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

Abstract

Let [Formula: see text] be an [Formula: see text] real matrix whose entries are independent identically distributed standard normal random variables [Formula: see text]. 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 [Formula: see text] are the real eigenvalues of [Formula: see text], then for any even polynomial function [Formula: see text] and even [Formula: see text], we have the convergence in distribution to a normal random variable [Formula: see text] as [Formula: see text], where [Formula: see text].