I'm reading up on some continuous-time branching process theory and I came across a statement in slide 36 of the following:
http://www.stats.ox.ac.uk/?a=5508
If I may be brief, the author basically singles out a class of branching processes called Malthusian processes, defined by the fact that there is a number $\alpha$ satisfying $$ \int_0^{\infty} e^{-\alpha t} \mu(dt) = 1 $$ Where $\mu$ is the reproduction function of the branching process.
Question: But when is a process not Malthusian?
The only case in which I can think of is the case where the branching process "explodes," i.e. the population size is infinite for some $t$. This is apparently possible with positive probability if $\mu(0) > 1$.
But if we assume that $\mu(0) < 1$, $\mu(t)<\infty$ for some $t>0$, and $\mu(\infty)<\infty$ as is done at the top of that slide, then I am pretty sure this guarantees that the branching process is finite almost surely (from a theorem in Jagers, Branching Processes with Biological Applications). What case am I missing?
EDIT: Forgot the very important condition that $\mu(\infty)<\infty$. Sorry.