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Supercritical multitype branching processes: the ancestral types of typical individuals

Hans-Otto Georgii·Ellen Baake
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Abstract

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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Hans-Otto Georgii, Ellen Baake. 2003-06-10. Supercritical multitype branching processes: the ancestral types of typical individuals. https://arxiv.org/abs/math/0302049

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