Reduced critical Bellman-Harris branching processes for small populations

Abstract

Let $\left\{Z(t),t\ge 0\right\}$ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. Assuming that $0<Z(t)\le \phi (t)$, where either $\phi (t)=o(t)$ as $t\to \infty$ or $\phi (t)=at,a>0$, we study the structure of the process where $Z(s,t)$ is the number of particles in the process at moment $s$ in the initial process which either survive up to moment $t$ or have a positive offspring number at this moment.

14 pages