Reduced Two-Type Decomposable Critical Branching Processes with Possibly Infinite Variance

2016, v.22, №2, 311-358

ABSTRACT

We consider a Galton\tire 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
$n\rightarrow \infty .$

Keywords: decomposable branching processes, genealogy, two-type branching processes, limit theorems