A correction to Kallenberg's theorem for jointly exchangeable random measures

TL;DR

This paper identifies an overlooked condition in Kallenberg's 2005 theorem on jointly exchangeable random measures, clarifies its importance, and provides a counter-example illustrating its necessity.

Contribution

It corrects and refines Kallenberg's theorem by explicitly stating the missing condition and demonstrates its significance through a counter-example.

Findings

The additional condition is necessary for the theorem's validity.

A counter-example shows the theorem fails without the condition.

The correction improves the understanding of exchangeable random measures.

Abstract

Kallenberg (2005) provided a necessary and sufficient condition for the local finiteness of a jointly exchangeable random measure on $\R_+^2$. Here we note an additional condition that was missing in Kallenberg's theorem, but was implicitly used in the proof. We also provide a counter-example when the additional condition does not hold.