Vol. 31, No. 2, 1969

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Convergence of a sequence of transformations of distribution functions

William Leonard Harkness and R. Shantaram

Vol. 31 (1969), No. 2, 403–415

Let F be the distribution function (d.f.) of a nonnegative random variable (r.v.) X all of whose moments μn = 0xndF(x) exist and are finite. Define, recursively, the sequence {Gn} of absolutely continuous d.f.’s as follows: put

G1(x) = μ11 0x[1 F(y)]dy for x > 0 and
G1(x) = 0 for x 0; for n > 1, let
Gn(x) = μ1,n11 0x[1 G n1(y)]dy for x > 0 and
Gn(x) = 0 for x 0, where
μ1,n1 = 0[1 G n1(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∕cn1} 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∕cn1 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.

Mathematical Subject Classification
Primary: 60.30
Received: 11 May 1967
Revised: 1 May 1969
Published: 1 November 1969
William Leonard Harkness
R. Shantaram