This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We consider matrices of the form $B_n=(1/n)T_p^{ 1/2}X_nX_n^{ *}T_p^{ 1/2},$ where $X_n= (x_{ i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex random variables, and $T_p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. By using Bernstein polynomial approximation, we show that, under $\mathbb{ E}|x_{ ij}|^{ 8}<\infty$, the centered LSSs of $B_n$ have Gaussian limits. Under the stronger $\mathbb{ E}|x_{ ij}|^{ 10}<\infty$, we further establish convergence rates $O(n^{ -1/2+\kappa})$ in Kolmogorov--Smirnov $O(n^{ -1/2+\kappa})$, for any fixed $\kappa>0$.
Categories: math.ST math.PR stat.TH