Final probabilities for modified branching processes

Any modified branching process $\mathcal B^*$ is constructed by means of two Galton–Watson processes $\mathcal B_0$, $\mathcal B_1$, and a fixed finite set $S$ of positive integers. The number of particles $\mu^*(t)$ of the process $\mathcal B^*$ at time instants $t=0,1,2,\dots$ evolves as follows. If $\mu^*(t)\in S$, then each of the $\mu^*(t)$ particles independently of each other produces an offspring according to the law of the branching process $\mathcal B_1$, and if $\mu^*(t)\notin S$, then the birth of particles obeys the law of the process $\mathcal B_0$. Along with active, breeding particles, in the processes $\mathcal B_0$ and $\mathcal B_1$ a random amount of final particles emerges, which do not participate in the process evolution but accumulate and constitute some final amount $\eta_n$ after the process extinction, where $n$ is the initial number of active particles. It is known that in a critical branching process, under some conditions, the distribution of the random variable $\eta_n/n^2$ as $n\to\infty$ converges to the stable distribution law with parameter $\alpha=1/2$. In this paper, we demonstrate that this property of the distribution of final particles remains true for the modified branching process $\mathcal B^*$. We also show that in this limit theorem the number of final particles can be replaced by a certain final non-negative random variable $\eta_n$ that characterises the final state of the branching process.
This research was supported by the Russian Foundation for Basic Research, grants 99–01–00012, 00–15–96136, and by INTAS–RFBR, grant 99–01317.