Suppose a positive random variable

with an absolutely continouse distribution

is our model for duration of life of an individual; consider the so called "excess life distribution"

This is a very interesting object to study when we speak abut longevity: what is the probability that the individual will live longer than

, given that
this individual already lived for longer than

. It is especially interesting to study when

is large.

According to the so-called Balkemaa-de Haan-Pickands theorem, if there is a sequence of constants

such that for

there is a limit,

then this

can only be of the following form:

Now we follow what is said in the book

DemgraphyBook, Lecture 15.1*:

Denote

is the force of mortality, corresponding to the distribution

. If the asymptotic relation

holds, the only choice for

is

with

a generic notation for constant, and the only possible limit for

is the exponential distribution:

What is unusual and needs investigation is this: the (1) is true for

*all* distributions we discussed in the book, and yet the longevity data (for those over 90) for New Zealand population, does not quite agree with the exponential distribution it has to follow. Moreover, serious researchers studied similar data on longevity in other countries and also suggested long tail distribution above. But ... (1) is true and the limit must be exponential :-). Of course, one should understand what is happening.