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Week 12: Branching processes Sren Asmussen
 

Summary: Simulation
Week 12: Branching processes
Søren Asmussen
November 17, 2004
The Galton-Watson 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. X0 is the
number of present carriers of the name, X1 the number of their male children,
X2 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 Zn,i be the number of (in this case male) children of
individual i in the nth generation. Then
Xn+1 =
Xn
i=1
Zn,i
The Zn,i are assumed i.i.d. and we write fj = P(Zn,i = j), j = 0, 1, 2, . . .. When
X0 = 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