Gaussian fluctuations for non-Hermitian random matrix ensembles

Consider an ensemble of $N\times N$ non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and absolute mean-square one. If the entry distributions also possess bounded densities and finite $(4+\epsilon)$ moments, then Z. D. Bai [Ann. Probab. 25 (1997) 494--529] has shown the ensemble to satisfy the circular law: after scaling by a factor of $1/\sqrt{N}$ and letting $N\to \infty$, the empirical measure of the eigenvalues converges weakly to the uniform measure on the unit disk in the complex plane. In this note, we investigate fluctuations from the circular law in a more restrictive class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. The main result is a central limit theorem for linear statistics of type $X_N(f)=\sum_{k=1}^Nf(\lambda_k)$ where $\lambda_1,\lambda_2,...,\lambda_N$ denote the ensemble eigenvalues and the test function $f$ is analytic on an appropriate domain. The proof is inspired by Bai and Silverstein [Ann. Probab. 32 (2004) 533--605], where the analogous result for random sample covariance matrices is established.