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Week 12: Branching processes Sren Asmussen #
 

Summary: Simulation
Week 12: Branching processes
Søren Asmussen #
November 17, 2004
The Galton­Watson process
The process in question goes back to the late 19th century and the initial mo­
tivation was to compute the probability of survival of heir names. X 0 is the
number of present carriers of the name, X 1 the number of their male children,
X 2 the number of sons sons and so on. The name will die out (the family be­
come extinct) if Xn = 0 for some n (then also Xm = 0 for all m # n) so the
extinction probability is
P # Xn for some n # = P # Xn for all large n # .
For the modeling, let Z n,i be the number of (in this case male) children of
individual i in the nth generation. Then
Xn+1 =
Xn
# i=1
Z n,i
The Z n,i are assumed i.i.d. and we write f j = P(Z n,i = j), j = 0, 1, 2, . . .. When
X 0 = 1, a graphical illustration is often given in terms of a family tree, see the

  

Source: Asmussen, Søren - Department of Mathematical Sciences, Aarhus Universitet

 

Collections: Mathematics