Supercritical multitype branching processes: the backward view

For supercritical multitype Markov branching processes in continuous time, I consider the type evolution along the lineage leading to a randomly chosen individual living at time $t$.

The main results are almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on non- extinction), and the identification of the mutation process describing the type evolution along such lineages. Important tools are a representation of the family tree in terms of a suitable size-biased tree with trunk, and large deviation theory.