Martingale convergence and the functional equation in the multi-type branching random walk

A generalization of Biggins's martingale convergence theorem is proved for the multi-type branching random walk. The proof appeals to modern techniques involving the construction of size-biased measures on the space of marked trees generated by the branching process. As a simple consequence we obtain existence and uniqueness of solutions (within a specified class) to a system of functional equations.