Electronic Communications in Probability

About Doob’s inequality, entropy and Tchebichef

Emmanuel Rio

Full-text: Open access

Abstract

In this note we give upper bounds on the quantiles of the one-sided maximum of a nonnegative submartingale in the class $L\log L$ or the maximum of a submartingale in $L^p$. Our upper bounds involve the entropy in the case of nonnegative martingales in the class $L\log L$ and the $L^p$-norm in the case of submartingales in $L^p$. Starting from our results on entropy, we also improve the so-called bounded differences inequality. All the results are based on optimal bounds for the conditional value at risk of real-valued random variables.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 78, 12 pp.

Dates
Received: 6 December 2017
Accepted: 7 October 2018
First available in Project Euclid: 24 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1540346602

Digital Object Identifier
doi:10.1214/18-ECP178

Mathematical Reviews number (MathSciNet)
MR3873785

Zentralblatt MATH identifier
1401.60027

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G42: Martingales with discrete parameter

Keywords
Doob’s inequality Hardy-Littlewood maximal function $L\log L$ entropy binomial rate function covariance inequalities Cantelli’s inequality subGaussian random variables bounded differences inequality McDiarmid’s inequality conditional value at risk

Rights
Creative Commons Attribution 4.0 International License.

Citation

Rio, Emmanuel. About Doob’s inequality, entropy and Tchebichef. Electron. Commun. Probab. 23 (2018), paper no. 78, 12 pp. doi:10.1214/18-ECP178. https://projecteuclid.org/euclid.ecp/1540346602


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