 
Summary: Development of Populations 177
6.6 Growth of Populations with Sexual Reproduction
G. Alsmeyer
In Section 5.9, we studied the effect of sexual reproduction on criticality and ex
tinction risk of branching processes. Here, we consider the ultimately supercritical
case (m > 1) and take a look at the question of how such populations grow in the
event of survival. Since m describes the asymptotic growth rate per generation
if the population becomes large, it is not unreasonable to believe that Zn grows as
mn
in the event of survival. However, even for the simple GaltonWatson process,
the famous KestenStigum theorem has already shown that this is true only under
an additional condition on the offspring distribution. Defining the normalized pro
cess
Wn =
Zn
mn
, n 0 , (6.100)
we have that
E[Wn+1Wn = imn
