Functional limit theorems for Galton-Watson processes with very active immigration

TL;DR

This paper establishes weak convergence of Galton-Watson processes with immigration to extremal shot noise processes under logarithmic tail decay assumptions, extending previous marginal distribution results.

Contribution

It introduces a new weak convergence result for Galton-Watson processes with immigration with logarithmic tail decay, linking them to extremal shot noise processes.

Findings

Weak convergence on the Skorokhod space is proven.

Limits are characterized as extremal shot noise processes.

Results recover and extend previous marginal distribution findings.

Abstract

We prove weak convergence on the Skorokhod space of Galton-Watson processes with immigration, properly normalized, under the assumption that the tail of the immigration distribution has a logarithmic decay. The limits are extremal shot noise processes. By considering marginal distributions, we recover the results of Pakes [Adv. Appl. Probab., 11(1979), 31--62].