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Unformatted text preview: ) = 1. Letting ⇢ = e↵ this means
1= X p k ⇢k X or 1 k p k ⇢k = ⇢ k which is the result we found in Lemma 1.30: the extinction probability is the
ﬁxed point of the generating function in [0, 1).
P (Zn > 0 for all n) > 0 is necessary for P (W > 0) > 0 but it is not
su cient. Kesten and Stigum (1966) have shown:
P (W > 0) > 0 X if and only if k1 pk (k log k ) < 1. That result has a sophisticated proof, but it is not hard to show that
P
P
Theorem 5.19. If k kpk > 1 and k k 2 pk < 1 then P (W = 0) = ⇢. Proof. We begin with the easy part: if P (W = 0) < 1 the P (W = 0) = ⇢. If
we have Zn /µn ! 0 then this must be true for the branching processes started
by the Z1 individuals in generation 1. Breaking things down according to the
value of Z1 and letting ✓ = P (W = 0) we have
✓= X pk ✓ k k so ✓ < 1 must be a ﬁxed point of theP
generating function.
P
2
To show that
kpk > 1 and
k
k k pk < 1 are su cient for P (W >
n
2
0) > 0,...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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