Improving Hoeffding's inequality using higher moments information

In this paper, we generalize and improve Hoeffding's inequality using information on the random variables’ first p moments for any fixed integer p. Importantly, our generalized Hoeffding's inequality is tighter than Hoeffding's inequality and is given in a simple closed-form expression for any fixed 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. We demonstrate the usefulness of the generalized Hoeffding's inequality in obtaining refined confidence intervals when there is some information on the random variables’ high moments.