the finiteness of a sufficiently high moment would be enough. (For a necessary and sufficient condition for the edge universality we refer to [@ding2018necessary].)

The sample covariance matrix corresponding to data matrix $ X $ is defined by $ H := X^* X $. Throughout this paper, to avoid trivial eigenvalues, we will be working in the regime $$\xi=\xi(N) := N/M,\ \ \ \lim_{N \to \infty} \xi \in (0,1) \  \mbox{or}\ \xi \equiv 1.$$ We will mainly work with the rectangular case $ 0< \xi <1 $, but will also show how to adapt the arguments to the square case $ M \equiv N $. (The reason why we do not discuss the general case $ \lim \xi =1 $ is merely technical due to the lack of local laws at the hard edge. In particular, the rigidity estimate at the hard edge is only known for a fixed $ \xi \equiv 1 $ but not for $ \xi = \xi(N) \to 1 $.)

We order the eigenvalues of $ H $ as $ \lambda_1 {\leqslant}\cdots {\leqslant}\lambda_N $, and use $ \lambda_+ $ to denote the typical location of the largest eigenvalue (see for the definition). For the main result 