Uniform approximation of eigenvalues in Laguerre and Hermite $\beta$-ensembles by roots of orthogonal polynomials

Abstract:

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 $n,s\to \infty$ with $n\leq s$, and the rate is $\sqrt {\log n/n}$, if $n,s\to \infty$ with $n\leq s\leq n+K$. In the latter case we also show the a.s. convergence of the $\lfloor nt \rfloor$ largest eigenvalue of $W(I_n,s)/s$ to the corresponding quantile of the Marc̆enko-Pastur law.