Fluctuations of the Traces of Complex-Valued Random Matrices

The aim of this paper is to provide a central limit theorem for complex random matrices \((X_{i,j})_{i,j\geq 1}\) with i.i.d. entries having moments of any order. Tao and Vu (Ann. Probab. 38(5):2023–2065, 2010) showed that for large renormalized random matrices, the spectral measure converges to a circular law. Rider and Silverstein (Ann. Probab. 34(6):2118–2143, 2006) studied the fluctuations around this circular law in the case where the imaginary part and the real part of the random variable X _{i, j} have densities with respect to Lebesgue measure which have an upper bound, and their moments of order k do not grow faster than \({k}^{\alpha k}\), with α > 0. Their result does not cover the case of real random matrices. Nourdin and Peccati (ALEA 7:341–375, 2008) established a central limit theorem for real random matrices using a probabilistic approach. The main contribution of this paper is to use the same probabilistic approach to generalize the central limit theorem to complex random matrices.