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

Authors:
Holger Dette and Lorens A. Imhof

Journal:
Trans. Amer. Math. Soc. **359** (2007), 4999-5018

MSC (2000):
Primary 60F15, 15A52; Secondary 82B10

DOI:
https://doi.org/10.1090/S0002-9947-07-04191-8

Published electronically:
May 7, 2007

MathSciNet review:
2320657

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**Holger Dette**

Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany

Email:
holger.dette@ruhr-uni-bochum.de

**Lorens A. Imhof**

Affiliation:
Department of Statistics, Bonn University, D-53113 Bonn, Germany

Email:
limhof@uni-bonn.de

Keywords:
Gaussian ensemble,
random matrix,
rate of convergence,
Weyl’s inequality,
Wishart matrix

Received by editor(s):
March 11, 2005

Received by editor(s) in revised form:
September 2, 2005

Published electronically:
May 7, 2007

Additional Notes:
The authors would like to thank Isolde Gottschlich, who typed parts of the paper with considerable technical expertise, and Z.D. Bai and J. Silverstein for some help with the references. We are also grateful to an unknown referee for his/her constructive comments on an earlier version of this paper and to I. Dumitriu for sending us the paper of Dumitriu and Edelman (2004) before publication. The work of the first author was supported by the Deutsche Forschungsgemeinschaft (SFB 475, Komplexitätsreduktion in multivariaten Datenstrukturen). Parts of this paper were written during a visit of the first author at Purdue University and this author would like to thank the Department of Statistics for its hospitality.

Article copyright:
© Copyright 2007
American Mathematical Society