By Janos Galambos, Samuel Kotz

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F(x) is continuous but not strictly increasing on a semiline. F(a) > 0 and F(b) = F(a). z < a. ) is continuous, we can choose two values z I < z 2 49 such that 0 < F(Zl) < F(z 2) < F(a) and a - z I < ยข, where ~ > 0 is arbitrary. We shall impose one more restriction on the choice of z I and z 2 later. Now if we start with (19), we again get (20), but the condition that "almost all z > 0" refers here to the measure Q(D) on the Borel sets D of the real line for which Q(D) = F(d) - F(c) for intervals D = (c,d).

Ahsanullah and M. S. Huang (1974a). For a characterization of the uniform distribution by distributional properties of dr: n of the type that it does not depend on r and F(x) is either super-additive or subadditive, see J,S. C. Arnold and M. Ghosh (1977). 3. Independence of functions of order s t a t i s t i c s . We present in this section a number of characterization theorems based on the independence of certain functions of order staitstics. In addition to the actual results, we also wish to draw attention to the methods of proof.

From the proof, it is clear that the theorem can be modified in the same manner as the lack of memory characterizations were modified Chapter. in the beginning of this For example, if "all s,t ~ 0" is changed to "all s ~ 0 and two values of t, t I and t2, say, for which tl/t 2 is irrational," the conclusion remains to hold. Other variants may be deduced from previous discussions. The assumption of contin- uity at zero cannot be dropped, however, because a jump of the distribution of X at zero has no effect on the other conditions.