Reduced critical Bellman-Harris branching processes for small populations

Abstract: Let $\left{ Z(t), t\geq 0\right} $ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. Assuming that $0<Z(t)\leq \varphi (t)$, where either $\varphi (t)=o(t)$ as $t\rightarrow \infty $ or $\varphi (t)=at,\, a\>0$, we study the structure of the process $% \left{ Z(s,t),0\leq s\leq t\right} ,$ 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.