Critical branching as a pure death process coming down from infinity

Abstract We consider the critical Galton–Watson process with overlapping generations stemming from a single founder. Assuming that both variance of offspring number and average generation length are finite, we establish convergence finite-dimensional distributions, conditioned on non-extinction at remote time observation. The limiting is identified as pure death coming down infinity. This result brings new perspective Vatutin’s dichotomy, claiming in regime age-dependent reproduction, an extant population either contains large short-living individuals or consists few long-living individuals.