We study the asymptotic behavior of a mutation--selection genetic
algorithm on the integers with finite population of size $p\ge 1$. The
mutation is defined by the steps of a simple random walk and the
fitness function is linear. We prove that the normalized population
satisfies an invariance principle, that a large-deviations principle
holds and that the relative positions converge in law. After $n$
steps, the population is asymptotically around $\sqrt{n}$ times the
position at time $1$ of a Bessel process of dimension $2p-1$.
Publié le : 2003-11-14
Classification:
Genetic algorithm,
invariance principle,
large-deviations,
population dynamics,
random walks,
interacting particle systems,
60F05,
60F10,
60F17,
92D15
@article{1069786510,
author = {B\'erard, J. and Bienven\"ue, A.},
title = {Sharp asymptotic results for simplified mutation-selection algorithms},
journal = {Ann. Appl. Probab.},
volume = {13},
number = {1},
year = {2003},
pages = { 1534-1568},
language = {en},
url = {http://dml.mathdoc.fr/item/1069786510}
}
Bérard, J.; Bienvenüe, A. Sharp asymptotic results for simplified mutation-selection algorithms. Ann. Appl. Probab., Tome 13 (2003) no. 1, pp. 1534-1568. http://gdmltest.u-ga.fr/item/1069786510/