Global Fluctuations for Linear Statistics of \beta-Jacobi Ensembles

We study the global fluctuations for linear statistics of the form ${\sum }_{i=1}^{n}f({\lambda }_i)$ as $n\to \mathrm{\infty }$, for $C^1$ functions $f$, and ${\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_n$ being the eigenvalues of a (general) $\beta$-Jacobi ensemble, for which tridiagonal models were given by Killip and Nenciu as well as Edelman and Sutton. The fluctuation from the mean (${\sum }_{i=1}^{n}f({\lambda }_i)-\text{Exp}{\sum }_{i=1}^{n}f({\lambda }_i)$) is given asymptotically by a Gaussian process. We compute the covariance matrix for the process and show that it is diagonalized by a shifted Chebyshev polynomial basis; in addition, we analyze the deviation from the predicted mean for polynomial test functions, and we obtain a law of large numbers.