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.
"Convergence of the Age Distribution in the One-Dimensional Supercritical Age-Dependent Branching Process." Ann. Probab. 4 (1) 38 - 50, February, 1976. https://doi.org/10.1214/aop/1176996179