Uniform approximation of eigenvalues in Laguerre and Hermite 𝛽-ensembles by roots of orthogonal polynomials

Abstract

Abstract. We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite β-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 → ∞ with n ≤ s, and the rate is log n/n, if n, s → ∞ 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.

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

“…The arguments presented in Dette and Imhof [24] now show that the difference between the eigenvalues λ (n,p) j and the zeros x (n,p) j satisfies the inequality in Theorem 5.3 and conclude the proof.…”

“…The following theorem makes this statement more precise and provides a rate for the convergence. The proof follows by similar arguments as the proof of Theorem 2.2 in [24] and is therefore omitted. holds for all n ≥ 2.…”

“…The following theorem makes this statement more precise and is a generalization of Theorem 2.5 of [24] to the matrix case. We outline the proof in the Appendix.…”

“…Now the two probabilities in (A.1) can be considered separately and the arguments in Dette and Imhof [24] show that…”

Random Block Matrices and Matrix Orthogonal Polynomials

“…For the proof we establish the bound 5) then the assertion follows along the lines of the proof of Theorem 2.2 in Dette and Imhof (2007).…”