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Measure change in multitype branching

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

J. D. Biggins  and
A. E. Kyprianou
J. D. Biggins*
Affiliation:
University of Sheffield
A. E. Kyprianou*
Affiliation:
University of Utrecht
*
∗ Postal address: Department of Probability and Statistics, Hicks Building, University of Sheffield, Sheffield S3 7RH, UK. Email address: j.biggins@sheffield.ac.uk
∗∗ Postal address: University of Utrecht, Department of Mathematics, Buadapestlaan 6, 3584CD, The Netherlands. Email address: kyprianou@math.uu.nl
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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 the X log X condition 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.

Keywords

Branchingmeasure changemultitypebranching random walkvarying environmentrandom environmentmartingalesharmonic functionsCrump-Mode-Jagers processoptional lines

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G42: Martingales with discrete parameter
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 544 - 581
Copyright
Copyright © Applied Probability Trust 2004 

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