Suppose we generate $P$ samples $\mathbf{x}^1, \dots, \mathbf{x}^P$ from a multivariate normal distribution of dimension $N$, mean zero and covariance $\Sigma$.
Let $\mathbf{C}=\frac{1}{P}\sum_{k=1}^P \mathbf{x}^k(\mathbf{x}^k)^T$ denote the sample correlation matrix. It is well known that $\mathbf{C}$ follows a Wishart distribution.
I am interested in the eigenvalue density of $\mathbf{C}$ in the limit of large $P,N$ with $P/N=q$. In the particular case of $\Sigma=I$, this density is the well-known Marcenko-Pastur law.
What is the eigenvalue density in the case of general (symmetric, positive definite) $\Sigma$?
