Exchangeable Hoeffding decompositions over finite sets: a characterization and counterexamples

TL;DR

This paper characterizes Hoeffding decomposability in exchangeable sequences over finite sets, revealing new classes of such sequences beyond Pólya and i.i.d. cases, and addresses an open question in the field.

Contribution

It provides a new combinatorial characterization of Hoeffding decomposability and constructs non-Pólya, non-i.i.d. exchangeable sequences that are Hoeffding decomposable.

Findings

Existence of non-Pólya, non-i.i.d. Hoeffding decomposable sequences for sets with more than two elements

A new combinatorial criterion for Hoeffding decomposability

Answers an open question from previous research

Abstract

We study Hoeffding decomposable exchangeable sequences with values in a finite set D. We provide a new combinatorial characterization of Hoeffding decomposability and use this result to show that, if the cardinality of D is strictly greater than 2, then there exists a class of neither P\'{o}lya nor i.i.d. D-valued exchangeable sequences that are Hoeffding decomposable. The construction of such sequences is based on some ideas appearing in Hill, Lane and Sudderth [1987] and answers a question left open in El-Dakkak and Peccati [2008].