Remarks on the tail order on moment sequences

TL;DR

This paper investigates a partial order on moment sequences of measures, showing that the order is not total even for smooth densities, and discusses implications for distributional game payoffs.

Contribution

It introduces a partial order on moment sequences, identifies classes where the order is total, and disproves claims that all such measures are comparable.

Findings

The order is total on certain classes of measures.

Counterexamples show measures with identical moments but different distributions.

Disproves previous claims about the totality of the order.

Abstract

We consider positively supported Borel measures for which all moments exist. On the set of compactly supported measures in this class a partial order is defined via eventual dominance of the moment sequences. Special classes are identified on which the order is total, but it is shown that already for the set of distributions with compactly supported smooth densities the order is not total. In particular we construct a pair of measures with smooth density for which infinitely many moments agree and another one for which the moments alternate infinitely often. This disproves some recently published claims to the contrary. Some consequences for games with distributional payoffs are discussed.