On Non-Gaussian Limiting Laws for the Certain Statistics of the Wigner Matrices

2012/01/14 by Anna Lytova, Lytova, Anna
Mathematics · #Advanced Algebra and Geometry #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1201.3027

openalex publication_date 2012/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We continue investigations of our previous papers, in which there were proved central limit theorems (CLT) for linear eigenvalue statistics Tr f(M_n) and there were found the limiting probability laws for the normalised matrix elements of differential functions f(M_n) of nxn real symmetric Wigner matrices M_n. Here we consider another spectral characteristic of Wigner matrices, namely, Tr f(M_n)A_n, where A_n, n=1,2,... is a certain sequence of non-random matrices. We show first that if M_n belongs to the Gaussian Orthogonal Ensemble, then Tr f(M_n)A_n satisfies CLT. Then we consider Wigner matrices with i.i.d. entries possessing entire characteristic function and find the limiting probability law for Tr f(M_n)A_n, which in general is not Gaussian.

Citations

• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices

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