Supercritical multitype branching processes: the ancestral types of typical individuals

Details Supercritical multitype branching processes: the ancestral types of typical individuals Supercritical multitype branching processes: the ancestral types of typical individuals

2003-02-05

arxiv.org/abs/math/0302049v2

Adv. Appl. Prob. 35 (2003), 1090-1110

Mathematics

Probability

23 pages, 1 figure; minor additions, added references

Scientific paper

For supercritical multitype branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on non-extinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a `conceptual proof' (in the sense of Kurtz, Lyons, Pemantle, Peres 1997) of the continuous-time version of the Kesten-Stigum theorem.

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