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Supercritical multitype branching processes: the ancestral types of typical individuals

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Hans-Otto Georgii  and
Ellen Baake
Hans-Otto Georgii*
Affiliation:
Universität München
Ellen Baake*
Affiliation:
Universität Greifswald
*
∗ Postal address: Institut für Mathematik, Universität München, Theresienstraße 39, D-80333 München, Germany. Email address: georgii@mathematik.uni-muenchen.de
∗∗ Postal address: Institut für Mathematik und Informatik, Universität Greifswald, Jahnstraße 15a, D-17487 Greifswald, Germany.
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Abstract

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

Keywords

Multitype branching processtype historyancestral distributionsize-biased treeempirical processlarge deviationsKesten-Stigum theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 1090 - 1110
Copyright
Copyright © Applied Probability Trust 2003 

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