A Sharper Hoeffding Bound for Weighted Sums of Exchangeable Random Variables

Abstract

We prove a Hoeffding-type moment generating function bound for weighted sums of bounded exchangeable random variables centered by their finite-population average. The bound reduces the excess inflation above one in a recent weighted exchangeable Hoeffding inequality from order $(\log N)/N$ to the rate-optimal order $1/N$, with an explicit constant. The proof reduces the problem to Hamming slices, identifies two-level extremizers for the relevant symmetric variational problem, and applies a hypergeometric martingale bound. We also give a lower bound showing that an excess inflation of order $1/N$ is unavoidable.

arXiv comment14 pages