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Unformatted text preview: as k children. If µ = 1 and
p1 < 1 then P (Zn > 0) ! 0.
Proof. When µ = 1, Zn is martingale and hence by Theorem 5.17 converges to
a limit. Since Zn is integer valued, if Zn (! ) ! j then we must have Zn (! ) = j
for n N (! ) but this has probability 0 if p1 < 1.
Supercritical. If µ > 1 then Zn /µn ! W as n ! 1. If we can show that P (W > 0) > 0 then we can conclude that the population grows exponentially. The ﬁrst step is to show that if µ > 1 then
P (Zn > 0 for all n) > 0. To approach this problem, we will consider a version of 174 CHAPTER 5. MARTINGALES the branching process in which at each time only one individual is chosen to reproduce. In this case as long as the population size Sn > 0, Sn+1 = Sn 1+Yn+1
where P (Yn+1 = k ) = pk . Since Yn+1 0 this is a left-continuous random walk
with steps Xm = 1 + Ym run until T0 = min : Sn = 0}. EXn = µ 1 so if
µ > 1, it follows from Theorem 5.16 that
P1 (T0 < 1) = e↵
where ↵ < 0 is the solution of E exp(↵Xi...
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