Comparison between two types of large sample covariance matrices

Abstract

Let {Xij}, i, j = · · · , be a double array of independent and identically distributed (i.i.d.) real random variables with EX11= μ, E|X11 − μ|2 = 1 and E|X11|4 < ∞. Consider sample covariance matrices (with/without empirical centering) S = 1/n nΣj=1 (sj− s)(sj −¯s)T and S = 1/n nΣj=1 sjsTj, where ¯s =1/n nΣj=1 sj and sj = T1/2 n (X1j , · · · ,Xpj)T with (T1/2 n )2 = Tn, 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.