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 of Lyons, Peres and Pemantle (1995) to this theorem, which exploits a change of measure argument, is extended to martingales defined on Galton-Watson processes with a general type space through non-negative functions that are harmonic for the mean kernel. Many examples satisfy stochastic domination conditions on the offspring distributions that combine with the measure change argument to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem; a general treatment of this phenomenon is given. The application of the approach to branching processes in varying environments and random environments is indicated; the results also apply to the general (Crump-Mode-Jagers) branching process once suitable results on what are called optional lines are obtained. However, the main reason for developing the theory was to obtain martingale convergence results in branching random walk that did 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.