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

Abstract

We consider n × n real symmetric random matrices n ^−1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ⁻¹ A ^T A with independent entries of m × n matrix A. Assuming first that the 4th cumulant (excess) κ ₄ 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 → ∞, ${m/n\rightarrow c\in[0,\infty)}$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class ${\mathbb{C}^5}$). 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 ${\mathbb{C}^4}$ test function. Here the variance contains additional term proportional to κ ₄. The proofs of all limit theorems follow essentially the same scheme.