Decomposable branching processes with two types of particles

A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable $\Xi _{2},$ the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as $N\rightarrow \infty $) the conditional distribution of the amount of the first type particles in different generations given that either $\Xi _{2}=N$ or $\Xi _{2}>N.$