Electronic Communications in Probability

A multivariate version of Hoeffding's inequality

Peter Major

Full-text: Open access

Abstract

In this paper a multivariate version of Hoeffding's inequality is proved about the tail distribution of homogeneous polynomials of Rademacher functions with an optimal constant in the exponent of the upper bound. The proof is based on an estimate about the moments of homogeneous polynomials of Rademacher functions which can be considered as an improvement of Borell's inequality in a most important special case.

Article information

Source
Electron. Commun. Probab., Volume 11 (2006), paper no. 24, 220-229.

Dates
Accepted: 9 October 2006
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058867

Digital Object Identifier
doi:10.1214/ECP.v11-1221

Mathematical Reviews number (MathSciNet)
MR2266713

Zentralblatt MATH identifier
1110.60010

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60H05: Stochastic integrals

Keywords
Hoeffding's inequality Borell's inequality multiple Wiener–It^o integrals diagram formula

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Major, Peter. A multivariate version of Hoeffding's inequality. Electron. Commun. Probab. 11 (2006), paper no. 24, 220--229. doi:10.1214/ECP.v11-1221. https://projecteuclid.org/euclid.ecp/1465058867


Export citation

References

  • Bonami, A. Étude des coefficients de Fourier des fonctions de $L^p(G)$. Ann. Inst. Fourier (Grenoble) 20 (1970), 335–402
  • Borell, C. On the integrability of Banach space valued Walsh polynomials. Séminaire de Probabilités XIII, Lecture Notes in Math. 721 Springer, Berlin. (1979) 1–3.
  • Dudley, R. M. Uniform Central Limit Theorems. Cambridge University Press, Cambridge U.K. (1998)
  • Dynkin, E. B. and Mandelbaum, A. (1983) Symmetric statistics, Poisson processes and multiple Wiener integrals. Annals of Statistics 11 (1983), 739–745
  • Gross, L. (1975) Logarithmic Sobolev inequalities. Amer. J. Math. 97, (1975) 1061–1083
  • Ito K. Multiple Wiener integral. J. Math. Soc. Japan 3 (1951), 157–164
  • Janson, S. Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129. Cambridge University Press, Cambridge (1997)
  • Major, P. Multiple Wiener–It integrals. Lecture Notes in Mathematics 849, Springer Verlag, Berlin Heidelberg, New York, (1981)
  • Major, P. Tail behaviour of multiple random integrals and $U$-statistics. Probability Reviews. (2005) 448–505
  • Major, P. An estimate on the maximum of a nice class of stochastic integrals. Probability Theory and Related Fields., 134, (2006) 489–537
  • Major, P. On a multivariate version of Bernstein's inequality. Submitted to Journal of Electronic Probability (2006)