If Bₙ denotes the time of the first birth in the nth generation of an age-dependent branching process of Crump-Mode type, then under a weak condition there is a constant γ such that Bₙ/n → γ as n → ∞, almost surely on the event of ultimate survival. This strengthens a result of Hammersley, who proved convergence in probability for the more special Bellman-Harris process. The proof depends on a class of martingales which arise from a `collective marks' argument.
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J. F. C. Kingmán (1975) studied this question.