Quantitative Version of a Silverstein’s Result

Chapter
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 2116)

Abstract

We prove a quantitative version of a Silverstein’s Theorem on the 4-th moment condition for convergence in probability of the norm of a random matrix. More precisely, we show that for a random matrix with i.i.d. entries, satisfying certain natural conditions, its norm cannot be small.

Keywords

High Probability Covariance Matrix Matrix Theory Euclidean Norm Discrete Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We are grateful to A. Pajor for useful comments and to S. Sodin for bringing reference [5] to our attention. Research partially supported by the E.W.R. Steacie Memorial Fellowship.

References

  1. 1.
    R. Adamczak, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Am. Math. Soc. 23, 535–561 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    R. Adamczak, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling. Constr. Approx. 34, 61–88 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. Adamczak, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Sharp bounds on the rate of convergence of empirical covariance matrix. C.R. Math. Acad. Sci. Paris 349, 195–200 (2011)Google Scholar
  4. 4.
    G.W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118 (Cambridge University Press, Cambridge, 2010)Google Scholar
  5. 5.
    A. Auffinger, G. Ben Arous, S. Péché, Sandrine Poisson convergence for the largest eigenvalues of heavy tailed random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 45, 589–610 (2009)CrossRefMATHGoogle Scholar
  6. 6.
    Z.D. Bai, J.W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd edn. Springer Series in Statistics (Springer, New York, 2010)Google Scholar
  7. 7.
    Z.D. Bai, J. Silverstein, Y.Q. Yin, A note on the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivar. Anal. 26(2), 166–168 (1988)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    O. Guédon, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Restricted isometry property for random matrices with heavy tailed columns. C.R. Math. Acad. Sci. Paris 352, 431–434 (2014)Google Scholar
  9. 9.
    O. Guédon, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, On the interval of fluctuation of the singular values of random matrices (submitted)Google Scholar
  10. 10.
    Y. Gordon, On Dvoretzky’s theorem and extensions of Slepian’s lemma, in Israel Seminar on Geometrical Aspects of Functional Analysis (1983/84), II (Tel Aviv University, Tel Aviv, 1984)Google Scholar
  11. 11.
    Y. Gordon, Some inequalities for Gaussian processes and applications. Isr. J. Math. 50, 265–289 (1985)CrossRefMATHGoogle Scholar
  12. 12.
    R. Latala, Some estimates of norms of random matrices. Proc. Am. Math. Soc. 133, 1273–1282 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    S. Mendelson, G. Paouris, On generic chaining and the smallest singular values of random matrices with heavy tails. J. Funct. Anal. 262, 3775–3811 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    S. Mendelson, G. Paouris, On the singular values of random matrices. J. Eur. Math. Soc. 16, 823–834 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    L. Pastur, M. Shcherbina, Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs, vol. 171 (American Mathematical Society, Providence, 2011)Google Scholar
  16. 16.
    Y. Seginer, The expected norm of random matrices. Combin. Probab. Comput. 9, 149–166 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    J. Silverstein, On the weak limit of the largest eigenvalue of a large dimensional sample covariance matrix. J. Multivar. Anal. 30(2), 307–311 (1989)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    N. Srivastava, R. Vershynin, Covariance estimation for distributions with 2+epsilon moments. Ann. Probab. 41, 3081–3111 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    S.J. Szarek, Condition numbers of random matrices. J. Complex. 7(2), 131–149 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Y.Q. Yin, Z.D. Bai, P.R. Krishnaiah, On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Relat. Fields 78, 509–527 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations