On the rate of escape of a mutation-selection algorithm

We consider a genetic algorithm associated with mutation and selection, modeled as a measure valued dynamical system on the integers. A simple symmetric random walk induces the mutation and the fitness is linear. We prove that the rate of escape of the fitness is of order the number of iteration steps, in a strong sense, since a.s. convergence and L-p convergence hold. Furthermore, the normalized algorithm converges in law, and a large deviations principle hold.