Sharp asymptotic results for simplified mutation-selection algorithms

Abstract

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 $2p-1$.