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We consider the critical Galton-Watson process with overlapping generationsstemming from a single founder. Assuming that both the variance of theoffspring number and the average generation length are finite, we establish theconvergence of the finite-dimensional distributions, conditioned onnon-extinction at a remote time of observation. The limiting process isidentified as a pure death process coming down from infinity.  This result brings a new perspective on Vatutin's dichotomy claiming that inthe critical regime of age-dependent reproduction, an extant population eithercontains a large number of short-living individuals or consists of fewlong-living individuals.

Critical branching as a pure death process coming down from infinity