Galton-Watson process

The Galton-Watson process is a arising from 's statistical investigation of the extinction of.

History


There was concern amongst the s that surnames were becoming extinct. Galton originally posed the question regarding the probability of such an event in the of, and the Reverend  replied with a solution. Together, they then wrote an paper entitled On the probability of extinction of families. However, the concept was previously discussed by ; see Heyde and Seneta 1977; though it appears that Galton and Watson derived their process independently. For a detailed history see Kendall (1966 and 1975).

Concepts
Assume (as was taken quite for granted in Galton's time and is still the most frequent occurrence in many countries), that surnames are passed on to all male children by their father. Suppose the number of a man's sons to be a  on the set { 0, 1, 2, 3, ...}. Further suppose the numbers of different men's sons to be random variables, all having the same distribution.

Then the simplest substantial mathematical conclusion is that if the average number of a man's sons is 1 or less, then their surname will surely die out, and if it is more than 1, then there is more than zero probability that it will survive forever.

Modern applications include the survival probabilities for a new gene, or the initiation of a, or the dynamics of s in their first generations of spread, or the chances of  of small  of s; as well as explaining (perhaps closest to Galton's original interest) why only a handful of males in the deep past of humanity now have any surviving male-line descendants, reflected in a rather small number of distinctive.

A corollary of high extinction probabilities is that if a lineage has survived, it is likely to have experienced, purely by chance, an unusually high growth rate in its early generations at least when compared to the rest of the population.

Mathematical definition
A Galton-Watson process is a stochastic process {Xn} which evolves according to the recurrence formula X0 = 1 and


 * $$X_{n+1} = \sum_{j=1}^{X_n} \xi_j^{(n+1)}$$

where for each n, $$\xi_j^{(n)}$$ is a sequence of IID natural number-valued random variables. The extinction probability is given by


 * $$\lim_{n \to \infty} \Pr(X_n = 0)$$

and is equal to one if E{&xi;1} &le; 1 and strictly less than one if E{&xi;1} &gt; 1.

The process can be treated analytically using the method of s.

If the number of children &xi; j at each node follows a Poisson distribution, a particularly simple recurrence can be found for the total extinction probability xn for a process starting with a single individual at time n = 0:


 * $$x_{n+1} = e^{\lambda (x_n - 1)} $$

giving the curves plotted above.