Artikel,
Limit distributions of the distance to the nearest common ancestor
Akademija Nauk SSSR. Teorija Verojatnoste\uı\ i ee Primenenija, 20 (3): 614--623 (1975)
Zusammenfassung
For a Galton-Watson process μ(t), t=0,1,⋯, with μ(0)=1, define τ(t) to be the last generation where there was a common ancestor of all members of the tth generation. If the process is supercritical then limt→∞t−τ(t)=τ exists a.s. on the set of nonextinction. It has an exponential distribution Pτ<x|μ(t)↛0=1−e−f′(q)x, where q is the extinction probability and f the p.g.f. of the reproduction law. If the process is subcritical, then limt→∞Pτ(t)≤x|μ(t)>0=p1/Pμ(x)=1|μ(x)>0, where p1=limx→∞Pμ(x)=1|μ(x)>0. Corresponding results are deduced for the critical case and also for continuous time Markovian branching processes, as well as for the multitype case.
