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,a = n^（-1/2）Xn ＋ n^（-...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,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （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.展开更多
基金Much of the work was done when the author was visiting the Department of Mathematics, Harvard University, under the project from the Y. C. Tang Disciplinary Development Fund, Zhejiang University. The author thanks Professor H. T. Yau and Professor S. T. Yau for their hospitality during the visit. The referees＇ careful reading helps to improve the presentation of the paper. This work was Partially supported by the National Natural Science Foundation of China （Grant No. 11071213）, the Natural Science Foundation of Zhejiang Province （No. R6090034）, and the Doctoral Program ~and of Ministry of Education （No. J20110031）.
文摘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,a = n^（-1/2）Xn ＋ n^（-a/2）diag（y1,...,yn）, where 0 〈 a 〈 1. It is well known that the average spectral distribution is the classical Wigner semicircle law, i.e., the Stieltjes transform mn,a（z） converges in probability to the corresponding Stieltjes transform rn（z）. In this paper, we shall give the asymptotic estimate for the expectation Emn,a （z） and variance Var（mn,a （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.
