Narrow Results

Let $G$ be an $N\times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij}\sim {𝒩}(0,1)$. 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 Probability26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if ${\lambda }_1,\dots ,{\lambda }_{N_{ℝ}}$ 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