The First Birth Problem for an Age-dependent Branching Process

If $B_n$ denotes the time of the first birth in the $n$th generation of an age-dependent branching process of Crump-Mode type, then under a weak condition there is a constant $\gamma$ such that $B_n/n \rightarrow \gamma$ as $n \rightarrow \infty$, 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.