Sharp Khinchin-type inequalities for symmetric discrete uniform random variables

TL;DR

This paper derives optimal inequalities for moments of sums of symmetric discrete uniform variables, providing sharp constants and new convexity results, advancing understanding of moment comparison in discrete probability distributions.

Contribution

The paper introduces new sharp Khinchin-type inequalities for symmetric discrete uniform variables, including constants for moments and Schur-convexity results, extending classical inequalities to discrete settings.

Findings

Sharp constants for second and higher moments established.

Convex dominance by Gaussian variables demonstrated.

Schur-convexity result for three-atom case proved.

Abstract

We establish several optimal moment comparison inequalities (Khinchin-type inequalities) for weighted sums of independent identically distributed symmetric discrete random variables which are uniform on sets of consecutive integers. Specifically, we obtain sharp constants for the second moment and any moment of order at least 3 (using convex dominance by Gaussian random variables). In the case of only 3 atoms, we also establish a Schur-convexity result. For moments of order less than 2, we get sharp constants in two cases by exploiting Haagerup's arguments for random signs.