Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices
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- 作者：Zhigang Bao (1)
Guangming Pan (2)
Wang Zhou (3) - 关键词：Wigner matrices ; Central limit theorem ; Partial linear eigenvalue statistics ; Partial sum process
- 刊名：Journal of Statistical Physics
- 出版年：2013
- 出版时间：January 2013
- 年：2013
- 卷：150
- 期：1
- 页码：88-129
- 全文大小：1218KB
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Guangming Pan (2)
Wang Zhou (3)
1. Department of Mathematics, Zhejiang University, Hangzhou, P.R. China
2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, Singapore
3. Department of Statistics and Applied Probability, National University of Singapore, Singapore, 117546, Singapore - ISSN：1572-9613
文摘
In this paper, we study the complex Wigner matrices $M_{n}=\frac{1}{\sqrt{n}}W_{n}$ whose eigenvalues are typically in the interval [?,2]. Let λ 1?em class="a-plus-plus">λ 2?≤λ n be the ordered eigenvalues of?M n . Under the assumption of four matching moments with the Gaussian Unitary Ensemble (GUE), for test function f 4-times continuously differentiable on an open interval including [?,2], we establish central limit theorems for two types of partial linear statistics of the eigenvalues. The first type is defined with a threshold u in the bulk of the Wigner semicircle law as $\mathcal{A}_{n}[f; u]=\sum_{l=1}^{n}f(\lambda_{l})\mathbf{1}_{\{\lambda_{l}\leq u\}}$ . And the second one is $\mathcal{B}_{n}[f; k]=\sum_{l=1}^{k}f(\lambda_{l})$ with positive integer k=k n such that k/n?em class="a-plus-plus">y?0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from $\mathcal{B}_{n}[f; \lfloor nt\rfloor]$ . The main difficulty is to deal with the linear eigenvalue statistics for the test functions with several non-differentiable points. And our main strategy is to combine the Helffer-Sj?strand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on $\mathcal{A}_{n}[f;u]$ for the real Wigner matrices will also be briefly discussed.