Extinction of population-size-dependent branching chains in random environments

We consider a population-size-dependent branching chain in a general random environment. We give sufficident conditions for certain extinction and for non-certain extinction. The chain exhibits different asymptotic according to supk,θmk,θ < 1, mk,θn → 1 as k → ∞, n → ∞, infk,θmk,θ > 1.