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A note on some random orthogonal polynomials on a compact interval

Authors: Melanie Birke and Holger Dette
Journal: Proc. Amer. Math. Soc. 137 (2009), 3511-3522
MSC (2000): Primary 60F15, 33C45, 44A60
Published electronically: June 3, 2009
MathSciNet review: 2515420
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Abstract: We consider a uniform distribution on the set $ \mathcal{M}_k$ of moments of order $ k \in \mathbb{N}$ corresponding to probability measures on the interval $ [0,1]$. To each (random) vector of moments in $ \mathcal{M}_{2n-1}$ we consider the corresponding uniquely determined monic (random) orthogonal polynomial of degree $ n$ and study the asymptotic properties of its roots if $ n \to \infty$.

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  • 1. Z.D.  Bai, Methodologies in spectral analysis of large dimensional random matrices, a review. Statistica Sinica 9 (1999), 611-677. MR 1711663 (2000e:60044)
  • 2. P. Billingsley, Convergence of Probability Measures. 2nd Ed., Wiley Series in Probability and Statistics, Chichester, 1999. MR 1700749 (2000e:60008)
  • 3. F. C. Chang, J. H. B. Kemperman and W. J. Studden, A normal limit theorem for moment sequences. Ann. Probab. 21 (1993), 1295-1309. MR 1235417 (94m:60041)
  • 4. T. Chihara, An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
  • 5. B. Collins, Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probability Theory and Related Fields 133 (2005), 315-344. MR 2198015 (2007b:60045)
  • 6. H. Dette, F. Gamboa, Asymptotic properties of the algebraic moment range process. Acta Math. Hung. 116 (2007), 247-264. MR 2322953 (2008g:60068)
  • 7. H. Dette, W. J. Studden, The theory of canonical moments with applications in statistics, probability, and analysis. John Wiley & Sons Inc., New York, 1997. MR 1468473 (98k:60020)
  • 8. J. Dumitriu, A. Edelman, Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics. Ann. I. Henri Poincaré, Probabilités et Statistiques 41 (2005), 1083-1099. MR 2172210 (2006g:15016)
  • 9. F. Gamboa and L.V. Lozada-Chang, Large deviations for random power moment problem. Ann. Probab. 32 (2004), 2819-2837. MR 2078558 (2005e:60059)
  • 10. F. Gamboa and A. Rouault, Canonical moments and random spectral measures. Preprint: arXiv:0801.4400v2. (2008)
  • 11. I. M. Johnstone, Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy-Widom limits and rates of convergence. Annals of Statistics 36 (2008), 2638-2716.
  • 12. S. Karlin, L. S. Shapeley, Geometry of moment spaces. Amer. Math. Soc. Memoir No. 12, Amer. Math. Soc., Providence, RI, 1953. MR 0059329 (15:512c)
  • 13. R. Killip, I. Nenciu. Matrix models for circular ensembles. Int. Math. Res. Not. 50 (2004), 2665-2701. MR 2127367 (2006h:82003)
  • 14. M. L. Mehta, Random Matrices. Academic Press, 2004. MR 2129906 (2006b:82001)
  • 15. M. Skibinsky, The range of the $ (n+1)$-th moment for distributions on $ [0, 1]$. J. Appl. Probability 4 (1967), 543-552. MR 0228040 (37:3624)
  • 16. M. Skibinsky, Extreme $ n$th moments for distributions on $ [0, 1]$ and the inverse of a moment space map. J. Appl. Probability 5 (1968), 693-701. MR 0246351 (39:7655)
  • 17. M. Skibinsky, Some striking properties of binomial and beta moments. Ann. Math. Statist. 40 (1969), 1753-1764. MR 0254899 (40:8106)
  • 18. G. Szegö, Orthogonal Polynomials. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.

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Additional Information

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

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

Keywords: Moment space, random moment sequence, random orthogonal polynomial, arcsine distribution, Chebyshev polynomials, random matrices
Received by editor(s): June 20, 2008
Received by editor(s) in revised form: February 19, 2009
Published electronically: June 3, 2009
Additional Notes: The authors are grateful to Martina Stein, who typed most of this paper with considerable technical expertise. The work of the authors was supported by the Sonderforschungsbereich Tr/12, Fluctuations and universality of invariant random matrix ensembles (project C2), and in part by an NIH grant award IR01GM072876:01A1.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2009 American Mathematical Society

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