Total progeny in the near-critical multi-type Galton-Watson processes with immigration

In this paper, we study Galton-Watson branching processes with immigration. These processes are an extension of the classical Galton-Watson model, incorporating an additional mechanism where new individuals, called immigrants, enter the population independently of the reproduction dynamics of existing individuals. We focus on the multi-type case, where individuals are classified into several distinct types, and the reproduction law depends on the type.A crucial role in the study of multi-type Galton-Watson processes is played by the matrix $M$, which represents the expected number of descendants of different particle types, and its largest positive eigenvalue, $\rho$. Sequences of branching processes with primitive matrices $M$ and eigenvalues $\rho$ converging to $1$ are referred to as near-critical. Our focus is on the random vector $Y_n$, representing the total number of particles across all generations up to generation $n$, commonly called the total progeny, in near-critical multi-type Galton-Watson processes with immigration. Assuming the double limit $n(\rho - 1)$ exists as $n \to \infty$ and $\rho \to 1$, we establish the limiting distribution of the properly normalized vector~$Y_n$. This result is derived under standard conditions imposed on the probability generating functions of the offspring and immigration component.