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