Skip to main content

Sharp deviation bounds for quadratic forms

Abstract

This paper presents sharp inequalities for deviation probability of a general quadratic form of a random vector ξ with finite exponential moments. The obtained deviation bounds are similar to the case of a Gaussian random vector. The results are stated under general conditions and do not suppose any special structure of the vector ξ. The obtained bounds are exact (non-asymptotic), all constants are explicit and the leading terms in the bounds are sharp.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Y. Baraud, “A Bernstein-Type Inequality for Suprema of Random Processes with Applications to Model Selection in Non-Gaussian Regression”, Bernoulli 16(4), 1064–1085 (2010).

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    S. Boucheron and P, Massart, “A High-Dimensional Wilks Phenomenon”, Probab. Theory and Rel. Fields 150(3), 405–433 (2011) 10.1007/s00440-010-0278-7.

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    J. Bretagnolle, “A New Large Deviation Inequality for U-Statistics of Order 2”, ESAIM, Probab. Statist. 3, 151–162 (1999).

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    J. Fan and T. Huang, “Profile Likelihood Inferences on Semiparametric Varying-Coefficient Partially Linear Models”, Bernoulli 11(6), 1031–1057 (2005).

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    J. Fan, C. Zhang, and J. Zhang, “Generalized Likelihood Ratio Statistics and Wilks Phenomenon”, Ann. Statist. 29(1), 153–193 (2001).

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    E. Giné, R. Latała, and J. Zinn, “Exponential and Moment Inequalities for U-Statistics”, in Birkhäuser Prog. Probab., Vol. 47: High Dimensional Probability II. 2nd Internat. Conf., Univ. of Washington, DC, USA, August 1–6, 1999, Ed. by E. Giné et al. (Birkhäuser, Boston, MA, 2000), pp. 13–38.

    Google Scholar 

  7. 7.

    F. Götze and A. N. Tikhomirov, “Asymptotic Distribution of Quadratic Forms”, Ann. Statist. 27(2), 1072–1098 (1999).

    MATH  Google Scholar 

  8. 8.

    L. Horváth and Q.-M. Shao, “Limit Theorems for Quadratic Forms with Applications to Whittle’s Estimate”, Ann. Appl. Probab. 9(1), 146–187 (1999).

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    C. Houdré and P. Reynaud-Bouret, “Exponential Inequalities, with Constants, for U-Statistics of Order Two”, in Birkhäuser Prog. Probab., Vol. 56: Stochastic Inequalities and Applications. Selected Papers Presented at the Euroconference on “Stochastic Inequalities and Their Applications”, Barcelona, June 18–22, 2002, Ed. by E. Giné et al. (Birkhäuser, Basel, 2003), pp. 55–69.

    Google Scholar 

  10. 10.

    D. Hsu, S.M. Kakade and T. Zhang, “A Tail Inequality for Quadratic Forms of Subgaussian Random Vectors”, Electron. Commun. Probab. 17(52), 6 (2012).

    MathSciNet  Google Scholar 

  11. 11.

    P. Massart, Concentration Inequalities and Model Selection. Ecole d’Eté de Probabilités de Saint-Flour XXXIII-2003, in Lecture Notes in Mathematics (Springer, 2007).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to V. Spokoiny.

About this article

Cite this article

Spokoiny, V., Zhilova, M. Sharp deviation bounds for quadratic forms. Math. Meth. Stat. 22, 100–113 (2013). https://doi.org/10.3103/S1066530713020026

Download citation

Keywords

  • quadratic forms
  • deviation bounds

2000 Mathematics Subject Classification

  • primary 60F10
  • secondary 62F10