Simm, N J (2017) Central limit theorems for the real eigenvalues of large Gaussian random matrices. Random Matrices: Theory and Applications, 6. ISSN 2010-3263

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Abstract

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 λ1,…,λNR 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)−−−−−√⎛⎝∑j=1NRP(λj/2n−−√)−E∑j=1NRP(λj/2n−−√)⎞⎠→N(0,σ2(P))
as n→∞, where σ2(P)=2−2√2∫1−1P(x)2dx.

Item Type:	Article
Schools and Departments:	School of Mathematical and Physical Sciences > Mathematics
Subjects:	Q Science > QA Mathematics
Related URLs:	https://www.worldscientific.com/worldsci...
Depositing User:	Nicholas Simm
Date Deposited:	13 Jun 2018 09:29
Last Modified:	13 Jun 2018 10:50
URI:	http://srodev.sussex.ac.uk/id/eprint/76460

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