Reduced Critical Branching Processes for Small Populations

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

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