Sharp High-Entropy Bounds for Sums of Independent Discrete Random Variables

Abstract

Sharp high-entropy lower bounds for the entropy of a sum were known for identically distributed summands in torsion-free abelian groups and in prime cyclic groups. For arbitrary independent summands, Gavalakis, Goh and Kontoyiannis obtained an additive constant of $1/8$ and conjectured that the sharp constant is $1/2$. We prove that independent discrete random variables $X,Y$ with finite Shannon entropies satisfy $H(X+Y)\ge (H(X)+H(Y))/2+1/2-o(1)$ in every torsion-free abelian group as $\max{H(X),H(Y)}\to\infty$. The same conclusion holds in the prime cyclic group $\mathbb F_p$ when both $\max{H(X),H(Y)}$ and $\log_2 p-\max{H(X),H(Y)}$ tend to infinity. We give explicit error bounds in both settings. The proof extracts a component with paired point probabilities while controlling the entropy of the remainder independently of its support. Discrete rearrangement and uniform perturbation then transfer the continuous entropy power inequality to this component. In prime cyclic groups, an additional estimate controls the entropy lost under modular reduction. Binomial distributions show that the constant $1/2$ is optimal.