Reduced two-type decomposable critical branching processes with possibly infinite variance

We consider a Galton-Watson process $\mathbf{Z}% (n)=(Z_{1}(n),Z_{2}(n))$ with two types of particles. Particles of type 2 may produce offspring of both types while particles of type 1 may produce particles of their own type only. Let $Z_{i}(m,n)$ be the number of particles of type $i$ at time $m<n$ having offspring at time $n$. Assuming that the process is critical and that the variance of the offspring distribution may be infinite we describe the asymptotic behavior, as $m,n\rightarrow \infty $ of the law of $\mathbf{Z}(m,n)=(Z_1(m,n),Z_2(m,n))$ given $\mathbf{Z}(n)\neq \mathbf{0}$. We find three different types of coexistence of particles of both types. Besides, we describe, in the three cases, the distributions of the birth time and the type of the most recent common ancestor of individuals alive at time $n\rightarrow \infty .$