We consider a population with non-overlapping generations, whose size goes to infinity. It is described by a discrete genealogy which may be time non-homogeneous and we pay special attention to branching trees in varying environments. A Markov chain models the dynamic of the trait of each individual along this genealogy and may also be time non-homogeneous. Such models are motivated by transmission processes in the cell division, reproduction-dispersion dynamics or sampling problems in evolution. We want to determine the evolution of the distribution of the traits among the population, namely the asymptotic behavior of the proportion of individuals with a given trait. We prove some quenched laws of large numbers which rely on the ergodicity of an auxiliary process. A central limit is also established in the transient case.
DOI : 10.1051/ps/2014027
Mots clés : Markov chain, random environment, branching processes, law of large numbers
@article{PS_2015__19__307_0, author = {Bansaye, Vincent and Huang, Chunmao}, title = {Weak law of large numbers for some {Markov} chains along non homogeneous genealogies}, journal = {ESAIM: Probability and Statistics}, pages = {307--326}, publisher = {EDP-Sciences}, volume = {19}, year = {2015}, doi = {10.1051/ps/2014027}, mrnumber = {3412648}, zbl = {1333.60026}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps/2014027/} }
TY - JOUR AU - Bansaye, Vincent AU - Huang, Chunmao TI - Weak law of large numbers for some Markov chains along non homogeneous genealogies JO - ESAIM: Probability and Statistics PY - 2015 SP - 307 EP - 326 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2014027/ DO - 10.1051/ps/2014027 LA - en ID - PS_2015__19__307_0 ER -
%0 Journal Article %A Bansaye, Vincent %A Huang, Chunmao %T Weak law of large numbers for some Markov chains along non homogeneous genealogies %J ESAIM: Probability and Statistics %D 2015 %P 307-326 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2014027/ %R 10.1051/ps/2014027 %G en %F PS_2015__19__307_0
Bansaye, Vincent; Huang, Chunmao. Weak law of large numbers for some Markov chains along non homogeneous genealogies. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 307-326. doi : 10.1051/ps/2014027. http://www.numdam.org/articles/10.1051/ps/2014027/
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