Let F be the distribution
function (d.f.) of a nonnegative random variable (r.v.) X all of whose moments
μ_{n} = ∫
_{0}^{∞}x^{n}dF(x) exist and are finite. Define, recursively, the sequence {G_{n}} of
absolutely continuous d.f.’s as follows: put
G_{1}(x)  = μ_{1}^{−1} ∫
_{0}^{x}[1 − F(y)]dy for x > 0 and  
 G_{1}(x)  = 0 for x ≤ 0; for n > 1, let  
 G_{n}(x)  = μ_{1,n−1}^{−1} ∫
_{0}^{x}[1 − G_{
n−1}(y)]dy for x > 0 and  
 G_{n}(x)  = 0 for x ≤ 0, where  
 μ_{1,n−1}  = ∫
_{0}^{∞}[1 − G_{
n−1}(y)]dy.   
It is shown that if F is distributed on a finite interval, then the sequence
{G_{n}(x∕n)} converges to the simple exponential d.f. On the other hand, if F(x) < 1
for all x > 0 and G_{n}(c_{n}x) → G(x), where G is a proper d.f. and {c_{n}} is a sequence
of constants such that {c_{n}∕c_{n−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) c_{n} is asymptotically equal to μ_{n+1}∕b(n + 1)μ_{n} where
b = ∫
_{0}^{∞}[1 −G(u)]du and (d) limc_{n}∕c_{n−1} exists. Finally, criteria for the existence of
a sequence {c_{n}} such that {G_{n}(c_{n}x)} 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.
