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.
- Z. D. Bai, Methodologies in spectral analysis of large-dimensional random matrices, a review, Statist. Sinica 9 (1999), no. 3, 611–677. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. MR 1711663
- Z. D. Bai, Baiqi Miao, and Jian-Feng Yao, Convergence rates of spectral distributions of large sample covariance matrices, SIAM J. Matrix Anal. Appl. 25 (2003), no. 1, 105–127. MR 2002902, DOI https://doi.org/10.1137/S0895479801385116
- Z. D. Bai and Y. Q. Yin, Convergence to the semicircle law, Ann. Probab. 16 (1988), no. 2, 863–875. MR 929083
- Z. D. Bai and Y. Q. Yin, Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix, Ann. Probab. 16 (1988), no. 4, 1729–1741. MR 958213
- Z. D. Bai and Y. Q. Yin, Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix, Ann. Probab. 21 (1993), no. 3, 1275–1294. MR 1235416
- T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
- P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
- Ioana Dumitriu and Alan Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830–5847. MR 1936554, DOI https://doi.org/10.1063/1.1507823
- Ioana Dumitriu and Alan Edelman, Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 6, 1083–1099 (English, with English and French summaries). MR 2172210, DOI https://doi.org/10.1016/j.anihpb.2004.11.002
- Freeman J. Dyson, The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Mathematical Phys. 3 (1962), 1199–1215. MR 177643, DOI https://doi.org/10.1063/1.1703863
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695
- R. A. Fisher, The sampling distribution of some statistics obtained from non-linear equations, Ann. Eugenics 9 (1939), 238–249. MR 1499
- Luigi Gatteschi, Asymptotics and bounds for the zeros of Laguerre polynomials: a survey, J. Comput. Appl. Math. 144 (2002), no. 1-2, 7–27. MR 1909981, DOI https://doi.org/10.1016/S0377-0427%2801%2900549-0
- Jonas Gustavsson, Gaussian fluctuations of eigenvalues in the GUE, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 151–178 (English, with English and French summaries). MR 2124079, DOI https://doi.org/10.1016/j.anihpb.2004.04.002
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183
- P. L. Hsu, On the distribution of roots of certain determinantal equations, Ann. Eugenics 9 (1939), 250–258. MR 1500
- Mourand E. H. Ismail and Xin Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115 (1992), no. 1, 131–140. MR 1079891, DOI https://doi.org/10.1090/S0002-9939-1992-1079891-5
- Alan T. James, Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964), 475–501. MR 181057, DOI https://doi.org/10.1214/aoms/1177703550
- Kurt Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476. MR 1737991, DOI https://doi.org/10.1007/s002200050027
- Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), no. 2, 295–327. MR 1863961, DOI https://doi.org/10.1214/aos/1009210544
- V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. (N.S.) 72 (114) (1967), 507–536 (Russian). MR 0208649
- Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
- Robb J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. MR 652932
- Jack W. Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab. 13 (1985), no. 4, 1364–1368. MR 806232
- Alexander Soshnikov, A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices, J. Statist. Phys. 108 (2002), no. 5-6, 1033–1056. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. MR 1933444, DOI https://doi.org/10.1023/A%3A1019739414239
- G. Szegö, Orthogonal Polynomials, 4th ed. Amer. Math. Soc., Providence, RI, 1975.
- Craig A. Tracy and Harold Widom, The distribution of the largest eigenvalue in the Gaussian ensembles: $\beta =1,2,4$, Calogero-Moser-Sutherland models (Montréal, QC, 1997) CRM Ser. Math. Phys., Springer, New York, 2000, pp. 461–472. MR 1844228
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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