Weighted moments of the limit of a branching process in a random environment

Let (Z n ) be a supercritical branching process in an independent and identically distributed random environment ζ = (ζ 0, ζ 1,…), and let W be the limit of the normalized population size Z n / E (Z n |ζ). We show a necessary and sufficient condition for the existence of weighted moments of W of the form E , where α ≥ 1 and ℓ is a positive function slowly varying at ∞.