We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineage. Our approach relies on ergodicity arguments for this time-inhomogeneous Markov process. We apply this technique on the example of a size-structured population with exponential growth in varying environment.
Accepté le :
DOI : 10.1051/ps/2018029
Mots-clés : Branching Markov processes, law of large numbers, time-inhomogeneous Markov process, ergodicity
@article{PS_2019__23__638_0,
author = {Marguet, Aline},
title = {A law of large numbers for branching {Markov} processes by the ergodicity of ancestral lineages},
journal = {ESAIM: Probability and Statistics},
pages = {638--661},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2018029},
mrnumber = {4011569},
zbl = {1506.60097},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2018029/}
}
TY - JOUR AU - Marguet, Aline TI - A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages JO - ESAIM: Probability and Statistics PY - 2019 SP - 638 EP - 661 VL - 23 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2018029/ DO - 10.1051/ps/2018029 LA - en ID - PS_2019__23__638_0 ER -
%0 Journal Article %A Marguet, Aline %T A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages %J ESAIM: Probability and Statistics %D 2019 %P 638-661 %V 23 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2018029/ %R 10.1051/ps/2018029 %G en %F PS_2019__23__638_0
Marguet, Aline. A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 638-661. doi : 10.1051/ps/2018029. http://www.numdam.org/articles/10.1051/ps/2018029/
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