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

Let Bₙ=(1/N)Tₙ1/2XₙXₙ^*Tₙ1/2 where Xₙ=(Xᵢⱼ) is n× N with i.i.d. complex standardized entries having finite fourth moment, and Tₙ1/2 is a Hermitian square root of the nonnegative definite Hermitian matrix Tₙ. The limiting behavior, as n→∞ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of Bₙ, where 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 nonrandom quantity. This paper shows their rate of convergence to be $1/n$ by proving, after proper scaling, that they form a tight sequence. Moreover, if X²₁₁=0 and |X₁₁|⁴=2, or if X₁₁ and Tₙ are real and X₁₁⁴=3, they are shown to have Gaussian limits.