Ancestral Lineages and Limit Theorems for Branching Markov Chains in Varying Environment

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Abstract

We consider branching processes in discrete time for structured population in varying environment. Each individual has a trait which belongs to some general state space and both its reproduction law and the trait inherited by its offsprings may depend on its trait and the environment. We study the long-time behavior of the population and the ancestral lineage of typical individuals under general assumptions. We focus on the mean growth rate and the trait distribution among the population. The approach relies on many-to-one formulae and the analysis of the genealogy, and a key role is played by well-chosen (possibly non-homogeneous) Markov chains. The applications use large deviations principles and the Harris ergodicity for these auxiliary Markov chains.

Keywords

Branching processes Markov chains Varying environment Genealogies 

Mathematics Subject Classification (2010)

60J80 60J05 60F05 60F10 

Notes

Acknowledgements

This work was partially funded by Chair Modelisation Mathematique et Biodiversite VEOLIA-Ecole Polytechnique-MNHN-F.X, the professorial chair Jean Marjoulet, the project MANEGE ‘Modèles Aléatoires en Écologie, Génétique et Évolution’ 09-BLAN-0215 of ANR (French National Research Agency). The author is also grateful to Clément Dombry for mentioning [48].

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMAP, Ecole Polytechnique, CNRSPalaiseau CedexFrance

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