Limit theorems for stochastic growth models. I

Abstract

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_nρ⁻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.