On Branching Processes in Random Environments

\ζₙ\ is a sequence of iid "environmental" variables in an abstract space Θ. Each point ζ ε Θ is associated with a pgf φ_ζ(s). The branching process ₙ\ is defined as a Markov chain such that Z₀ = k, a finite integer, and given Zₙ and ζₙ, Zₙ₊₁ is distributed as the sum of Zₙ iid random variables, each with pgf φζₙ(s). Set ξ(ζₙ) = φ'ζₙ(1) and assume that E|logξ(ζₙ)| < ∞. Then: (i) Pₙ = 0\ → 1 if E log ξ(ζₙ) 0; (ii) qk = def lim Pₙ = 0\ < 1 if E log ξ(ζₙ) > 0 and E|log (1 - φζₙ(0))| < ∞. Furthermore ₖ\, k = 1, 2, ⋯, forms a moment sequence.