The logarithmic law of sample covariance matrices near singularity

Let B = (bjk)p×n = (Y1, Y2, · · · , Yn) be a collection of independent real random variables with mean zero and variance one. Suppose that Σ is a p by p population covariance matrix. Let Xk = Σ Yk for k = 1, 2, · · · , n and Σ̂1 = 1 n ∑n k=1XkX T k . Under the moment condition supp,n max1≤j≤p,1≤k≤n Ebjk < ∞, we prove that the log determinant of the sample covariance matrix Σ̂1 satisfies log det Σ̂1 − ∑p k=1 log ( 1− k n ) − log det Σ √ −2 log ( 1− p n ) d −−−→ N(0, 1), when p/n→ 1 and p < n. For p = n we prove that log det Σ̂1 + n log n− log(n− 1)!− log det Σ √ 2 log n d −−−→ N(0, 1).