Global Fluctuations for Linear Statistics of β-Jacobi Ensembles

We study the global fluctuations for linear statistics of the form $\sum_{i=1}^n f(\lambda_i)$ as $n \rightarrow \infty$, for $C^1$ functions $f$, and $\lambda_1, ..., \lambda_n$ 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 ($\sum_{i=1}^n f(\lambda_i) - \Exp \sum_{i=1}^n f(\lambda_i)$) 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.