The age distribution for a supercritical Bellman-Harris process is proven to converge in probability to a deterministic distribution under assumptions slightly more than finite first moment. If the usual "$j \log j$" condition holds, then the convergence can be strengthened to hold w.p. 1. As a corollary to this result, the population size, properly normalized is shown to converge w.p. 1 to a nondegenerate random variable under the "$j \log j$" assumption.
@article{1176996179,
author = {Athreya, K. B. and Kaplan, N.},
title = {Convergence of the Age Distribution in the One-Dimensional Supercritical Age-Dependent Branching Process},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 38-50},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996179}
}
Athreya, K. B.; Kaplan, N. Convergence of the Age Distribution in the One-Dimensional Supercritical Age-Dependent Branching Process. Ann. Probab., Tome 4 (1976) no. 6, pp. 38-50. http://gdmltest.u-ga.fr/item/1176996179/