On the equivalence of various definitions of mixed Poisson processes

TL;DR

This paper proves the equivalence of various definitions of mixed Poisson processes under mild conditions, providing characterizations and examples, and emphasizing the importance of assumptions for these equivalences.

Contribution

It establishes the equivalence of different mixed Poisson process definitions and offers characterizations via disintegrations, with illustrative examples and essential assumptions.

Findings

Proves equivalence of mixed Poisson process definitions under mild assumptions

Provides characterizations of mixed Poisson processes using disintegrations

Offers examples of probability spaces satisfying the equivalence

Abstract

Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing distribution as well as to the mixed Poisson process in the sense of Huang is obtained, and a characterization of each one of the above mixed Poisson processes in terms of disintegrations is provided. Moreover, some examples of "canonical" probability spaces admitting counting processes satisfying the equivalence of all above statements are given. Finally, it is shown that our assumptions are essential for the characterization of mixed Poisson processes in terms of disintegrations.