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

Abstract

Let $B_n=(1/N)T_n^{1/2}X_nX_n^*T_n^{1/2}$ where $X_n=(X_{ij})$ is $n\times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. The limiting behavior, as $n\to\infty$ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of $B_n$, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of $B_n$, 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 $\expp X^2_{11}=0$ and $\expp|X_{11}|^4=2$, or if $X_{11}$ and $T_n$ are real and $\expp X_{11}^4=3$, they are shown to have Gaussian limits.