Please use this identifier to cite or link to this item: https://doi.org/10.1007/s10955-012-0663-y
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dc.titleCentral Limit Theorem for Partial Linear Eigenvalue Statistics of Wigner Matrices
dc.contributor.authorBao, Z.
dc.contributor.authorPan, G.
dc.contributor.authorZhou, W.
dc.date.accessioned2014-10-28T05:10:38Z
dc.date.available2014-10-28T05:10:38Z
dc.date.issued2013
dc.identifier.citationBao, 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. https://doi.org/10.1007/s10955-012-0663-y
dc.identifier.issn00224715
dc.identifier.urihttp://scholarbank.nus.edu.sg/handle/10635/105048
dc.description.abstractIn 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.
dc.description.urihttp://libproxy1.nus.edu.sg/login?url=http://dx.doi.org/10.1007/s10955-012-0663-y
dc.sourceScopus
dc.subjectCentral limit theorem
dc.subjectPartial linear eigenvalue statistics
dc.subjectPartial sum process
dc.subjectWigner matrices
dc.typeArticle
dc.contributor.departmentSTATISTICS & APPLIED PROBABILITY
dc.description.doi10.1007/s10955-012-0663-y
dc.description.sourcetitleJournal of Statistical Physics
dc.description.volume150
dc.description.issue1
dc.description.page88-129
dc.identifier.isiut000314276400004
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