A note on some random orthogonal polynomials on a compact interval

Birke, Melanie; Dette, Holger

doi:10.1090/S0002-9939-09-09933-X

A note on some random orthogonal polynomials on a compact interval
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by Melanie Birke and Holger Dette PDF

Proc. Amer. Math. Soc. 137 (2009), 3511-3522 Request permission

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 $n \to \infty$.

References

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
Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749, DOI 10.1002/9780470316962
Fu Chuen Chang, J. H. B. Kemperman, and W. J. Studden, A normal limit theorem for moment sequences, Ann. Probab. 21 (1993), no. 3, 1295–1309. MR 1235417
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
Benoît Collins, Product of random projections, Jacobi ensembles and universality problems arising from free probability, Probab. Theory Related Fields 133 (2005), no. 3, 315–344. MR 2198015, DOI 10.1007/s00440-005-0428-5
H. Dette and F. Gamboa, Asymptotic properties of the algebraic moment range process, Acta Math. Hungar. 116 (2007), no. 3, 247–264. MR 2322953, DOI 10.1007/s10474-007-6047-0
Holger Dette and William J. Studden, The theory of canonical moments with applications in statistics, probability, and analysis, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1997. A Wiley-Interscience Publication. MR 1468473
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 10.1016/j.anihpb.2004.11.002
Fabrice Gamboa and Li-Vang Lozada-Chang, Large deviations for random power moment problem, Ann. Probab. 32 (2004), no. 3B, 2819–2837. MR 2078558, DOI 10.1214/009117904000000559
F. Gamboa and A. Rouault, Canonical moments and random spectral measures. Preprint: arXiv:0801.4400v2. (2008)
I. M. Johnstone, Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy-Widom limits and rates of convergence. Annals of Statistics 36 (2008), 2638–2716.
S. Karlin and L. S. Shapley, Geometry of moment spaces, Mem. Amer. Math. Soc. 12 (1953), 93. MR 59329
Rowan Killip and Irina Nenciu, Matrix models for circular ensembles, Int. Math. Res. Not. 50 (2004), 2665–2701. MR 2127367, DOI 10.1155/S1073792804141597
Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
Morris Skibinsky, The range of the $(n+1)$th moment for distributions on $[0,\,1]$, J. Appl. Probability 4 (1967), 543–552. MR 228040, DOI 10.1017/s0021900200025535
Morris 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 246351, DOI 10.1017/s0021900200114500
Morris Skibinsky, Some striking properties of binomial and beta moments, Ann. Math. Statist. 40 (1969), 1753–1764. MR 254899, DOI 10.1214/aoms/1177697387
G. Szegö, Orthogonal Polynomials. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.