Uniform approximation of eigenvalues in laguerre and hermite β-ensembles by roots of orthogonal polynomials

We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite beta-ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence. In the special case of a normalized real Wishart matrix W(I-n, s) / s, where n denotes the dimension and s the degrees of freedom, the rate is (log n/s) 1/4, if n, s -> infinity with n <= s, and the rate is root log n/n, if n, s -> infinity with n <= s <= n + K. In the latter case we also show the a. s. convergence of the [nt] largest eigenvalue of W(I-n, s) / s to the corresponding quantile of the Marcenko-Pastur law.