Martingale Convergence and the Functional Equation in the Multi-Type Branching Random Walk

A generalization of Biggins' 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 . Keywords and phrases. Multi-type branching random walk, functional equation, size-biased measures. AMS 1991 subject classification. Primary 60J80. 1 Introduction A multi-type branching random walk with p-types is defined as follows. An initial ancestor, f; of type i 2 f1; :::; pg resides at the origin of the real line. This individual gives birth to a random number of offspring scattered on R according to the point process Z i = (Z i1 ; :::; Z ip ) where Z ij is the point process counting the number of individuals of type j 2 f1; :::; pgborn to the individual of type i: These of...