Comparison between two types of large sample covariance matrices

Let {X ij }, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX 11 = µ, E|X 11 − µ| 2 = 1 and E|X 11 | 4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1 n n j=1 (s j − ¯ s)(s j − ¯ s) T and S = 1 n n j=1 s j s T j , where ¯ s = 1 n n j=1 s j and s j = T 1/2 n (X 1j , · · · , X pj) T with (T 1/2 n) 2 = T n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of S and S are different as n → ∞ with p/n approaching a positive constant. Moreover , it is also proved that such a different behavior is not observed in the average behavior of eigenvectors.