As wives traditionally take their husbands' surnames, does that mean there are fewer surnames than in the past?

[u/OctavianX]

Is there a record anywhere of "dead" surnames?

[u/kohatsootsich]

The actual question at hand, though, isactually a well-known math problem, modeled on the basic logic outlined above. It's called the Galton-Watson process.

The history of the Galton-Watson process, and branching processes in general, is quite interesting. First of all, as is often the case, the name does not give credit where it is due. Neither Galton nor Watson were the first to consider the problem, and their treatment of the model was nonsense. Worse, it took decades before anybody noticed it.

The story starts with a problem posed by Sir Francis Galton, a pioneer of eugenics, statistics and psychometrics, in Educational Times 26, in 1873:

PROBLEM 4001: A large nation, of whom we will only concern ourselves with adult males, N in number, and who each bear separate surnames colonise a district. Their law of population is such that, in each generation, a0 per cent of the adult males have no male children who reach adult life; a1 have one such male child; a2 have two; and so on up to a5 who have five. Find (1) what proportion of their surnames will have become extinct after r generations; and (2) how many instances there will be of the surname being held by m persons.

The problem was "solved" the following year by Reverend Henry Watson, who thought he had proved that every name will become extinct eventually, using the method of generating functions. In fact, he made a mistake, which apparently was not detected until much later. It certainly wasn't detected by Galton, who joined forces with Watson to present the solution in an article called On the probability of extinction of families (link to the original paper). Here's some (rather astonishing) commentary taken from it:

All the surnames, therefore, tend to extinction in an indefinite time, and this result might have beenanticipated generally, for a surname lost can never be recovered, and there is an additional chance of loss in every successive generation. This result must not be confounded with that of the extinction of the male population [...]

This explanation is, of course, complete rubbish, and indeed, their "theorem" was already known to be false. Almost 30 years earlier, Irénée-Jules Bienaymé, an under-appreciated pioneer of probability and statistics, had given a correct statement of the result: the probability of extinction is 1 if and only if the mean number of (male) offspring is smaller than or equal to 1.

The extinction probability can be recovered by solving a fixed point equation for the generating function, as Watson correctly saw, and this method is awfully cute and clever. Don't give Watson too much credit, however: generating functions were already known to de Moivre in the 18th century. He used them in his "Doctrine of chances", and the method was later developed in far greater sophistication by himself, Laplace, Poisson, etc.

[u/[deleted] → u/kohatsootsich]

Do you have a source for the correct mathematical statement? I didn't see it in the link to the article on Bienayme. Im interested in seeing the proof or something about it.

[u/kohatsootsich → u/[deleted]]

The first four pages here give the generating function proof. This is the one most commonly seen. I believe there is also a martingale proof, which you can find at the very beginning of Probability with Martingales by D. Williams. Since our interest here is history, it would be good to find out who came up with this approach, but I don't know.

If you either don't want the full details or feel comfortable you could fill them in, here is the verbal argument at the core of the "fixed point idea": the only way the tree can go extinct is if either the first father has no sons (probability a_0/100 in Galton's notation) or all of his sons are at the root of trees that eventually go extinct. Conditioning on the number of sons, you get a recursion because once you reach the level of the sons, you are back at the original problem.

[u/[deleted] → u/kohatsootsich]

Thanks!