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

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.