# On aggregation of subcritical Galton-Watson branching processes with regularly varying immigration

## TL;DR

This paper investigates the limiting behavior of aggregated subcritical Galton-Watson processes with heavy-tailed immigration, revealing stable or deterministic limits depending on the tail index, through a two-stage limit process.

## Contribution

It provides a novel analysis of the asymptotic distribution of aggregated Galton-Watson processes with regularly varying immigration, highlighting the impact of heavy tails on the limit processes.

## Key findings

- Limit processes are $ ext{alpha}$-stable for $ ext{alpha}
eq 1$
- Deterministic line limit when $ ext{alpha} = 1$
- Limits exist when first taking $N o inity$ then $n o inity$

## Abstract

We study an iterated temporal and contemporaneous aggregation of $N$ independent copies of a strongly stationary subcritical Galton-Watson branching process with regularly varying immigration having index $\alpha \in (0, 2)$. Limits of finite dimensional distributions of appropriately centered and scaled aggregated partial sum processes are shown to exist when first taking the limit as $N \to \infty$ and then the time scale $n \to \infty$. The limit process is an $\alpha$-stable process if $\alpha \in (0, 1) \cup (1, 2)$, and a deterministic line with slope $1$ if $\alpha = 1$.