Branching processes in nearly degenerate varying environment

We investigate branching processes in varying environment, for which f̄ₙ → 1 and ∑ₙ₌₁^∞ (1-f̄ₙ)₊ = ∞ , ∑ₙ₌₁^∞ (f̄ₙ - 1)₊ < ∞ , where f̄ₙ stands for the offspring mean in generation n . Since subcritical regimes dominate, such processes die out almost surely, therefore to obtain a nontrivial limit we consider two scenarios: conditioning on nonextinction, and adding immigration. In both cases we show that the process converges in distribution without normalization to a nondegenerate compound-Poisson limit law. The proofs rely on the shape function technique, worked out by Kersting (2020).