Multi-type branching processes with immigration generated by point processes

TL;DR

The paper advances the theory of continuous-time branching processes with immigration by embedding immigration times into a general point-process framework, moving beyond time-homogeneous Poisson arrivals. It develops explicit generating-function and Laplace-functional representations for both single- and multitype processes, and analyzes immigration spanned by determinantal point processes (DPPs) as a key illustration, alongside Cox and fractional Poisson immigration. The main contributions include moment formulas, asymptotic results for subcritical, critical, and supercritical regimes, and detailed limit theorems for DPP-driven immigration, with corrections to prior results in the literature. This framework enables flexible modeling of immigration patterns (repulsion, clustering, heavy tails) and provides tractable tools for analyzing long-run behavior in complex population systems.

Abstract

Following the pivotal work of Sevastyanov, who considered branching processes with homogeneous Poisson immigration, much has been done to understand the behaviour of such processes under different types of branching and immigration mechanisms. Recently, the case where the times of immigration are generated by a non-homogeneous Poisson process was considered in depth. In this work, we try to demonstrate how one can use the framework of point processes in order to go beyond the Poisson process. As an illustration, we show how to transfer techniques from the case of Poisson immigration to the case where it is spanned by a determinantal point process.

Theorems & Definitions (14)

• Theorem 4.1
• proof
• Theorem 4.2
• Theorem 5.1
• proof
• Theorem 5.2
• Theorem 5.3
• Remark 5.4
• Lemma 5.5
• proof