I am talking about world records of course.  Tyler Cowen linked to this Boston Globe piece about the declining rate at which world records are broken in athletic events, especially Track and Field.  (Usain Bolt is the exception.)

How quickly should we expect the rate of new world records to decline?  Suppose that long jumps are independent draws from a Normal distribution.  Very quickly the world record will be in the tail.  At that point breaking the record becomes very improbable.  But should the rate decline quickly from there?  Two forces are at work.

First, every new record pushes us further into the tail and reduces the probability, and hence freqeuncy, of new records.  But, because of the thin tail property of the Normal distribution, new records will with very high probability be tiny advances.  So the new record will be harder to beat but not by very much.

So the rate will decline and asymptotically it will be zero, but how fast will it converge to zero?  Will there be a constant K such that we will have to wait no more than nK years for the nth record to be broken or will it be faster than that?

I am sure there is an easy answer to this question for the Normal distribution and probably a more general result, but my intuition isn’t taking me very far.  Probably this is a standard homework problem in probability or statistics.

The Boston Globe piece is about humans ceasing to progress physically.  The theory could shed light on this conclusion.  If the answer above is that the arrival rate increases exponentially, I wonder what rate the mean of the distribution can grow and still give rise to the slowdown.  If the mean grows logarithmically?