Universality for the largest eigenvalue of sample covariance matrices with general population

In this paper, we will derive the universality of the largest eigenvalue of a class of large dimensional real or complex sample covariance matrices in the form of $\mathcal{W}_N=\Sigma^{1/2} XX^*\Sigma^{1/2}$. Here $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij},1\leq i\leq M, 1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^2=1/N$. We say $\mathcal{W}_N$ is a classical complex sample covariance matrix if there also exists $\mathbb{E}x_{ij}^2=0,1\leq i\leq M, 1\leq j\leq N$. Moreover, on dimensions we assume that $M=M(N)$ and $N/M\rightarrow d\in (0,\infty)$ as $N\rightarrow \infty$. For a class of highly general deterministic positive definite $M\times M$ matrices $\Sigma$, we show that the limiting behavior of the largest eigenvalue of $\mathcal{W}_N$ is universal under some additional assumptions on the distribution of $(x_{ij})^\prime$s via pursuing a Green function comparison strategy raised in \cite{EYY2012, EYY20122} by Erd\"{o}s, Yau and Yin for Wigner matrices and extended by Pillai and Yin \cite{PY2012} to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the classical complex case, combing this universality property and the results known for Gaussian matrices derived by El Karoui in \cite{Karoui2007} (nonsingular case) and Onatski in \cite{Onatski2008} (singular case) we show that after appropriate normalization the largest eigenvalue of $\mathcal{W}_N$ converges weakly to the type 2 Tracy-Widom distribution $\mathrm{TW_2}$. Moreover, in the real case, we show that when $\Sigma$ is spiked with fixed number of sub-critical spikes, the type 1 Tracy-Widom distribution $\mathrm{TW}_1$ holds for the largest eigenvalue of $\mathcal{W}_N$, which extends a result of F\'{e}ral and P\'{e}ch\'{e} in \cite{FP2009} to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$.