Erasing a branching tree. (English) Zbl 0613.60078

Neveu, J.

Adv. Appl. Probab., Spec. Suppl. 1986, 101-108 (1986).

The author considers a critical or subcritical continuous-time Markov branching process, \(Z_ t\) with \(Z_ 0=1\), having infinitesimal generating function \(a(s)=\alpha [f(s)-s]\), where 1/\(\alpha\) is the mean lifetime and f(s) is the probability generating function of the offspring distribution.
For fixed \(p\geq 0\) the family tree of the process is erased as follows. The branches of the tree are erased at unit speed for a time p, starting from the tips of the branches; when a branch has been fully erased the erasure is continued on the parent branch only when all the sibling branches, if any, have been erased.
It is shown that this leads with probability \(\lambda\) to another continuous-time Markov branching process with infinitesimal generating function a(1-\(\lambda\) (1-s))/\(\lambda\) and with probability 1-\(\lambda\) to complete erasure, where 1-\(\lambda\) is the probability of extinction by time p for \(Z_ t\).

Reviewer: D.P.Kennedy

MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J65	Brownian motion

Keywords:

Galton-Watson process; critical; subcritical; branching process