2012/06/04 by Zhigang Bao, Guangming Pan, Wang Zhou
Computer Science · Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Bayesian Methods and Mixture Models #Central limit theorem #Combinatorics #Differentiable function #Eigenvalues and eigenvectors #Integer (computer science) #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Physics #Quantum mechanics #Random Matrices and Applications #Random matrix #Statistics #math-ph #math.MP #math.PR #msc:15B52 #msc:60F05 #msc:60F17
paper · pdf · doi:10.1007/s10955-012-0663-y
39 pages
arxiv created 2012/06/04 · openalex publication_date 2012/12/11 · arxiv updated 2015/06/05 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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=1nf(λl)1_\λl≤ u\. And the second one is Bn[f; k]=∑l=1kf(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; \lfloor nt\rfloor].