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On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

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Abstract

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.

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References

  1. Anderson, G.W., Zeitouni, O.: CLT for a band matrix model. Probab. Theory Relat. Fields 134, 283–338 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, Z.D.: Methodologies in spectral analysis of large dimensional random matrices: a review. Stat. Sinica 9(3), 611–661 (1999)

    MATH  Google Scholar 

  3. 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  MathSciNet  Google Scholar 

  4. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)

    MATH  Google Scholar 

  5. Boutet de Monvel, A., Pastur, L., Shcherbina, M.: On the statistical mechanics approach to the random matrix theory: the integrated density of states. J. Stat. Phys. 79, 585–611 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Cabanal-Duvillard, T.: Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré, Probab. Stat. 37, 373–402 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chatterjee, S., Bose, A.: A new method for bounding rates of convergence of empirical spectral distributions. J. Theor. Probab. 17, 1003–1019 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Costin, O., Lebowitz, J.L.: Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75, 69–72 (1995)

    Article  ADS  Google Scholar 

  9. Diaconis, P., Evans, S.: Linear functionals of eigenvalues of random matrices. Trans. AMS 353, 2615–2633 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Forrester, P.: Log—gas and random matrices, available at http://www.ms.unimelb.edu.au/~matpjf/matpjf.html (2000)

  11. Girko, V.L.: Spectral Theory of Random Matrices. Nauka, Moscow (1988). (In Russian)

    MATH  Google Scholar 

  12. Guionnet, A.: Large deviations, upper bounds, and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré, Probab. Stat. 38, 341–384 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91, 151–204 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jonsson, D.: Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12, 1–38 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  15. Khoruzhenko, B., Khorunzhy, A., Pastur, L.: 1/n-corrections to the Green functions of random matrices with independent entries. J. Phys. A: Math. Gen. 28, L31–L35 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Koroljuk, V.S., Borovskich, Y.V.: Theory of U-statistics. Kluwer, Dordrecht (1993)

    Google Scholar 

  17. Lee, A.J.: U-statistics. Marcel Dekker, New York (1990)

    MATH  Google Scholar 

  18. Lytova, A., Pastur, L.: Central Limit Theorem for Linear Eigenvalue Statistics of Random Matrices with Independent Entries (2008). arXiv:0809.4698v1

  19. Mehta, L.: Random Matrices. Academic Press, New York (1991)

    MATH  Google Scholar 

  20. Marchenko, V.A., Pastur, L.A.: The eigenvalue distribution in some ensembles of random matrices. Math. USSR Sbor. 1, 457–483 (1967)

    Article  MATH  Google Scholar 

  21. Pastur, L.: On the spectrum of random matrices. Theor. Math. Phys. 10, 67–74 (1972)

    Article  MathSciNet  Google Scholar 

  22. Pastur, L.: A simple approach to the global regime of Gaussian ensembles of random matrices. Ukrainian Math. J. 57, 936–966 (2005)

    Article  MathSciNet  Google Scholar 

  23. Pastur, L.: Limiting laws of linear eigenvalue statistics for unitary invariant matrix models. J. Math. Phys. 47, 103303 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  24. Pastur, L., Shcherbina, M.: Bulk universality and related properties of hermitian matrix models. J. Stat. Phys. 130, 205 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Soshnikov, A.: The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28, 1353–1370 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Spohn, H.: Interacting Brownian particles: a study of Dyson’s model. In: Papanicolaou, G. (ed.) Hydrodynamic Behavior and Interacting Particle Systems, pp. 151–179. Springer, New York (1987)

    Google Scholar 

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  1. Mathematical Division, B. Verkin Institute for Low Temperature Physics, 47 Lenin Ave., 61103, Kharkov, Ukraine

    A. Lytova & L. Pastur

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  1. A. Lytova
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  2. L. Pastur
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Correspondence to A. Lytova.

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Lytova, A., Pastur, L. On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices. J Stat Phys 133, 871–882 (2008). https://doi.org/10.1007/s10955-008-9644-6

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  • DOI: https://doi.org/10.1007/s10955-008-9644-6

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