| Abstrakt: |
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, ρ. Sequences of branching processes with primitive matrices M and eigenvalues ρ converging to 1 are referred to as near-critical. Our focus is on the random vector Yn, 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(ρ-1) exists as n → 8 and ρ → 1, we establish the limiting distribution of the properly normalized vector Yn. This result is derived under standard conditions imposed on the probability generating functions of the offspring and immigration component. [ABSTRACT FROM AUTHOR] |
