Majorization, Entropy and Concentration Inequalities for Discrete $\alpha$-Concave Random Variables

Abstract

We establish a maximum-variance inequality for discrete $\alpha$-concave random variables in the regime $-1/3<\alpha<0$, extending corresponding results beyond the discrete log-concave setting. As applications, we obtain a reverse entropy power inequality and bounds for the L\'evy concentration function. We further establish polynomial-type concentration inequalities, reflecting the heavy-tailed nature of the class. A main tool in our approach is a convex majorization principle that reduces the relevant extremal problems to suitable $\alpha$-affine distributions.