On Seneta's constants for the supercritical Bellman-Harris process with $E(Z_+ \log Z_+) = \infty$

Abstract

For a finite mean supercriticial Bellman-Harris process, there exist numbers $\chi_t$ (the Seneta constants) such that $\chi_t$ times the size of the population at time $t$ converges almost surely to a non-degenerate limit. We obtain a characterisation of the slowly varying part of the Seneta constants under the assumption that the life-time distribution of particles is strongly non-lattice.