Some properties of stationary continuous state branching processes

TL;DR

This paper studies the genealogical structures of stationary continuous state branching processes with immigration, revealing their distributional properties and transition dynamics under various branching mechanisms.

Contribution

It characterizes the genealogical tree of such processes as a continuous-time Galton-Watson process with immigration for sub-critical cases and analyzes related birth-death processes.

Findings

Genealogical tree distribution as Galton-Watson process with immigration

Transition rates for birth-death processes in the model

Distributional equivalences under different branching mechanisms

Abstract

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.