It's correct, of course, as the problem is solved, but most people would attempt a ballpark approximation in the end!
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."
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.)
1) Parametric models for late life mortality:
A useful mortality model, which has a logistic form, is: h(x) = (alpha exp(betax))(1+alpha exp(beta*x)). One can also add an additive term, often denoted as gamma or c. Please see for a comparison for late life models:
In case you speak German, the German Society of Actuaries has an interesting comparison of models:
2) An article which estimates the constant hazard at advanced ages:
I hope this helps a bit!
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 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?")
I posted a historical response there, so hopefully that settles the issue.
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."
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.
tq = teaser question