Limit theorems for beta-Jacobi ensembles

Abstract

For a $\beta$-Jacobi ensemble determined by parameters $a_{1}$, $a_{2}$ and $n$, under the restriction that the three parameters go to infinity with $n$ and $a_{1}$ being of small orders of $a_{2}$, 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.