If the ospring distributions is pk and the generating

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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,...
View Full Document

This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

Ask a homework question - tutors are online