There's an excellent quotation attributed to Alexandre Grothendieck, one of the greatest mathematicians alive. At a seminar he was giving on analytic number theory, someone suggested that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck replied "All right, take 57."
From Davis and Hersh's excellent book "The Mathematical Experience":
Professor John Wermer tells the story of how, when he was an undergraduate, he took a course in projective geometry from Oscar Zariski, one of the foremost figures in the field of algebraic geometry. Zariski's course was exceedingly general and Wermer, as a young student, was occasionally in need of clarification. "What would you get," he asked his teacher, "if you specialized the field F to the complex numbers?" Zariski answered, "Yes, just take F as the complex numbers."
The student doesn't understand the general result, so asks for an explanation in a simple special case. The teacher fully understands the general result, and sees through to the special case like light goes through glass. So he just says yes, that's the special case, and sees no need to elaborate, because establishing the correspondence of the general to the specific is too obvious to bother putting in to words.
It seems to me that we can get a pretty decent approximation as follows:
1. At any given time, Pr(oldest person <= age x) = Pr(one person <= age x)^N. So (over time) the median oldest-person age is the (1-2^-1/N) quantile of the age distribution. (For large N, this is roughly 1-log(2)/N.) (You can get that from actuarial tables, or use the Gompertz-Makeham approximation.)
2. So, crudely, the time between oldest-person deaths is comparable to either the expected lifetime of a person of the age found in step 1, or 1/Pr(someone of that age dies in a given year). (Both are approximations. The former will give shorter inter-death times.)
3. According to Wikipedia (which is always right, except when it's wrong), once you get old enough it's a decent approximation to say that that a Very Old Person has about a 50% chance of making it through any given year, and that figure doesn't depend very much on exactly how old they are.
Which would suggest that we should get a new oldest person about once every two years, and that for decent-sized populations (say, 1000 or more) the figure should depend only very weakly on population size.
If #3 is correct and at very advanced ages the mortality rate is roughly independent of age, it seems like this result shouldn't actually depend much on the details of the probability distributions. (The oldest person alive will almost always be very old.)
(You'd get quite different results if, e.g., there were a hard divinely-appointed cutoff at some particular age.)
As pointed out in 3.) the mortality hazard h(x) does not increase exponentially at very high ages as it does for ages 30 up to let's 80 or 90. Until then the Gompertz-Makeham hazard h(x)= alpha * exp(betax) + gamma approximates adult mortality very well. At higher ages mortality begins to decelerate and, most likely, reaches a plateau at about age 110.
Do you have a link for 3? It would probably affect my answer as I was assuming that the probability of dying between ages x and x+1 increases as a power law after age 60 (which fits well for 60 <= x <= 100, but I don't have data for beyond 100).
All I have for #3 is a brief comment on a Wikipedia page. I wouldn't trust it much, and would in fact expect your model to be nearer the truth.
In fact, I'd expect your model to be nearer the truth than it actually is. It seems that the true oldest-person death rate is on the order of 2 per year; my argument yields one per 2 years, your simulation yields one per 1.6 years; is it possible that there's a bug still lurking in your simulation code? (I know you fixed one already! ... and I see you've already responded to my asking the same question over on math.se. [EDIT to add: and I think I may have found a bug, which would make your code underreport oldest-death events. I've reported it on math.se.])
What happens as you twiddle the parameters of your power law to make mortality increase more sharply with increasing age? What do you need to do (if it's possible at all) to get the oldest-person death rate up to 2 per year?
It's conceivable that there's some bias in the reporting of deaths of oldest living people that causes too many to be reported -- but I haven't been able to think of any remotely plausible mechanism that would have that effect.
The latest iteration of the code (I fixed the bug you pointed out, and another one that I spotted myself) reports 1 death per 0.66 years, which is close to 2 per year. The remaining difference, as you said, could be due to the mortality rate in my model not accelerating fast enough past 100 years old. I'll have a play and let you know.
The person who asked the question on math.stackexchange.com referred to news reports, and is asking what is essentially a historical question, so the question really should have been asked on a question-and-answer site about historical research rather than on a site about mathematics. That's why the answers are so irrelevant to the nature of the question.
The Nexis commercial database of news stories may be comprehensive enough these days to answer a question like that in detail going back to your own birth year. It would cost money to do the Nexis search, and you'd probably have to pay someone to pore through the search results and edit a document that would accurately summarize the results, but this should be a solvable problem these days.
(As another comment here has already pointed out, the basic answer is "Every time someone becomes the oldest person in the world, that person eventually dies," but I take it that the question actually asked means "How often does the identity of the 'oldest person in the world' change to being a new individual?")
Thank you for that. I see the French woman Jeanne Louise Calment (21 February 1875 – 4 August 1997) who long held the title of world's oldest person is an impressive outlier.
I heard a description of the life of Calment by a local researcher who participates in longitudinal studies of extreme old age. She outlived her husband, and all her (few) descendants. She only gave up smoking when she became so blind that she could no longer see the end of a cigarette to light it. The researcher told me this story (taken here from Wikipedia) about how she kept living in her house after she became aged and widowed:
"In 1965, aged 90 years and with no heirs, Calment signed a deal to sell her former apartment to lawyer André-François Raffray, on a contingency contract. Raffray, then aged 47 years, agreed to pay her a monthly sum of 2,500 francs until she died. Raffray ended up paying Calment the equivalent of more than $180,000, which was more than double the apartment's value. After Raffray's death from cancer at the age of 77, in 1995, his widow continued the payments until Calment's death."
If anyone is curious, I just updated the post with a quickie analysis of the GRG data (I'm on the mailing list and brought this and the post up and someone else posted the relevant table). Turns out the actual timing differs on whether you're looking at mean or median, because Calment screwed things up by living a ridiculously long time.
Since the question is about the average, it seems the question can be simply answered: the next-oldest person, on average, will die at the same age as the current oldest person. (Obviously, assuming an unchanging mortality rate over time, but this sounds like a valid approximation for very old people. There seem to be a wall around 122.) Thus the average waiting time between two oldest-person deaths is the average age difference between the two oldest living persons.
Edit: actually, an even simpler first-order approximation is possible. If we take at face-value that very old people have a 50% chance of living one more year, and that this statistics holds whatever the baseline date, then upon the death of the eldest person, the average life-span of the next eldest person is 1/2 + 1/4 + 1/8 ... IOW, 1 year.
ps. All are from Japan, US, Italy or UK. I suspect that may be down to record keeping as much as lifestyle.
For example, a friend's wife is from Turkey and doesn't know how old she is as her date of birth was never recorded; one year her parents just made a guess saying "well, you were born in summer and you look like an 8 year old, so we'll stick you down as 21st June 1965".