The Galton Watson Process – Part II

The derivation of this equation is not shown here for brevity. If we recursively write out this equation we obtain:

\displaystyle G_n (s) = G_{n-1} \left( G(s) \right) \\ = G_{n-2} \left( G (G(s)) \right) \\ = G_{n-3} \left( G (G (G(s))) \right) \\ = \dots \\ = G_{1} \left( G (G (G(\dots (s)\dots ) )) \right) \\ = G \left( G (G (G(\dots (s)\dots ) )) \right) \ \ \ \ \ (15)

From this equation we see:

\displaystyle G_n = G \left( G (G (G(\dots (s)\dots ) )) \right) \ \ \ \ \ (16)

Then {G_{n-1}} is simply {G_{n}} that is one less recursive function of {G}, i.e.

\displaystyle G_{n-1} = G (G (G(\dots (s)\dots ) )) \ \ \ \ \ (17)

Then by plugging Eq.17 into Eq.16 we obtain:

G_n \left( s \right) = G\left( G_{n-1} \left(s\right) \right) \ \ \ \ \ \mbox {for}\, n = 1,2,3, \dots (18)

which is much more handy for calculating the probability of extinction than the previous form, G_n (s) = G_{n-1} \left( G(s) \right). This relation is useful if we realize that the probability of extinction is identical to the probability generation function when s=0. This is because G_n(0) = P(X_n=0), where P(X_n=0) is the probability that the total number of people of the n-th generation (X_n) is equal to zero. Thus the linkage between the probability of extinction and the probability generation function is:

e_n = G_n \left(0 \right) = P \left(X_n = 0\right) \, \, \, \, \,(19)


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