Almost sure convergence in Markov branching processes with infinite mean

Abstract

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a functionL, slowly varying at 0, such thatconverges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such thatconverges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence offor some constant α > 0.