Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices

We consider n × n real symmetric random matrices n−1/2W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n−1ATA with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ4 of entries of W and A is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m/n\rightarrow c\in[0,\infty)}$$\end{document} with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}^5}$$\end{document}). This is done by using a simple “interpolation trick”. Then, by using a more elaborated techniques, we prove the CLT in the case of non-zero excess of entries for essentially \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}^4}$$\end{document} test function. Here the variance contains additional term proportional to κ4. The proofs of all limit theorems follow essentially the same scheme.