A note on asymptotic behavior of critical Galton-Watson processes with immigration

In this somewhat didactic note we give a detailed alternative proof of the known result of Wei and Winnicki (1989) which states that, under second-order moment assumptions on the offspring and immigration distributions, the sequence of appropriately scaled random step functions formed from a critical Galton-Watson process with immigration (not necessarily starting from zero) converges weakly towards a squared Bessel process. The proof of Wei and Winnicki (1989) is based on infinitesimal generators, while we use limit theorems for random step processes towards a diffusion process due to Ispany and Pap (2010). This technique was already used by Ispany (2008), who proved functional limit theorems for a sequence of some appropriately normalized nearly critical Galton-Watson processes with immigration starting from zero, where the offspring means tend to its critical value 1. As a special case of Theorem 2.1 of Ispany (2008) one can get back the result of Wei and Winnicki (1989) in the case of zero initial value. In the present note we handle nonzero initial values with the technique used by Ispany (2008), and further, we simplify some of the arguments in the proof of Theorem 2.1 of Ispany (2008) as well.

Automated summary

This note provides an alternative proof of the known result of Wei and Winnicki (1989) and utilizes limit theorems for random step processes towards a diffusion process by Ispany and Pap (2010). Handling nonzero initial values with the technique used by Ispany (2008) and simplifying some of the arguments in the proof of Theorem 2.1 of Ispany (2008).