Supercritical multitype branching processes: the ancestral types of typical individuals

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.