An ergodic theorem for evolution in a random environment

Abstract

Let w₁, w₁₂ and w₂ be the fitnesses of genotypes A₁A₁, A₁A₂ and A₂A₂ in an infinite diploid population, and let p_n be the A₁, gene frequency in the nth generation. If fitness varies independently from generation to generation, then p_n is a Markov process with a continuum of states. If E[In(w_i/w₁₂)] < 0 for i = 1, 2, then there is a unique stationary probability, and the distribution of p_n converges to it as n → ∞.