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

Let $G$ be an $N\times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij} \sim $\mathcal{N}$(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 note is to show that by appropriately adapting the methods of \cite{KPTTZ15}, we can prove a central limit theorem of the following form: if ${\lambda }_1,\dots ,{\lambda }_{N_{\mathbb{R}}}$ 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 \begin{equation} \frac{1}{\sqrt{$\mathbb{E}$(N_{$\mathbb{R}$})}}\left(\sum_{j=1}^{N_{$\mathbb{R}$}}P(\lambda_{j})-$\mathbb{E}$\sum_{j=1}^{N_{$\mathbb{R}$}}P(\lambda_{j})\right) \to $\mathcal{N}$(0,\sigma^{2}(P)) \end{equation} as $n\to \mathrm{\infty }$, where $\sigma^{2}(P) = \frac{2-\sqrt{2}}{2}\int_{-1}^{1}P(x)^{2},dx$.