Sharp asymptotic results for simplified mutation-selection algorithms

We study the asymptotic behavior of a mutation--selection genetic algorithm on the integers with finite population of size p.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 .n times the position at time 1 of a Bessel process of dimension 2p.1.