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n is a Hermitian square root of the non-negative deflnite Hermitian matrix Tn. The limiting behavior, as n !1 with n=N approaching a positive constant, of functionals of the eigenvalues of Bn ,w here each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of Bn, it is known that these linear spectral statistics converges a.s. to a non-random quantity. This paper shows their of rate of convergence to be 1=n by proving, after proper scaling, they form a tight sequence. Moreover, if EX2 11 =0 andEjX11j4 =2 , or if X11 and Tn are real and EX4 11= 3, they are shown to have Gaussian limits.

CLT for linear spectral statistics of large-dimensional sample covariance matrices