Electronic Communications in Probability

About Doob’s inequality, entropy and Tchebichef

Emmanuel Rio

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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