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:=min₁,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].