Asymptotics of nearly critical Galton-Watson processes with immigration

TL;DR

This paper studies the long-term behavior of nearly critical Galton-Watson processes with immigration, identifying conditions under which their asymptotics match simpler models and establishing the limits when these conditions fail.

Contribution

It provides a detailed analysis of the asymptotic behavior of inhomogeneous Galton-Watson processes with immigration, extending previous results and determining optimal conditions.

Findings

Asymptotics match INAR(1) case under certain conditions

Identifies the limit behavior when conditions are not met

Establishes the optimality of the assumptions

Abstract

We investigate the inhomogeneous Galton--Watson processes with immigration, where $\rho_n$ the offspring means in the $n^\textrm{th}$ generation tends to 1. We show that if the second derivatives of the offspring generating functions go to 0 rapidly enough, then the asymptotics are the same as in the INAR(1) case, treated by Gy\"orfi et al. We also determine the limit if this assumption does not hold showing the optimality of the conditions.