In this paper, we prove a necessary and sufficient condition for the edgeuniversality of sample covariance matrices with general population. We considersample covariance matrices of the form $\mathcal Q = TX(TX)^{*}$, where thesample $X$ is an $M_2\times N$ random matrix with $i.i.d.$ entries with meanzero and variance $N^{-1}$, and $T$ is an $M_1 \times M_2$ deterministic matrixsatisfying $T^* T$ is diagonal. We study the asymptotic behavior of the largesteigenvalues of $\mathcal Q$ when $M:=\min\{M_1,M_2\}$ and $N$ tends to infinitywith $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$. 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 Wignermatrices by Lee and Yin.
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机译：在本文中，我们证明了样本协方差矩阵与一般总体的边缘通用性的充要条件。我们考虑形式为$ \ mathcal Q = TX（TX）^ {*} $的样本协方差矩阵，其中样本$ X $是$ M_2 \乘以Ni个随机矩阵，其中$ iid $个条目具有均值零和方差$ N ^ { -1} $，并且$ T $是$ M_1 \ times M_2 $确定性矩阵，满足$ T ^ * T $是对角线。当$ M：= \ min \ {M_1，M_2 \} $和$ N $趋于无穷大且$ \ lim_ {N \ to \ infty} {N} /时，我们研究了$ \ mathcal Q $的最大特征值的渐近行为。 {M} = d \ in（0，\ infty）$。在$ T $的温和假设下，我们证明Tracy-Widom法则在且仅当$ \ lim_ {s \ rightarrow \ infty} s ^ 4 \ mathbb {P}（ | \ sqrt {N} x_ {ij} | \ geq s）= 0 $。这种条件最早是由Lee和Yin提出用于Wignermatrices的。
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