About Doob’s inequality, entropy and Tchebichef

Electronic Communications in Probability

Emmanuel Rio¹ 

关键词: 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;   

DOI  :  10.1214/18-ECP178

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

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.

【 授权许可】

CC BY