Reduced critical processes for small populations

Abstract

Let $\left\{Z(n),n\ge 1\right\}$ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. Assuming that $0<Z(n)\le \phi (n)$, where either $\phi (n)=an$ for some $a>0$ or $\phi (n)=o(n)$ as $n\to \infty$, we study the structure of the process where $Z(m,n)$ is the number of particles in the process at moment $m\le n$ having a positive number of descendants at moment $n$.

10 pages