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
![]() |
PDF (Electronic version of an article published © [copyright World Scientific Publishing Company]https://www.worldscientific.com/page/authors/author-rights)
- Accepted Version
Download (347kB) |
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: | |
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 |
View download statistics for this item
📧 Request an update