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

Reference:

Kyprianou, A. E. and Rahimzadeh Sani, A., 2001. Martingale convergence and the functional equation in the multi-type branching random walk. Bernoulli, 7 (4), pp. 593-604.

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Abstract

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.

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