Skip to main content
Log in

A Few Remarks on the Operator Norm of Random Toeplitz Matrices

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We present some results concerning the almost sure behavior of the operator norm of random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some concentration inequalities for suprema of empirical processes, which are refinements of recent results by Einmahl and Li.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamczak, R.: A tail inequality for suprema of unbounded empirical processes with applications to Markov chains. Electron. J. Probab. 13(34), 1000–1034 (2008)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. de Acosta, A., Kuelbs, J., Ledoux, M.: An inequality for the law of the iterated logarithm. In: Probability in Banach spaces, IV, Oberwolfach, 1982. Lecture Notes in Math., vol. 990, pp. 1–29. Springer, Berlin (1983)

    Chapter  Google Scholar 

  6. Einmahl, U., Li, D.: Characterization of LIL behavior in Banach space. Trans. Am. Math. Soc. 360, 6677–6693 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fuk, D.H., Nagaev, S.V.: Probabilistic inequalities for sums of independent random variables. Teor. Veroyatnost. Primenen. 16, 660–675 (1971)

    MathSciNet  Google Scholar 

  8. Giné, E., Latała, R., Zinn, J.: Exponential and moment inequalities for U-statistics. In: High Dimensional Probability II. Progr. Probab., vol. 47, pp. 13–38. Birkhäuser, Boston (2000)

    Google Scholar 

  9. Hammond, C., Miller, S.J.: Distribution of eigenvalues for the ensemble of real symmetric Toeplitz matrices. J. Theor. Probab. 18, 537–566 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Klein, T., Rio, E.: Concentration around the mean for maxima of empirical processes. Ann. Probab. 33, 1060–1077 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 23(3). Springer, Berlin (1991)

    MATH  Google Scholar 

  12. Meckes, M.: On the spectral norm of a random Toeplitz matrix. Electron. Commun. Probab. 12, 315–325 (2007)

    MATH  MathSciNet  Google Scholar 

  13. Talagrand, M.: New concentration inequalities in product spaces. Invent. Math. 126(3), 505–563 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, New York (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Radosław Adamczak.

Additional information

Research partially supported by MEiN Grant 1 PO3A 012 29.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adamczak, R. A Few Remarks on the Operator Norm of Random Toeplitz Matrices. J Theor Probab 23, 85–108 (2010). https://doi.org/10.1007/s10959-008-0201-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-008-0201-7

Keywords

Mathematics Subject Classification (2000)

Navigation