Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population

We consider sample covariance matrices of the form $\mathcal{Q}=({\mathrm{\Sigma }}^{1\mathrm{/}2}X)({\mathrm{\Sigma }}^{1\mathrm{/}2}X{)}^{\ast }$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1\mathrm{/}N$ and where $\mathrm{\Sigma }$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N\mathrm{/}M\to d\in (0,\mathrm{\infty })$. For a large class of populations $\mathrm{\Sigma }$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of $X$ are i.i.d. Gaussians or (2) that $\mathrm{\Sigma }$ is diagonal and that the entries of $X$ have a subexponential decay.