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 Q=TX(TX)., where X is an M2.N random matrix with Xij=N.1/2qij such that qij are i.i.d. random variables with zero mean and unit variance, and T is an M1.M2 deterministic matrix such that T.T is diagonal. We study the asymptotic behavior of the largest eigenvalues of Q when M:=min{M1,M2} and N tend to infinity with limN..N/M=d.(0,.). We prove that the Tracy.Widom law holds for the largest eigenvalue of Q if and only if lims..s4P(|qij|.s)=0 under mild assumptions of T. The necessity and sufficiency of this condition for the edge universality was first proved for Wigner matrices by Lee and Yin [Duke Math. J. 163 (2014) 117.173].