Fluctuations of deformed Wigner random matrices

摘要

Let X_n be a standard real symmetric (complex Hermitian) Wigner matrix, y_1,y_2, … ,y_n a sequence of independent real random variables independent of X_n. Consider the deformed Wigner matrix H_(n,α) = n~(-1/2)X_n + n~(-α/2)diag(y_1,…,y_n), where 0 ＜ α ＜ 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform m_(n,α)(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation Em_(n,α)(z) and variance Vax(m_(n,α)(z)), and establish the central limit theorem for linear statistics with sufficiently regular test function. A basic tool in the study is Stein's equation and its generalization which naturally leads to a certain recursive equation.