We investigate, in detail, the consequences of the assumption that ''scaling in the mean'' and Koba-Nielsen-Olesen (KNO) scaling remain valid at asymptotic energies for which p/sub L/ very-much-greater-than p/sub T/, m and very-much-greater-than 1. We argue that the scaling function phi (t) can be fit by the simple function e/sup -//sup t/ with no free parameters. We show that, asymptotically, the semi-inclusive distributions satisfy Feynman scaling and vanish at x = 0, and that the inclusive distributions satisfy scaling in the mean, vanish at x = 0, and break Feynman scaling through an implicit s dependence through the variable . For Slattery's fit to the KNO function, we obtain the inclusive distributions for various values of . We assume that scaling in the mean holds for two-particle semi-inclusive distributions and obtain the normalization conditions for the two-particle scaling function. We obtain an expression for the two-particle inclusive distribution and define correlation functions in terms of the scaling functions. We show explicitly that even if the semi-inclusive distributions factorize, the inclusive distributions do not.