Tail generating functions for extendable branching processes

We study branching processes of independently splitting particles in the continuous time setting. If time is calibrated such that particles live on average one unit of time, the corresponding transition rates are fully determined by the generating function $f$ for the offspring number of a single particle. We are interested in the defective case $f(1)=1-ε$, where each splitting particle with probability $ε$ is able to terminate the whole branching process. A branching process $\{Z_t\}_{t\ge0}$ will be called extendable if $f(q)=q$ and $f(r)=r$ for some $0\le q<r<\infty$. Specializing on the extendable case we derive an integral equation for $F_t(s)={\rm E} s^{Z_t}$. This equation is expressed in terms of what we call, tail generating functions. With help of this equation, we obtain limit theorems for the time to termination as $ε\to0$. We find that conditioned on non-extinction, the typical values of the termination time follow an exponential distribution in the nearly subcritical case, and require different scalings depending on whether the reproduction regime is asymptotically critical or supercritical. Using the tail generating function approach we also obtain new refined asymptotic results for the regular branching processes with $f(1)=1$.

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