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

Let G be an N x N real matrix whose entries are independent identically distributed standard normal random variables G(ij) similar to 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 lambda(1),...,.lambda(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 1/root E(N-R) (Sigma (j=1) (NR) P(lambda(j)/root 2n) - E Sigma (j=1) (NR) P(lambda(j)/root 2n) ) -> N (0, sigma(2) (P)) as n -> infinity, where sigma(2) (P) = 2-root 2/2 integral(-1) (1) P(x)(2) dx.