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.
"About Doob’s inequality, entropy and Tchebichef." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP178