Negative correlation and log-concavity

TL;DR

This paper explores the relationships between negative correlation and log-concavity in probability measures, providing counterexamples, positive results, and new proofs to clarify their properties and interconnections.

Contribution

It presents new counterexamples, positive results, and proofs regarding negative correlation and log-concavity, advancing understanding of these properties in probability measures.

Findings

Counterexamples to conjectures on negative correlation and log-concavity.

Proofs that 'almost exchangeable' measures satisfy the 'Feder-Mihail' property.

Identification of classes of measures with these properties.

Abstract

We give counterexamples and a few positive results related to several conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea et al.; prove that "almost exchangeable" measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions.