A martingale approach to supercritical (CMJ) branching processes
A supercritical Crump-Mode-Jagers branching process \(\{Z_ t^{\phi}\}\) counted with a general random characteristic \(\phi\) is considered. It is shown that convergence in probability of \(\{e^{-\alpha t}Z_ t^{\phi}\}\) is equivalent to convergence in distribution, where \(\alpha\) is the Malthusian parameter. Norming constants C(t) are identified such that \(\{C^{-1}(t)Z_ t^{\phi}\}\) converges almost surely to a nondegenerate limit under mild restrictions on \(\phi\). The key idea rests upon the identification of a martingale derived from a weakly convergent subsequence of \(\{C^{-1}(t)Z_ t^{\phi}\}\).
supercritical Crump-Mode-Jagers branching process
scientific article; zbMATH DE number 3942671