A limit theorem for the total progeny distribution of multi-type branching processes

A multi-type branching process is defined as a random tree with labeled vertices, where each vertex produces offspring independently according to the same multivariate probability distribution. We demonstrate that in realizations of the multi-type branching process, the relative frequencies of the different types in the whole tree converge to a fixed ratio, while the probability distribution for the total size of the process decays exponentially. The results hold under the assumption that all moments of the offspring distributions exist. The proof uses a combination of the arborescent Lagrange inversion formula, a measure tilting argument, and a local limit theorem. We illustrate our concentration result by showing applications to random graphs and multi-component coagulation processes.