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Title: Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices
Authors: Bao, Z.
Pan, G.
Zhou, W. 
Keywords: Central limit theorem
Partial linear eigenvalue statistics
Partial sum process
Wigner matrices
Issue Date: 2013
Citation: Bao, Z., Pan, G., Zhou, W. (2013). Central Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices. Journal of Statistical Physics 150 (1) : 88-129. ScholarBank@NUS Repository.
Abstract: In this paper, we study the complex Wigner matrices Mn=1/√n Wn whose eigenvalues are typically in the interval [-2,2]. Let λ1≤λ2⋯≤λn be the ordered eigenvalues of Mn. 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 [-2,2], 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 An[f; u]=∑l=1 nf(λl)1{λl≤u}. And the second one is Bn[f; k]=∑l=1 kf(λl) with positive integer k=kn such that k/n→y∈(0,1) as n tends to infinity. Moreover, we derive a weak convergence result for a partial sum process constructed from Bn[f; ⌊ nt⌋]. 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-Sjöstrand formula and a comparison procedure on the resolvents to extend the results from GUE case to general Wigner matrices case. Moreover, the results on An[f;u] for the real Wigner matrices will also be briefly discussed. © 2012 Springer Science+Business Media New York.
Source Title: Journal of Statistical Physics
ISSN: 00224715
DOI: 10.1007/s10955-012-0663-y
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