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Extinction 135 Example 5.5: Near-critical binary splitting. The theorem applies to binary splitting where
 

Summary: Extinction 135
Example 5.5: Near-critical binary splitting. The theorem applies to binary splitting where
the probability of division is p(z) = 1/2+1/(2z), and q(z) = 1- p(z) is the probability of
no children, if population size is z. Thus, m(z) = 1 + 1/z and the process is supercritical,
but approaches criticality as z increases.
The probability generating function of the offspring number is fz(s) = q(z) + p(z)s2
.
We take u = 2 in Equation (5.118) to obtain
E
1
Zn+1 + 2
|Zn = z =
1
0
(q(z) + p(z)s2
)z
s ds
= 1/2
1
0

  

Source: Alsmeyer, Gerold - Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster

 

Collections: Mathematics