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If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that 
converges 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 that 
converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of 
for some constant α > 0.
