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 X₈₉=N^-1/2q₈₉ such that q₈₉ 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: =\M₁, M₂\ and N tend to infinity with ₍N/M=d (0, ). We prove that the Tracy–Widom law holds for the largest eigenvalue of Q if and only if ₒs^4P (q₈₉ 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.