Sharp Khinchin-type inequalities for symmetric discrete uniform random variables
arXiv:1912.13345
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.
Revised (exposition shortened; L1-L2 inequality generalised to arbitrary symmetric distributions with large atom at 0; results for even moments will appear elsewhere). 12 pages