Limit theorems for stochastic growth models. I

We consider d-dimensional stochastic processes which take values in (R+) d . These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some Here τ: (R+) d → R+, |x| = Σ1 d |x(i)| A = {x ∈ (R+) d : |x| = 1} and T: A → A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that lim n→∞ Z n ρ−n = wp. This result is a generalization of the main limit theorem for super-critical branching processes; note, however, that in the present situation both p and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.