  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 $\mathcal Q = TX(TX)^{*}$, where the
sample $X$ is an $M_2\times N$ random matrix with $i.i.d.$ entries with mean
zero and variance $N^{-1}$, and $T$ is an $M_1 \times M_2$ deterministic matrix
satisfying $T^* T$ 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 $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$. 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.
