Total progeny in almost critical multi-type Galton-Watson processes

We consider a multi-type Galton-Watson branching processes, where the largest in magnitude positive eigenvalue $\rho$ of the first moments matrix is close to unity. Specifically, we examine the random vector representing the number of individuals preceding the generation $n$, often referred to as the total progeny. By conditioning on non-extinction or extinction at current time, and properly normalizing it, we derive the asymptotic distribution for this vector. Similar theorem is derived for the processes with immigration. The behavior of this distribution is primarily influenced by the limit of $n(\rho -1)$ as $n$ tends to infinity and $\rho$ tends to 1.