Limit theorems for beta-Jacobi ensembles

For a β-Jacobi ensemble determined by parameters a1, a2 and n, under the restriction that the three parameters go to infinity with n and a1 being of small orders of a2, we obtain some limit theorems about the eigenvalues. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the central limit theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues.