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Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles
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  • Published: 11 March 2008

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

  • Tiefeng Jiang1 

Probability Theory and Related Fields volume 144, pages 221–246 (2009)Cite this article

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Abstract

We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author’s earlier work in this direction, we obtain the Tracy–Widom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko–Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

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References

  1. Absil, P.A., Edelman, A., Koev, P.: On the largest principal angle between random subspaces. Preprint

  2. Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 2nd edn. Wiley, New York (1984)

    Google Scholar 

  3. Bai, Z.D.: Methodologies in spectral analysis of large dimensional random matrices, a review. Statist. Sin. 9, 611–677 (1999)

    MATH  Google Scholar 

  4. Bai, Z.D.: Circular law. Ann. Probab. 25, 494–529 (1997)

    Article  MATH  Google Scholar 

  5. Bai, Z.D., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices. Science Press, Marricville (2006)

    Google Scholar 

  6. Bai, Z.D., Silverstein, J.W.: CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32, 553–605 (2004)

    Article  MATH  Google Scholar 

  7. Bai, Z.D., Yin, Y.Q.: Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. Ann. Probab. 21, 1275–1294 (1993)

    Article  MATH  Google Scholar 

  8. Bai, Z.D., Yin, Y.Q.: Convergence to the semicircle law. Ann. Probab. 16, 863–875 (1988)

    Article  MATH  Google Scholar 

  9. Bhatia, R., Elsner, L., Krause, G.: Bounds for variation of the roots of a polynomial and the eigenvalues of a matrix. Linear Algebra Appl. 142, 195–209 (1990)

    Article  MATH  Google Scholar 

  10. Baranger, H.U., Mello, P.A.: Mesoscopic transport through chaotic cavities: a random S-matrix theory approach. Phys. Rev. Lett. 73, 142–145 (1994)

    Article  Google Scholar 

  11. Beenakker, C.W.J.: Random-matrix theory of quantum transport. Rev. Modern Phys. 69, 731–808 (1997)

    Article  Google Scholar 

  12. Bohigas, O.: Random matrix theories and chaotic dynamics. In: Giannoni, M., Voros, A., Zinn-Justin, J.(eds) Chaos and Quantum Physics, pp. 89–199. Elsevier, Amsterdam (1991)

    Google Scholar 

  13. Bryc, W., Dembo, A., Jiang, T.: Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34, 1–38 (2006)

    Article  MATH  Google Scholar 

  14. Capitaine, M., Casalis, M.: Asymptotic Freeness by generalized moments for Gaussian and Wishart matrices. Application to Beta random matrices. Indiana Univ. Math. J. 53, 397–431 (2004)

    Article  MATH  Google Scholar 

  15. Chow, Y.S., Teicher, H.: Probability Theory, Independence, Interchangeability, Martingales, 3rd edn. Springer Texts in Statistics, Heidelberg (1997)

    Google Scholar 

  16. Collins, B.: Intégrales Matricielles et Probabilitiés Non-commutatives. Thèse de Doctorat of Université Paris 6 (2003)

  17. Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133, 315–344 (2005)

    Article  MATH  Google Scholar 

  18. Conrey, J.B., Farmer, D.W., Mezzadri, F., Snaith, N.C.: Ranks of Elliptic Curves and Random Matrix Theory (London Mathematical Society Lecture Notes Series). Cambridge University Press, Cambridge (2007)

  19. Constantine, A.G.: Some non-central distribution problems in multivariate analysis. Ann. Math. Stat. 34, 1270–1285 (1963)

    Article  MATH  Google Scholar 

  20. D’Aristotle, A., Diaconis, P., Newman, C.: Brownian Motion and the Classical Groups, Probability, Statisitca and their applications. In: Athreya, K. et al. (eds.) Papers in Honor of Rabii Bhattacharaya, pp. 97–116. Institute of Mathematical Statistics, Beechwood, OH (2003)

  21. Diaconis, P.: Patterns in eigenvalues: the 70th Josiah Willard Gibbs Lecture. Bull. Am. Math. Soc. 40, 155–178 (2003)

    Article  MATH  Google Scholar 

  22. Diaconis, P., Eaton, M., Lauritzen, L.: Finite deFinetti theorem in linear models and multivariate analysis. Scand. J. Statist. 19(4), 289–315 (1992)

    MATH  Google Scholar 

  23. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach (Courant Lecture Notes), American Mathematical Society, Providence (2000)

  24. Dudley, R.M.: Real Analysis and Probability, 2nd edn. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  25. Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43, 5830–5847 (2002)

