Total family size
Section of Galton–Watson branching, part of Stochastic processes
Total family size
| Type | Section |
| Taught in | Galton–Watson branching |
| Part | Stochastic processes |
Let T=\sum_{n\ge 0} Z_n, including the initial ancestor. If m<1, the partial sums increase to T. The monotone convergence theorem, stated on Expectation and convergence, lets the expectation pass inside the sum:
\mathbb{E} T=\sum_{n\ge 0}\mathbb{E} Z_n=\sum_{n\ge 0}m^n=\frac{1}{1-m}.
At m\ge 1 the geometric series diverges, so the expected total size is infinite. An infinite expectation does not mean that an infinite family tree occurs with positive probability. In the critical non-degenerate case, extinction is certain, and the total family size still has infinite mean. An infinite expectation is not a typical outcome.
The same subcritical formula comes from a first-generation decomposition. Let T_1,T_2,\ldots be independent copies of T, one from each child of the ancestor. Then
T\;\stackrel{d}{=}\; 1+\sum_{i=1}^{\xi} T_i.
If the expectation is finite, \mathbb{E} T=1+m\,\mathbb{E} T, so \mathbb{E} T=1/(1-m). The algebra uses a finite expectation. At m\ge 1 that step would treat an infinite number as if it were finite.
Used in
See also
Offspring probability generating function · Extinction criterion · Galton–Watson process · Mean and variance by generation · Extinction probability · Subcritical decay, and what the model leaves out · Other offspring laws · Generations and dependence