[en] A generalization of the Koba-Nielsen-Olesen scaling law of the multiplicity distributions P(n) is presented. It consists of a change in the normalization point of P(n) compensated by a suitable change in the renormalized parameters and a rescaling. The iterative repetition of the transformation yields the sequence of higher-order moment distributions of P(n). Each member of this sequence may exhibit data collapsing behavior in case of violation of the original KNO scaling hypothesis. It is shown that the iterative procedure can be viewed as varying the collision energy, i.e. the moment distributions of P(n) can represent the pattern of pre-asymptotic KNO scaling violation. The fixed points of the iteration will be determined and a consistency test based on Feynman scaling is to be given. (author)