Multiplicative processes as a source of fat-tail distributions

Many works in the literature are in favour of the universality of power-laws. However, more recently, a significant amount of research has exposed more subtle details about this subject. In this paper we present two generalisations that aim to solve these possible antagonistic conceptions in a mathematical framework. The first generalisation allows us to show that a vast range of power-laws can be produced through multiplicative process (for example through a single positive feedback mechanism). The second generalisation shows, by solving Lambert's transcendental equations, that the space of solutions where the Pareto distribution is equal to the distribution produced by the power multiplicative transformation (for a number of input distributions) is infinite.