Branching processes in nearly degenerate varying environment

Abstract We investigate branching processes in varying environment, for which fₙ 1 and ₍=₁^ (1-fₙ) _+ =, ₍=₁^ (fₙ - 1) _+ <, where fₙ 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).