Critical age-dependent branching process diffusions

Two diffusions are derived as the limits in finite dimensional distributions of appropriately conditioned and scaled critical age-dependent branching processes. A technical lemma about the asymptotic behavior of the joint generating function is used to overcome the difficulties introduced by the non-Markov nature of the process. The results are extensions of those of Lamperti and Ney 5] for Galton-Watson processes. Also, the “age-dependent Q-process” is defined and its transition probabilities obtained.