  Consider a multi-dimensional supercritical branching process with offspring
distribution in a parametric family. Here, each vector coordinate corresponds
to the number of offspring of a given type. The process is observed under
family-size sampling: a random sample is drawn, each individual reporting its
vector of brood sizes. In this work, we show that the set in which no siblings
are sampled (so that the sample can be considered independent) has probability
converging to one under certain conditions on the sampling size. Furthermore,
we show that the sampling distribution of the observed sizes converges to the
product of identical distributions, hence developing a framework for which the
process can be considered iid, and the usual methods for parameter estimation
apply. We provide asymptotic distributions for the resulting estimators.