    Article  MATH  Google Scholar 

  26. Dyson, F.J.: The three fold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1962)

    Article  MATH  Google Scholar 

  27. Eaton, M.: Multivariate Statistics: A Vector Space Approach (Wiley Series in Probability and Statistics). Wiley, New York (1983)

    MATH  Google Scholar 

  28. Edelman, A.: The circular law and the probability that a random matrix has k real eigenvalues. J. Multivar. Anal. 60, 188–202 (1997)

    Article  Google Scholar 

  29. Forrester, P.: Quantum conductance problems and the Jacobi ensemble. J. Phys. A: Math. Gen. 39, 6861–6870 (2004)

    Article  Google Scholar 

  30. Forrester, P.: Log-gases and Random matrices, Prepint. http://www.ms.unimelb.edu.au/~matpjf/matpjf.html (2007)

  31. Geman, S.: The spectral radius of large random matrices. Ann. Probab. 14, 1318–1328 (1986)

    Article  MATH  Google Scholar 

  32. Girko, V.L.: Circle law. Theory Probab. Appl. 4, 694–706 (1984a)

    Google Scholar 

  33. Girko, V.L.: On the circle law. Theory Probab. Math. Statist. 28, 15–23 (1984b)

    Google Scholar 

  34. Grenander, U., Silverstein, J.W.: Spectral analysis of networks with random topologies. SIAM J. Appl. Math. 32, 499– (1977)

    Article  MATH  Google Scholar 

  35. Guhr, T., Müller-Groelling, A., Weidenmüller, H.: Random matrix theories in quantum physics: common concepts. Phys. Rep. 199, 189–425 (1998)

    Article  Google Scholar 

  36. Hiai, F., Petz, D.: Large deviations for functions of two random projection matrices. To appear “Acta Szeged”

  37. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  38. Hsu, P.L.: On the distribution of the roots of certain determinantal equations. Ann. Eugene 9, 250–258 (1939)

    Google Scholar 

  39. Hwang, C.R.: A brief survey on the spectral radius and the spectral distribution of large dimensional random matrices with i.i.d. entries. Random Matrices and Their Applications, Contemporary Mathematics, vol. 50, pp. 145–152. AMS, Providence (1986)

  40. James, A.T.: Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35, 475–501 (1964)

    Article  MATH  Google Scholar 

  41. Jiang, T.: A variance formula for quantum conductance. Preprint

  42. Jiang, T.: How many entries of a typical orthogonal matrix can be approximated by independent normals?. Ann. Probab. 34(4), 1497–1529 (2006)

    Article  MATH  Google Scholar 

  43. Jiang, T.: Maxima of entries of Haar distributed matrices. Probab. Theory Relat. Fields 131, 121–144 (2005)

    Article  MATH  Google Scholar 

  44. Jiang, T.: The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14(2), 865–880 (2004)

    Article  MATH  Google Scholar 

  45. Jonsson, D.: Some limit theorem for eigenvalues of a sample covariance matrices. J. Multivar. Anal. 12, 1–38 (1982)

    Article  MATH  Google Scholar 

  46. Johansson, K.: Shape fluctation and random matrices. Comm. Math. Phys. 209, 437–476 (2000)

    Article  MATH  Google Scholar 

  47. Johnstone, I.: On the distribution of the largest eigenvalue in principal component analysis. Ann. Stat. 29(2), 295–327 (2001)

    Article  MATH  Google Scholar 

  48. Katz, N., Sarnak, P.: Random Matrices, Frobenius Eigenvalues, and Monodromy. AMS, Providence (1999)

  49. Kilip, R.: Gaussian fluctuations for beta Ensembles. Preprint. http://xxx.lanl.gov/abs/math/0703140 (2007)

  50. Koev, P., Dumitriu, I.: Distribution of the extreme eigenvalues of the complex Jacobi random matrix ensemble. To appear in Siam J. Matrix Anal. Appl. (2005)

  51. Ledoux, M.: Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The continuous case. J. Theor. Probab. 9, 177–208 (2004)

    Google Scholar 

  52. Marchenko, V.A., Pastur, L.A.: Distribution of some sets of random matrices. Math. USSR-sb. 1, 457–483 (1967)

    Article  MATH  Google Scholar 

  53. Mezzadri, F., Snaith, N.C.: Recent Perspectives in Random Matrix Theory and Number Theory (London Mathematical Society Lecture Note Series). Cambridge University Press, Cambridge (2005)

  54. Mehta, M.L.: Random Matrices, 2nd edn. Academic Press, Boston (1991)

    MATH  Google Scholar 

  55. Muirhead, R.J.: Aspects of Multivariate Statistical Theory (Wiley Series in Probability and Statistics), 2nd edn. Wiley-Interscience, New York (2005)

    Google Scholar 

  56. Petz, D., Réffy, J.: Large deviations for the empirical eigenvalue density of truncated Haar unitary matrices. Probab. Theory Relat. Fields 133, 175–189 (2005)

    Article  MATH  Google Scholar 

  57. Phillips, D.: Improving spectral-variation bound with Chebyshev polynomials. Linear Algebra Appl. 133, 165–173 (1990)

    Article  MATH  Google Scholar 

  58. Pichard, J.L., Jalabert, R.A., Beenakker, C.W.J.: Universal signature of chaos in ballistic transport. Europhys. Lett. 27, 255–260 (1994)

    Google Scholar 

  59. Silverstein, J.W.: The smallest eigenvalue of a large dimensional Wishart matrix. Ann. Probab. 13, 1364–1368 (1985)

    Article  MATH  Google Scholar 

  60. Silverstein, J.W.: Comments on a result of Yin, Bai and Krishnaiah for large dimensional multivariate F matrices. J. Multivar. Anal. 15, 408–409 (1984a)

    Article  MATH  Google Scholar 

  61. Silverstein, J.W.: Some limit theorems on the eigenvectors of large dimensional sample covariance matrices. J. Multivar. Anal. 15, 295–324 (1984b)

    Article  MATH  Google Scholar 

  62. Sinai, Y., Soshnikov, A.: Central limit theorems for traces of large symmetric matrices with independent matrix elements. Bol. Soc. Brasil Mat. (N.S.) 29, 1–24 (1998)

    Article  MATH  Google Scholar 

  63. Tracy, C.A., Widom, H.: Level-spacing distributions and Airy kernal. Comm. Math. Phys. 159, 151–174 (1994)

    Article  MATH  Google Scholar 

  64. Tracy, C.A., Widom, H.: On the orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177, 727–754 (1996)

    Article  MATH  Google Scholar 

  65. Tracy, C.A., Widom, H.: The distribution of largest eigenvalue in the Gaussian ensembles. In: van Diejen, J., Vinet, L. (eds.) Calogero-Moser-Sutherland Models, vol. 4 of CRM, Series in Mathematical Physics, pp. 461–472. Springer, Berlin (2000)

  66. Tulino, A.M., Verdu, S.: Random Matrix Theory And Wireless Communications. Now Publishers, Hanover (2004)

    MATH  Google Scholar 

  67. Wachter, K.W.: The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6, 1–18 (1978)

    Article  MATH  Google Scholar 

  68. Wigner, E.P.: On the distribution of roots of certain symmetric matrices. Ann. Math. 67, 325–328 (1958)

    Article  Google Scholar 

  69. Yin, Y.Q.: LSD for a class of random matrices. J. Multivar. Anal. 20, 50–68 (1986)

    Article  MATH  Google Scholar 

  70. Yin, Y.Q., Bai, Z.D., Krishnaiah, P.R.: Limiting behavior of the eigenvalues of multivariate F matrix. J. Multivar. Anal. 13, 508–516 (1983)

    Article  MATH  Google Scholar 

  71. Yin, Y.Q., Bai, Z.D., Krishnaiah, P.R.: On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Relat. Fields 78, 509–521 (1988)

    Article  MATH  Google Scholar 

  72. Życzkowski K., Sommers, H.J.: Truncation of random unitary matrices. J. Phys. A: Math. Gen. 33, 2045–2057 (2000)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Statistics, University of Minnesota, 224 Church Street, Minneapolis, MN, 55455, USA

    Tiefeng Jiang

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  1. Tiefeng Jiang
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Correspondence to Tiefeng Jiang.

Additional information

Supported in part by NSF #DMS-0449365.

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Cite this article

Jiang, T. Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles. Probab. Theory Relat. Fields 144, 221–246 (2009). https://doi.org/10.1007/s00440-008-0146-x

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  • Received: 15 April 2007

  • Revised: 03 January 2008

  • Published: 11 March 2008

  • Issue Date: May 2009

  • DOI: https://doi.org/10.1007/s00440-008-0146-x

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Keywords

  • Haar measure
  • Eigenvalue
  • Random matrix
  • Largest eigenvalue
  • Empirical distribution
  • Limiting distribution

Mathematical Subject Classification (2000)

  • 15A33
  • 15A52
  • 60F05
  • 60F15
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