 
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
