Hoeffding's inequalities: A counterexample

Abstract

A sequence ψ: ℕ0 → ℝ satisfiesHoeffding's inequality of order n if wheneverX1,...,Xn are independent nonnegative integer-valued elementary random variables and\(\tilde X_1 , \ldots ,\tilde X_n \) are independent identically distributed nonnegative integer-valued elementary random variables, the common distribution of which is the average of those ofX1,...,Xn. We show that for each integerm greater than 2 there exists a sequence ψ satisfying Hoeffding's inequality of every order greater thanm but not that of orderm. This answers a question raised by Berg, Christensen, and Ressel.