A generalized age-dependent branching process and its limit distribution (Q1069570)

We study an age-dependent continuous time branching process \(\{X_ N(t),t\geq 0\}\), where \(X_ N(t)\) represents the population size at time t, and N is its initial size at time zero. By taking an appropriate dependence of the ''splits rates'' of the process on the initial size N, we show that the process \(\{X_ N(t)-N\), \(t\geq 0\}\) converges weakly as \(N\to \infty\), to a compound Poisson process \(\{\) X(t),t\(\geq 0\}.\) A similar result was proved earlier by \textit{H. H. Stratton} and \textit{H. G. Tucker} [Ann. Math. Stat. 35, 557-565 (1964; Zbl 0245.60064)] but for nonhomogeneous Markov branching processes.