Electronic Journal of Probability

Universality in Random Moment Problems

Holger Dette, Dominik Tomecki, and Martin Venker

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Let $\mathcal{M} _n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset \mathbb{R} $ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in the recent literature. For instance, it has been shown that a uniformly distributed moment sequence in $\mathcal M_n([0,1])$ converges in the large $n$ limit to the moment sequence of the arcsine distribution. In this article we provide a unifying viewpoint by identifying classes of more general distributions on $\mathcal{M} _n(E)$ for $E=[a,b],\,E=\mathbb{R} _+$ and $E=\mathbb{R} $, respectively, and discuss universality problems within these classes. In particular, we demonstrate that the moment sequence of the arcsine distribution is not universal for $E$ being a compact interval. Rather, there is a universal family of moment sequences of which the arcsine moment sequence is one particular member. On the other hand, on the moment spaces $\mathcal{M} _n(\mathbb{R} _+)$ and $\mathcal{M} _n(\mathbb{R} )$ the random moment sequences governed by our distributions exhibit for $n\to \infty $ a universal behaviour: The first $k$ moments of such a random vector converge almost surely to the first $k$ moments of the Marchenko-Pastur distribution (half line) and Wigner’s semi-circle distribution (real line). Moreover, the fluctuations around the limit sequences are Gaussian. We also obtain moderate and large deviations principles and discuss relations of our findings with free probability.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 15, 23 pp.

Received: 19 September 2017
Accepted: 15 January 2018
First available in Project Euclid: 23 February 2018

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 30E05: Moment problems, interpolation problems 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

random moment sequences universality CLT large deviations principles Stieltjes transform free probability

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Dette, Holger; Tomecki, Dominik; Venker, Martin. Universality in Random Moment Problems. Electron. J. Probab. 23 (2018), paper no. 15, 23 pp. doi:10.1214/18-EJP141. https://projecteuclid.org/euclid.ejp/1519354944

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  • [1] Akemann, G., Baik, J., and Di Francesco, P. (eds.), The Oxford handbook of random matrix theory, Oxford University Press, Oxford, 2011.
  • [2] Anshelevich, M., Free Meixner states, Commun. Math. Phys. 276 (2007), no. 3, 863–899.
  • [3] Bai, Z. and Silverstein, J. W., Spectral analysis of large dimensional random matrices, second ed., Springer Series in Statistics, Springer, New York, 2010.
  • [4] Berg, C., Markov’s theorem revisited, J. Approx. Theory 78 (1994), no. 2, 260–275.
  • [5] Castro, M. M. and Grünbaum, F. A., On a seminal paper by Karlin and McGregor, SIGMA Symmetry Integrability Geom. Methods Appl. 9 (2013), Paper 020, 11.
  • [6] Chang, F. C., Kemperman, J. H. B., and Studden, W. J., A normal limit theorem for moment sequences, Ann. Probab. 21 (1993), no. 3, 1295–1309.
  • [7] Chihara, T. S., An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978, Mathematics and its Applications, Vol. 13.
  • [8] Cohen, J. M. and Trenholme, A. R., Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis, J. Funct. Anal. 59 (1984), no. 2, 175–184.
  • [9] Dembo, A. and Zeitouni, O., Large deviations techniques and applications, second ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998.
  • [10] Dette, H. and Nagel, J., Distributions on unbounded moment spaces and random moment sequences, Ann. Probab. 40 (2012), no. 6, 2690–2704.
  • [11] Dette, H. and Studden, W. J., 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.
  • [12] Dumitriu, I. and Edelman, A., Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830–5847.
  • [13] Gamboa, F. and Lozada-Chang, L.-V., Large deviations for random power moment problem, Ann. Probab. 32 (2004), no. 3B, 2819–2837.
  • [14] Gamboa, F., Nagel, J., and Rouault, A., Sum rules via large deviations, J. Funct. Anal. 270 (2016), no. 2, 509–559.
  • [15] Gamboa, F., Nagel, J., and Rouault, A., Sum rules and large deviations for spectral measures on the unit circle, Random Matrices Theory Appl. 6 (2017), no. 1, 1750005, 49.
  • [16] Gao, F. and Zhao, X., Delta method in large deviations and moderate deviations for estimators, Ann. Statist. 39 (2011), no. 2, 1211–1240.
  • [17] Hamburger, H., über eine Erweiterung des Stieltjesschen Momentenproblems, Math. Ann. 81 (1920), no. 2–4, 235–319.
  • [18] Hiai, F. and Petz, D., The semicircle law, free random variables and entropy, Mathematical Surveys and Monographs, vol. 77, American Mathematical Society, Providence, RI, 2000.
  • [19] Karlin, S. and Shapley, L. S., Geometry of moment spaces, Mem. Amer. Math. Soc. No. 12 (1953), 93.
  • [20] Karlin, S. and Studden, W. J., Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966.
  • [21] Kesten, H., Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354.
  • [22] Kreĭn, M. G. and Nudel’man, A. A., The Markov moment problem and extremal problems, American Mathematical Society, Providence, R.I., 1977, Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development, Translated from the Russian by D. Louvish, Translations of Mathematical Monographs, Vol. 50.
  • [23] Krishnapur, M., Rider, B., and Virág, B., Universality of the stochastic Airy operator, Comm. Pure Appl. Math. 69 (2016), no. 1, 145–199.
  • [24] Lozada-Chang, L.-V., Asymptotic behavior of moment sequences, Electron. J. Probab. 10 (2005), no. 19, 662–690.
  • [25] McKay, B. D., The expected eigenvalue distribution of a large regular graph, Linear Algebra Appl. 40 (1981), 203–216.
  • [26] Meixner, J., Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc. S1-9 (1934), no. 1, 6.
  • [27] Nica, A. and Speicher, R., Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006.
  • [28] Saff, E. and Totik, V., Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997, Appendix B by Thomas Bloom.
  • [29] Saitoh, N. and Yoshida, H., The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory, Probab. Math. Statist. 21 (2001), no. 1, Acta Univ. Wratislav. No. 2298, 159–170.
  • [30] Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, American Mathematical Society Mathematical surveys, vol. I, American Mathematical Society, New York, 1943.
  • [31] Skibinsky, M., The range of the $(n+1)$th moment for distributions on $[0,\,1]$, J. Appl. Probability 4 (1967), 543–552.
  • [32] Skibinsky, M., Extreme $n$th moments for distributions on $[0,\,1]$ and the inverse of a moment space map, J. Appl. Probability 5 (1968), 693–701.
  • [33] Skibinsky, M., Some striking properties of binomial and beta moments, Ann. Math. Statist. 40 (1969), 1753–1764.
  • [34] Stahl, H. and Totik, V., General orthogonal polynomials, Encyclopedia of Mathematics and its Applications, vol. 43, Cambridge University Press, Cambridge, 1992.
  • [35] Verblunsky, S., On Positive Harmonic Functions, Proc. London Math. Soc. (2) 40 (1935), no. 4, 290–320.
  • [36] Verblunsky, S., On Positive Harmonic Functions: A Contribution to the Algebra of Fourier Series, Proc. London Math. Soc. (2) 38 (1935), 125–157.