A new method of tackling convergence properties of random processes turns out to be applicable to finite mean supercritical age-dependent branching processes. If $\{Z^\phi_t\}$ is a Crump-Mode-Jagers process counted with general characteristics $\phi$, convergence in probability of $\{e^{-\alpha t} Z^\phi_t\}$ follows from convergence in distribution. Under some mild restrictions on $\phi$, norming constants $\{C(t)\}$ are identified such that $\{C^{-1}(t)Z^\phi_t\}$ converges almost surely to a nondegenerate limit.
@article{1176992803,
author = {Cohn, Harry},
title = {A Martingale Approach to Supercritical (CMJ) Branching Processes},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 1179-1191},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992803}
}
Cohn, Harry. A Martingale Approach to Supercritical (CMJ) Branching Processes. Ann. Probab., Tome 13 (1985) no. 4, pp. 1179-1191. http://gdmltest.u-ga.fr/item/1176992803/