The Pólya sum process: a Cox representation

In [1], Zessin constructed the so-called Pólya sum process via partial integration. Here we use the technique of integration by parts to the Pólya sum process to derive representations of the Pólya sum process as an infinitely divisible point process and a Cox process directed by an infinitely divisible random measure. This result is related to the question of the infinite divisibilty of a Cox process and the infinite divisibility of its directing measure. Finally we consider a scaling limit of the Pólya sum process and show that the limit satisfies an integration by parts formula, which we use to determine basic properties of this limit.