On the Central Limit Theorem for the Eigenvalue Counting Function of Wigner and Covariance matrices

2010/11/17 by Sandrine Dallaporta, Dallaporta, Sandrine
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · doi:10.48550/arxiv.1011.4042

openalex publication_date 2010/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note presents some central limit theorems for the eigenvalue counting function of Wigner matrices in the form of suitable translations of results by Gustavsson and O'Rourke on the limiting behavior of eigenvalues inside the bulk of the semicircle law for Gaussian matrices. The theorems are then extended to large families of Wigner matrices by the Tao and Vu Four Moment Theorem. Similar results are developed for covariance matrices.

Related

• A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Central Limit Theorem for Linear Eigenvalue Statistics for Submatrices of Wigner Random Matrices
• Gaussian fluctuations of the determinant of Wigner Matrices
• Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices