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Fluctuations of eigenvalues and second order Poincaré inequalities
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  • Published: 30 November 2007

Fluctuations of eigenvalues and second order Poincaré inequalities

  • Sourav Chatterjee1 

Probability Theory and Related Fields volume 143, pages 1–40 (2009)Cite this article

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Abstract

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.

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References

  1. Anderson G., Zeitouni O. (2006) A CLT for a band matrix model. Probab. Theory Related Fields 134(2): 283–

    Article  MATH  MathSciNet  Google Scholar 

  2. Anderson, G., Zeitouni, O.: A law of large numbers for finite range dependent random matrices. Preprint. Available at http://arxiv.org/math/0609364, (2006)

  3. Avram F., Bertsimas D. (1993) On central limit theorems in geometrical probability. Ann. Appl. Probab. 3(4): 1033–

    Article  MATH  MathSciNet  Google Scholar 

  4. Bai Z.D. (1999) Methodologies in spectral analysis of large-dimensional random matrices, a review. Stat. Sinica 9(3): 611–

    MATH  Google Scholar 

  5. Bai, Z.D., Yao, J.-F.: On the convergence of the spectral empirical process of Wigner matrices. Preprint (2003)

  6. Bai Z.D., Silverstein J.W. (2004) CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32: 533–

    MathSciNet  Google Scholar 

  7. Basor, E.L.: Toeplitz determinants, Fisher–Hartwig symbols, and random matrices. Recent perspectives in random matrix theory and number theory, pp. 309–336. London Math. Soc. Lecture Note Ser., vol. 322. Cambridge University Press, Cambridge (2005)

  8. Basor E., Chen Y. (2005) Perturbed Hankel determinants. J. Phys. A 38(47): 10101–

    Article  MATH  MathSciNet  Google Scholar 

  9. Bickel P.J., Breiman L. (1983) Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11(1): 185–

    Article  MATH  MathSciNet  Google Scholar 

  10. Borovkov A.A., Utev S.A. (1983) An inequality and a characterization of the normal distribution connected with it. Teor. Veroyatnost. Primenen. 28(2): 209–

    MATH  MathSciNet  Google Scholar 

  11. Bose, A., Sen, A.: Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices. Electron. Commun. Probab. 12, 29–35 (2007) (electronic)

    Google Scholar 

  12. Boutetde Monvel A., Pastur L., Shcherbina M. (1995) On the statistical mechanics approach in the random matrix theory: Integrated density of states. J. Stat. Phys. 7: 585–

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Cabanal-Duvillard T. (2001) Fluctuations de la loi empirique de grande matrices aléatoires. Ann. Inst. H. Poincaré Probab. Stat. 37: 373–

    Article  MATH  MathSciNet  Google Scholar 

  15. Cacoullos T., Papathanasiou V., Utev S.A. (1994) Variational inequalities with examples and an application to the central limit theorem. Ann. Probab. 22(3): 1607–

    Article  MATH  MathSciNet  Google Scholar 

  16. Chatterjee, S.: A new method of normal approximation. Ann. Probab. Available at http://arxiv.org/abs/math/0611213, (2006) (to appear)

  17. Chen L.H.Y. (1988) The central limit theorem and Poincaré-type inequalities. Ann. Probab. 16(1): 300–

    Article  MATH  MathSciNet  Google Scholar 

  18. Chernoff H. (1981) A note on an inequality involving the normal distribution. Ann. Probab. 9(3): 533–

    Article  MATH  MathSciNet  Google Scholar 

  19. Costin O., Lebowtz J. (1995) Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75: 69–

    Article  Google Scholar 

  20. Diaconis P., Shahshahani M. (1994) On the eigenvalues of random matrices: Studies in applied probability. J.~Appl. Probab. 31A: 49–

    Article  MathSciNet  Google Scholar 

  21. Diaconis P., Evans S.N. (2001) Linear functionals of eigenvalues of random matrices. Trans. Am. Math. Soc. 353: 615–

    Article  MathSciNet  Google Scholar 

  22. Dumitriu I., Edelman A. (2006) Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models. J. Math. Phys. 47(6): 063302–

    Article  MathSciNet  Google Scholar 

  23. Friedrich K.O. (1989) A Berry–Esseen bound for functions of independent random variables. Ann. Stat. 17(1): 170–

    Article  MATH  MathSciNet  Google Scholar 

  24. Girko V.L. (1990) Theory of Random Determinants. Kluwer,

    Google Scholar 

  25. Goldstein L., Reinert G. (1997) Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7(4): 935–

    Article  MATH  MathSciNet  Google Scholar 

  26. Guionnet A. (2002) Large deviations upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Stat. 38: 341–

    Article  MATH  MathSciNet  Google Scholar 

  27. Guionnet A., Zeitouni O. (2000) Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5: 119–

    MATH  MathSciNet  Google Scholar 

  28. Hachem, W., Khorunzhiy, O., Loubaton, P., Najim, J., Pastur, L.: A new approach for capacity analysis of large dimensional multi-antenna channels. Preprint (2006)

