Universal emergence of local Zipf's law

Abstract: A plethora of natural and socio-economic phenomena share a striking statistical regularity, that is the magnitude of elements decreases with a power law as a function of their position in a ranking of magnitude. Such regularity is known as Zipf-Mandelbrot law (ZM), and plenty of problem-specific explanations for its emergence have been provided in different fields. Yet, an explanation for ZM ubiquity is currently lacking. In this paper we first provide an analytical expression for the cumulants of any ranked sample of i.i.d. random variables once sorted in decreasing order. Then we make use of this result to rigorously demonstrate that, whenever a small fraction of such ranked dataset is considered, it becomes statistically indistinguishable from a ZM law. We finally validate our results against several relevant examples.