Let F be the distribution
function (d.f.) of a nonnegative random variable (r.v.) X all of whose moments
μn=∫0∞xndF(x) exist and are finite. Define, recursively, the sequence {Gn} of
absolutely continuous d.f.’s as follows: put
G1(x)
= μ1−1∫0x[1 − F(y)]dy for x > 0 and
G1(x)
= 0 for x ≤ 0; for n > 1, let
Gn(x)
= μ1,n−1−1∫0x[1 − Gn−1(y)]dy for x > 0 and
Gn(x)
= 0 for x ≤ 0, where
μ1,n−1
=∫0∞[1 − Gn−1(y)]dy.
It is shown that if F is distributed on a finite interval, then the sequence
{Gn(x∕n)} converges to the simple exponential d.f. On the other hand, if F(x) < 1
for all x > 0 and Gn(cnx) → G(x), where G is a proper d.f. and {cn} is a sequence
of constants such that {cn∕cn−1} is bounded, then (among other things)
it is shown that (a) the convergence is uniform, (b) G is continuous and
concave on [0,∞), (c) cn is asymptotically equal to μn+1∕b(n + 1)μn where
b =∫0∞[1 −G(u)]du and (d) limcn∕cn−1 exists. Finally, criteria for the existence of
a sequence {cn} such that {Gn(cnx)} converges to a proper d.f. are given. In
particular, it is shown that this sequence converges if F is absolutely continuous
with probability density function (p.d.f.) f and F has increasing hazard
rate.
