CONVERGENCE OF A SEQUENCE OF TRANSFORMATIONS OF DISTRIBUTION FUNCTIONS

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) = μ1−1 –∫0x[1 – Gn−1(y)]dy for x > 0 and Gn(x) = 0 for x ≤ 0, where μ1n−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) lim cn/cn−1 exists. Finally, criteria for the existence of a sequence {cn} such that {Gn(cnx)} converges to aproper 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. © 1969 by Pacific Journal of Mathematics.