Convergence of state distributions in multi-type Bellman-Harris and Crump Mode-Jagers branching processes

Abstract

Athreya and Kaplan proved recently that the age distribution in a one?type supercritical Bellman–Harris process converges with probability one to a deterministic distribution, assuming either a j log j hypothesis on the offspring distribution or a tightness condition on the lifetime distribution. In this thesis, the ideas contained in that work are exploited further to prove the convergence of the state distribution in a multi?type Crump–Mode–Jagers branching process in which, unlike in the Bellman–Harris case, the particles can produce offspring throughout their lifetime, thereby making the process more realistic (also, the offspring production measures are not assumed to be independent of the lifetime variables). A rigorous construction of the Crump–Mode–Jagers branching model is first provided using the Ikeda–Nagasawa–Watanabe theory. Next, the convergence in probability of the state distribution in the supercritical case is proved under a finite first?moment hypothesis, and the convergence with probability one under a j log j hypothesis. In the critical case, conditioned on non?extinction, the convergence in probability is proved under the hypothesis that the population size, conditioned on non?extinction, goes to infinity in distribution, and the hypothesis lim?t??q(t)s=0\lim_{t \to \infty} q(t)^s = 0limt???q(t)s=0 for each fixed sss, where q(t)q(t)q(t) is the probability of extinction by time ttt. (These hypotheses are shown to hold automatically for a multi?type Bellman–Harris process under a tightness condition on the lifetime distributions.) The above results are then specialized to the multi?type Sevastyanov process (in which the particles produce offspring only at the time of their death, but the offspring distributions are not necessarily independent of the lifetime distributions) and the multi?type age?dependent birth?and?death process (introduced by Kendall), by translating the hypotheses in terms of the lifetime and offspring distributions in the former case, and the birth and death rates in the latter. Finally, for the supercritical multi?type Bellman–Harris process, the j log j hypothesis is eliminated and the almost sure convergence of the age and type distribution is proved under a tightness condition on the lifetime distributions. Also, in the critical case, conditioned on non?extinction, the convergence of the age and type distributions is established under the same tightness condition on the lifetime distributions, thus extending fully to the multi?type case the results of Athreya and Kaplan.