Abstract. We investigate the inhomogeneous Galton--Watson processes with immigration, where $\rho_n$, the offspring means in the $n^{\rm th}$ 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 in [4]. We also determine the limit if this assumption does not hold showing the optimality of the conditions.

AMS Subject Classification (1991): 60J80

Keyword(s): nearly critical Galton--Watson process, immigration, compound Poisson distribution, negative binomial distribution

Received October 20, 2010, and in revised form July 10, 2011. (Registered under 72/2010.)