Editing Galton–Watson process (section)

== Mathematical definition ==

A Galton–Watson process is a stochastic process {''X''<sub>''n''</sub>} which evolves according to the recurrence formula ''X''<sub>0</sub> = 1 and

:<math>X_{n+1} = \sum_{j=1}^{X_n} \xi_j^{(n)}</math>

where <math>\{\xi_j^{(n)} : n,j \in \mathbb{N}\}</math> is a set of [[Independent and identically-distributed random variables|independent and identically-distributed]] natural number-valued random variables.

In the analogy with family names, ''X''<sub>''n''</sub> can be thought of as the number of descendants (along the male line) in the ''n''th generation, and <math>\xi_j^{(n)}</math> can be thought of as the number of (male) children of the ''j''th of these descendants. The recurrence relation states that the number of descendants in the ''n''+1st generation is the sum, over all ''n''th generation descendants, of the number of children of that descendant.

The [[extinction probability]] (i.e. the probability of final extinction) is given by

:<math>\lim_{n \to \infty} \Pr(X_n = 0).\, </math>

This is clearly equal to zero if each member of the population has exactly one descendant.  Excluding this  case (usually called the trivial case) there exists
a simple necessary and sufficient condition, which is given in the next section.