Galton–Watson process
Section of Galton–Watson branching, part of Stochastic processes
Galton–Watson process
| Type | Section |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
Let Z_0=1. For n\ge 0,
Z_{n+1}=\sum_{i=1}^{Z_n}\xi_{n,i},
where the offspring variables \xi_{n,i} are independent and identically distributed non-negative integers, with \mathbb{P}(\xi=k)=p_k, mean m, and variance \sigma^2. The empty sum is 0, so the state 0 is absorbing: once the lineage is extinct it stays extinct.
Definition 1 (Offspring probability generating function) The offspring probability generating function is
f(s)=\mathbb{E}(s^{\xi})=\sum_{k\ge 0}p_k s^k, \qquad 0\le s\le 1.
It satisfies f(1)=1. If the mean is finite, f'(1-)=m. If \mathbb{E}[\xi(\xi-1)] is finite, f''(1-)=\mathbb{E}[\xi(\xi-1)].
Condition on the present generation. Each of the Z_n individuals contributes an independent factor f(s), so
\mathbb{E}(s^{Z_{n+1}}\mid Z_n)=f(s)^{Z_n}.
Write f_n(s)=\mathbb{E}(s^{Z_n}). Taking the expectation of the display gives f_{n+1}(s)=f_n\bigl(f(s)\bigr). With f_0(s)=s, the transform f_n is the n-fold composition f\circ f\circ\cdots\circ f. Composition, rather than multiplication, is the generating-function expression of generations nested inside generations. A sum of a fixed number of independent copies would multiply the transforms. Here the number of copies is itself the previous generation.
Used in
- Taught in Galton–Watson branching
- Also used in Stochastic processes · Discrete-time Markov chains
- All 3 questions on Galton–Watson branching
See also
Extinction criterion · Offspring probability generating function · Mean and variance by generation · Extinction probability · Subcritical decay, and what the model leaves out · Other offspring laws · Total family size · Generations and dependence