# If the ospring distributions is pk and the generating

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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 &gt; 0 for all n) &gt; 0 is necessary for P (W &gt; 0) &gt; 0 but it is not su cient. Kesten and Stigum (1966) have shown: P (W &gt; 0) &gt; 0 X if and only if k1 pk (k log k ) &lt; 1. That result has a sophisticated proof, but it is not hard to show that P P Theorem 5.19. If k kpk &gt; 1 and k k 2 pk &lt; 1 then P (W = 0) = ⇢. Proof. We begin with the easy part: if P (W = 0) &lt; 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 ✓ &lt; 1 must be a ﬁxed point of theP generating function. P 2 To show that kpk &gt; 1 and k k k pk &lt; 1 are su cient for P (W &gt; n 2 0) &gt; 0,...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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