
190 Right-tail behavior of
b
F
n
.
This implies that a
n
actually converges to 0 as c/x
k−1
0n
and that G is
again the exponential distribution function.
For the Gompertz distribution, with the force of mortality (2.4), the
same is true again with a
n
∼ e
−cx
0n
.
Hence, for all distributions that we have considered, with asymp-
totically constant or increasing failure rates, the only possible limit for
the excess life distribution
˜
F
n
is the exponential distribution.
Exercise. It is very interesting to see what happens if the rate of
mortality increases even more rapidly than exponential function.
For example, try the function
µ(x) = exp(exp(cx)),
with, say, c = 1