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

We consider multi-type Galton-Watson branching processes, where the Perron root (ρ) of the first moments matrix approaches unity. Specifically, we examine the random vector representing the number of individuals preceding 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 process with immigration. The key determinant of this distributions is the limit of n(ρ-1) as n tends to infinity and ρ approaches 1.