Extremal eigenvalues of sample covariance matrices with general population

We analyze the behavior of the largest eigenvalues of sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^*$. The sample $X$ is an $M\times N$ rectangular random matrix with real independent entries and the population covariance matrix $\Sigma$ is a positive definite diagonal matrix independent of $X$. In the limit $M, N \to \infty$ with $N/M\rightarrow d\in[1,\infty)$, we prove the relation between the largest eigenvalues of $\mathcal{Q}$ and $\Sigma$ that holds when $d$ is above a certain threshold. When the entries of $\Sigma$ are i.i.d., the limiting distribution of the largest eigenvalue of $\mathcal{Q}$ is given by a Weibull distribution.