Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 2, 256--264

On Seneta's Constants for the Supercritical Bellman-Harris Process with $E(Z_+ \log Z_+) = \infty$

Wolfgang P. Angerer, Universidad Aut\'onoma Metropolitana, M\'exico D.F.

SUMMARY. For a finite mean supercriticial Bellman-Harris process, let $Z_t$ be the number of particles at time $t$. There exist numbers $\chit$ (the Seneta constants) such that $\chit Z_t$ converges almost surely to a non-degenerate limit. Furthermore, $\chit \propto e^{-\beta t} \cL(e^{-\beta t})$, where $\beta$ is the Malthusian parameter, and $\cL$ is slowly varying at zero. We obtain a characterization of the slowly varying part of the Seneta constants under the assumption that the life-time distribution of particles is strongly non-lattice.

AMS (2000) subject classification. Primary 60J80.

Key words and phrases. Branching processes, renewal theory.

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