  29. Hachem, W., Loubaton, P., Najim, J.: A CLT for information-theoretic statistics of Gram random matrices with a given variance profile. Preprint (2007)

  30. Hughes C.P., Keating J.P., O’Connell N. (2000) On the characteristic polynomial of a random unitary matrix. Commun. Math. Phys. 220: 429–

    Article  MathSciNet  Google Scholar 

  31. Israelson S. (2001) Asymptotic fuctuations of a particle system with singular interaction. Stoch. Process. Appl. 93: 25–

    Article  Google Scholar 

  32. Jiang, T.: Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi Ensembles. Preprint (2006)

  33. Johansson K. (1997) On random matrices from the classical compact groups. Ann. Math. 145(2): 519–

    Article  MATH  MathSciNet  Google Scholar 

  34. Johansson K. (1998) On the fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91: 151–

    Article  MATH  MathSciNet  Google Scholar 

  35. Johnstone, I.M.: High dimensional statistical inference and random matrices. Proc. ICM 2006 (to appear) (2006)

  36. Jonsson D. (1982) Some limit theorems for the eigenvalues of a sample covariance matrix. J. Mult. Anal. 12: 1–

    Article  MATH  MathSciNet  Google Scholar 

  37. Keating J.P., Snaith N.C. (2000) Random matrix theory and ζ(1/2 + it). Commun. Math. Phys. 214: 57–

    Article  MATH  MathSciNet  Google Scholar 

  38. Khorunzhy A.M., Khoruzhenko B.A., Pastur L.A. (1996) Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37: 5033–

    Article  MATH  MathSciNet  Google Scholar 

  39. Ledoux, M.: The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001)

  40. Meckes, M.W.: On the spectral norm of a random Toeplitz matrix. Preprint. Available at http://arxiv.org/pdf/math/0703134, (2006)

  41. Mingo J.A., Speicher R. (2006) order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces. J. Funct. Anal. 235(1): 226–

    Article  MATH  MathSciNet  Google Scholar 

  42. Muckenhoupt B. (1972) Hardy’s inequality with weights. Studia Math. 44: 31–

    MATH  MathSciNet  Google Scholar 

  43. Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field. Available at http://arxiv.org/math.PR/0606663, (2006)

  44. Rider, B., Silverstein, J.W.: Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. (to appear) (2006)

  45. Rüschendorf L.: Projections and iterative procedures. In: Krishnaiah, P.R. (ed.) Multivariate Analysis VI, pp. 485–493. North-Holland, Amsterdam (1985)

  46. Sinaĭ Ya., Soshnikov A. (1998) Central limit theorems for traces of large random matrices with independent entries. Bol. Soc. Brasil. Mat. 29: 1–

    Article  MATH  MathSciNet  Google Scholar 

  47. Sinaĭ Ya., Soshnikov A. (1998) A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Funct. Anal. Appl. 32(2): 114–

    Article  MathSciNet  Google Scholar 

  48. Soshnikov A. (2002) Gaussian limits for determinantal random point fields. Ann. Probab. 28: 171–

    MathSciNet  Google Scholar 

  49. Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proc. of the Sixth Berkeley Symp. on Math. Statist. and Probab., Vol. II: Probability theory, pp. 583–602 (1972)

  50. Stein, C.: Approximate computation of expectations. IMS Lecture Notes—Monograph Series, vol. 7 (1986)

  51. Zwet W.R. (1984) A Berry–Esseen bound for symmetric statistics. Z. Wahrsch. Verw. Gebiete 66(3): 425–

    Article  MATH  MathSciNet  Google Scholar 

  52. Wieand K. (2002) Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields 123: 202–

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, University of California at Berkeley, 367 Evans Hall#3860, Berkeley, CA, 94720-3860, USA

    Sourav Chatterjee

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  1. Sourav Chatterjee
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Corresponding author

Correspondence to Sourav Chatterjee.

Additional information

The author’s research was partially supported by NSF grant DMS-0707054 and a Sloan Research Fellowship.

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Chatterjee, S. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Relat. Fields 143, 1–40 (2009). https://doi.org/10.1007/s00440-007-0118-6

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  • Received: 16 May 2007

  • Revised: 18 October 2007

  • Published: 30 November 2007

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0118-6

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Keywords

  • Central limit theorem
  • Random matrices
  • Linear statistics of eigenvalues
  • Poincaré inequality
  • Wigner matrix
  • Wishart matrix
  • Toeplitz matrix

Mathematical Subject Classification (2000)

  • 60F05
  • 15A52
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