Abstract
The Bellman–Harris branching processbranching processBellman-Harris is more general than the processes considered in the preceding chapters. Lifetimes of particles are nonnegative random variables with arbitrary distributions. It is described as follows. A single ancestor particle is born at t = 0. It lives for time τ which is a random variable with cumulative distribution function \documentclass[12pt]{minimal}
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$G(\tau)$
\end{document}. At the moment of death, the particle produces a random number of progeny according to a probability distribution with pgf f(s). Each of the first generation progeny behaves, independently of each other and the ancestor, as the ancestor particle did, i.e., it lives for a random time distributed according to \documentclass[12pt]{minimal}
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$G(\tau)$
\end{document} and produces a random number of progeny according to f(s). If we denote Z(t) the particle count at time t, we obtain a stochastic process \documentclass[12pt]{minimal}
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$\{Z(t),\ t\geq 0\}$
\end{document}. This so-called age-dependent process is generally non-Markov, but two of its special cases are Markov: the Galton–Watson process and the age-dependent branching process with exponential lifetimes. The Bellman–Harris process is more difficult to analyze, but it has many properties similar to these two processes.