## Electronic Communications in Probability

### About Doob’s inequality, entropy and Tchebichef

Emmanuel Rio

#### 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
Accepted: 7 October 2018
First available in Project Euclid: 24 October 2018

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

#### 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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