A Martingale Approach to Supercritical (CMJ) Branching Processes

Abstract

A new method of tackling convergence properties of random processes turns out to be applicable to finite mean supercritical age-dependent branching processes. If $\{Z^\phi_t\}$ is a Crump-Mode-Jagers process counted with general characteristics $\phi$, convergence in probability of $\{e^{-\alpha t} Z^\phi_t\}$ follows from convergence in distribution. Under some mild restrictions on $\phi$, norming constants $\{C(t)\}$ are identified such that $\{C^{-1}(t)Z^\phi_t\}$ converges almost surely to a nondegenerate limit.