Concentration inequalities using higher moments information (2006.05130v6)

Abstract: In this paper, we generalize and improve some fundamental concentration inequalities using information on the random variables' higher moments. In particular, we improve the classical Hoeffding's and Bennett's inequalities for the case where there is some information on the random variables' first $p$ moments for every positive integer $p$. Importantly, our generalized Hoeffding's inequality is tighter than Hoeffding's inequality and is given in a simple closed-form expression for every positive integer $p$. Hence, the generalized Hoeffding's inequality is easy to use in applications. To prove our results, we derive novel upper bounds on the moment-generating function of a random variable that depend on the random variable's first $p$ moments and show that these bounds satisfy appropriate convexity properties.