Measure change in multitype branching

Abstract

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like theXlogXcondition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.