Preservation of some life length classes for age distributions associated with age-dependent branching processes

Under a Bellman-Harris age-dependent branching process, a parent lives a random length of time and at death produces a random number of offspring. The parent lifetimes have the common cumulative distribution G([dot operator]) and the number of offspring per parent has a common probability distribution. The current age distribution is known to converge to a limiting distribution A([dot operator]), called the Lotka limit law, which depends only on G([dot operator]) and the mean number of offspring. Besides the age distribution, we also consider a residual life distribution and a total life distribution. We investigate the preservation of classes of life lengths for these distributions and their limits.