2003, Volume 65, Pt. 2, 229--248

On The Supercritical Bellman-Harris Process With Finite Mean


K.B. ATHREYA, Cornell University, Ithaca, N.Y., USA and H.-J. SCHUH, Johannes Gutenberg-University, Germany

SUMMARY. Let $(Z_t)_{0\le t<\infty}$ be a supercritical age-dependent branching process with offspring distribution $(p_j)_{j=0,1,2,\ldots}$, offspring mean $1<m:=\sum_{j=1}^\infty j\,p_j<\infty$ and non-lattice life-length distribution $G$ with $(G(0+)=0)$. It is known (Cohn, 1982 and Schuh, 1982) that there exist constants $(C_t)_{t\ge0}$ such that $Z_t/C_t\; \mathop{\stackrel{{\rm a.s.}}{\lra}}\limits_{t\to\infty}\,\wt W$ with $0<P(0<\wt W<\infty)$, whose Laplace transform $\vp(s):= E(e^{-s\wt W})$ satisfies the functional equation $$\vp(s)=\int_0^\infty f(\vp(s e^{-\al y})) G(dy),$$%\eqno(*) where $f(s):= \sum_{j=0}^\infty p_js^j$ is the offspring generating function and $\al>0$ is the Malthusian parameter defined by $m\int_0^\infty e^{-\al y} G(dy)=1$. In the case $\sum_{j=1}^\infty p_j \,j\log j<\infty$ Athreya (1969) has proven analytically that $\vp(s)$ is (up to a multiple constant) the unique solution of the above functional equation. In this paper we extend this result by a probabilistic argument assuming only that $1<m<\infty$. We also extend Athreya's (1969) result on the absolute continuity of $\wt W$ to the present case under a mild restriction on $G$.

AMS (1991) subject classification. 60J80.

Key words and phrases. Bellman-Harris branching process, supercritical, Seneta constants, Laplace transforms, functional equation, absolute continuity.

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