Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process. (English) Zbl 0356.60048

Athreya, K. B.; Kaplan, N.

Ann. Probab. 4, 38-50 (1976).

A supercritical age-dependent Bellman-Harris branching process with offspring distribution \( \left\{p_{j}\right\} \), mean \( m \equiv \Sigma j p_{j}>1 \) and lifetime distribution \( G, G(0+)=0, G \) non-lattice, has, for any family tree \( \omega, Z(x, t, \omega) \equiv \) number of objects living at time \( t \) with age \( \leq x \). Let \( A(x, t, w) \equiv(z(x, t, w))(z(\infty, t, w))^{-1} \) denote the age distribution. Denote the Malthusian parameter \( \alpha: m \int_{0}^{\infty} \mathrm{e}^{-\alpha t} \mathrm{dG}(t) \) and the asymptotic age distribution \( A(x) \equiv\left(\int_{0}^{x} e^{-\alpha t}(1-G(t)) d t\right)\left(\int_{0}^{\infty} e^{-\alpha t}(1-G(t)) d t\right)^{-1} \). The authors have three major results: Theorem A: Under a slightly stronger than first mean assumption on \( G \), that \( A(x, t, w) \rightarrow A(x) \) in distribution as \( t \rightarrow \infty \). Theorem \( B \) : If \( \Sigma_{j p_{j}} \log j<\infty \), then \( A(x, t, w) \rightarrow A(x) \) with probability one as \( t \rightarrow \infty \). Theorem \( C \) : If \( \sum \mathrm{jp}_{j} \log j<\infty \), then \( \lim _{t \rightarrow \infty} Z(t, \omega) e^{-\alpha t}=W(\omega) \) exists with probability. one and \( P(W(w)>0)=1 \). The method of proof consists in writing \( A(x, t+s, w) \) in terms of the age chart at time \( t \), and using the law of large numbers and a martingale convergence theorem.

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MSC:

60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J85	Applications of branching processes