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Central limit theorems for the real eigenvalues of large Gaussian random matrices

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posted on 2023-06-09, 13:43 authored by Nicholas SimmNicholas Simm
Let G be an N×N real matrix whose entries are independent identically distributed standard normal random variables Gij~N(0,1). The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability 26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if ?1,…,?NR are the real eigenvalues of G, then for any even polynomial function P(x) and even N=2n, we have the convergence in distribution to a normal random variable 1E(NR)-----v???j=1NRP(?j/2n--v)-E?j=1NRP(?j/2n--v)???N(0,s2(P)) as n?8, where s2(P)=2-2v2?1-1P(x)2dx.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Random Matrices: Theory and Applications

ISSN

2010-3263

Publisher

World Scientific

Volume

6

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Probability and Statistics Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-06-13

First Open Access (FOA) Date

2018-06-13

First Compliant Deposit (FCD) Date

2018-06-13

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