Electronic Communications in Probability

Extensions of the Hoeffding-Azuma inequalities

Emmanuel Rio

Full-text: Open access


In this paper we give extensions of the Hoeffding-Azuma inequalities for weighted sums of uniformly bounded martingale differences.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 54, 6 pp.

Accepted: 6 July 2013
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings

Hoeffding inequality Azuma inequality Discrete time martingales McDiarmid inequality

This work is licensed under a Creative Commons Attribution 3.0 License.


Rio, Emmanuel. Extensions of the Hoeffding-Azuma inequalities. Electron. Commun. Probab. 18 (2013), paper no. 54, 6 pp. doi:10.1214/ECP.v18-2690. https://projecteuclid.org/euclid.ecp/1465315593

Export citation


  • Antonov, S. N. Probability inequalities for a series of independent random variables. (Russian) Teor. Veroyatnost. i Primenen. 24 (1979), no. 3, 632-636.
  • Azuma, Kazuoki. Weighted sums of certain dependent random variables. TĂ´hoku Math. J. (2) 19 1967 357-367.
  • Bentkus, Vidmantas. On Hoeffding's inequalities. Ann. Probab. 32 (2004), no. 2, 1650-1673.
  • Devroye, Luc; Lugosi, GÄ‚Ä„bor. Combinatorial methods in density estimation. Springer Series in Statistics. Springer-Verlag, New York, 2001. xii+208 pp. ISBN: 0-387-95117-2
  • Hoeffding, Wassily. Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 1963 13-30.
  • Krafft, O. A note on exponential bounds for binomial probabilities. Annals of the Institute of Statistical Mathematics 21, (1969), 219-220. Zbl 0176-49106.
  • Rio, E. On McDiarmid's concentration inequality. Electron. Commun. Probab. 18, (2013), no. 44, 1-11.
  • Whittle, P. Refinements of Kolmogorov's inequality. Teor. Verojatnost. i Primenen. 14 1969 315-317.