A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal{Q}=TX(TX{)}^{\ast }$, where the sample $X$ is an $M_2\times N$ random matrix with $i\mathrm{.}i\mathrm{.}d\mathrm{.}$ entries with mean zero and variance $N^{-1}$, and $T$ is an $M_1\times M_2$ deterministic matrix satisfying $T^{\ast }T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal{Q}$ when $M:=\min \{M_1,M_2\}$ and $N$ tends to infinity with ${\lim }_{N\to \mathrm{\infty }}N\mathrm{/}M=d\in (0,\mathrm{\infty })$. Under mild assumptions of $T$, we prove that the Tracy-Widom law holds for the largest eigenvalue of $\mathcal{Q}$ if and only if $\lim_{s \rightarrow \infty}s^4 $\mathbb{P}$(| \sqrt{N} x_{ij}| \geq s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.