Functional CLT for general sample covariance matrices

TL;DR

This paper establishes Gaussian central limit theorems for linear spectral statistics of general sample covariance matrices with test functions in $C^3$, providing convergence rates under finite eighth and tenth moments of entries.

Contribution

It extends CLTs for spectral statistics to more general covariance matrices with smooth test functions, including convergence rates and moment conditions.

Findings

Gaussian limits for centered linear spectral statistics

Convergence rates of $O(n^{-1/2+\, ext{kappa}})$ under higher moment conditions

Applicability to matrices with bounded spectral norm and i.i.d. entries

Abstract

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 $\kappa>0$.