5. The Bellman–Harris Process

The Bellman–Harris branching process 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 \(G(\tau)\). 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 \(G(\tau)\) 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 \(\{Z(t),\ t\geq 0\}\). 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.