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Fluctuations of deformed Wigner random matrices |
Zhonggen SU() |
Department of Mathematics, Zhejiang University, Hangzhou 310027, China |
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Abstract Let Xn be a standard real symmetric (complex Hermitian) Wigner matrix, y1, y2, . . . , yn a sequence of independent real random variables independent of Xn. Consider the deformed Wigner matrix Hn,α = n-1/2Xn + n-α/2 diag (y1, . . . , yn), where 0<α<1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,α(z) converges in probability to the corresponding Stieltjes transform m(z). In this paper, we shall give the asymptotic estimate for the expectation Emn,α(z) and varianceVar(mn,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.
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Keywords
Asymptotic expansion
deformed Wigner matrice
Gaussian fluctuation
linear statistics
Stein’s equation
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Corresponding Author(s):
SU Zhonggen,Email:suzhonggen@zju.edu.cn
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Issue Date: 01 June 2013
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