Regeneration of critical multi-type branching process with immigration

In this paper, we study a two-type linear fractional branching process with immigration. For n \geq 0, let \boldsymbolZ_n=\left(Z_n, 1, Z_n, 2\right)^\top denote the population size of the nth generation excluding immigrants arriving at time n. We call time n as a regeneration epoch if Z_n=0. In the critical case, we derive the limiting distribution of regeneration counts up to time n.