Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 23 (2018), paper no. 78, 12 pp.
About Doob’s inequality, entropy and Tchebichef
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.
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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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
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