Critical multitype branching: a functional limit theorem and ancestral processes

We consider the long-term behaviour of critical multitype branching processes conditioned on non-extinction, both with respect to the forward and the ancestral processes. Forward in time, we prove a functional limit theorem in the space of trajectories of the linearly-scaled $h$-transformed process; the change of measure allows us to work on the same probability space for all times. Backward in time, we trace the lines of descent of individuals sampled from a (stationary) population and analyse the ancestral type distribution, that is, the type distribution of the ancestors in the distant past, as well as the type process along the ancestral line.