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

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 $\alpha =1$.

46 pages