  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.
