'm in interested in. The original theorem would be the special case where the spectral distribution of $\Sigma$ is a Dirac measure at each dimension.
Reminder: the empirical spectral distribution of a matrix is the sum of dirac measures in the eigenvalues, i.e. it is a discrete measure on the positive half line. If $\Sigma=I_d$ the spectral distribution of the sample covariance matrix (a random measure) converges almost surely to a continuous Marchenko--Pastur distribution with one free parameter, $\lambda=d/n$, as $d,n\to\infty$. I'm aware that in my case, I'll have to specify what it means for $d$ to change, if I talk about $\Sigma$ fixed. I'd be happy for an answer that assumes the spectral distribution of $\Sigma$ converge to a fixed spectral distribution as the dimension grows.
EDIT: To avoid the X-Y problem: what I really want is to be able to decide, by looking at a sample covariance matrix with $n\ll d$, whether the full covariance $\Sigma$ has a dominant eigenvalue or not, i.e., whether $\lambda_d/\sum_{0<i<d}\lambda_i$ is large.