A generalization of Goldstein's comparison lemma and the exponential limit law in critical Crump-Mode-Jagers branching processes

Abstract

Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.

Conditional limit theorems for general branching processes

“…Our proof is quite different from that of Holte (1976). His argument is a strengthening of that of Goldstein (1971) for the Bellman-Harris process involving a comparison with the Galton-Watson process governed by the same family-size distribution.…”

“…In the subcritical case, the finiteness and the value of the expectation of the limit law was not known, except for the Bellman-Harris process. In the critical case, the exponential limit was obtained by Durham (1971) under the condition that all moments of~(oo) are finite, and by Holte (1976), who required an unnatural assumption on the rate of convergence of E(Z(t» to (3-1f tL(dt).…”

“…Conditionallimit theorems for the population size in the general process were obtained by Ryan (1968) for the subcritical case, and by Durham (1971) and Holte (1976) in the critical case. The theorems we present are, however, stronger than these even when restricted to the special case of the population size.…”