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Fluctuations of deformed Wigner random matrices. (English) Zbl 1274.60022

Summary: Let \(X_n\) be a standard real symmetric (complex Hermitian) Wigner matrix, \(y_1,y_2,\dots ,y_n\) a sequence of independent real random variables independent of \(X_n\). Consider the deformed Wigner matrix \(H_{n,\alpha}=n^{-1/2}X_n+n^{-\alpha /2}\text{diag}(y_1,\dots ,y_n)\), where \(0<\alpha <1\). It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform \(m_{n,\alpha}(z)\) converges in probability to the corresponding Stieltjes transform \(m(z)\). In this paper, we shall give the asymptotic estimate for the expectation \(\mathbb{E}m_{n,\alpha}(z)\) and variance \(\text{Var}(m_{n,\alpha}(z))\), 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.

MSC:

60B20 Random matrices (probabilistic aspects)
60F10 Large deviations
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
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