Open Access
2018 About Doob’s inequality, entropy and Tchebichef
Emmanuel Rio
Electron. Commun. Probab. 23: 1-12 (2018). DOI: 10.1214/18-ECP178


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.


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Emmanuel Rio. "About Doob’s inequality, entropy and Tchebichef." Electron. Commun. Probab. 23 1 - 12, 2018.


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

zbMATH: 1401.60027
MathSciNet: MR3873785
Digital Object Identifier: 10.1214/18-ECP178

Primary: 60E15
Secondary: 60G42

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

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