This is part III in a series on scale invariance, power laws, and regular variation, so you should definitely click on over to parts I and II if you haven’t read those yet.

In part II, we showed that the class of regularly varying distributions formalizes the notion of “approximately scale-invariant,” just like power-law tails formalize the notion of scale-invariant. The fact that regularly-varying distributions are exactly those distributions that are asymptotically scale-free suggests that they, in a sense, should be able to be analyzed (at least asymptotically) like they are simply power-law distributions. In fact, this can be formalized explicitly, and regularly varying distributions can be analyzed nearly as if they were power-law (Pareto) distributions as far as the tail is concerned. This makes them remarkably easy to work with and highlights that the added generality from working with the class of regularly-varying distributions, as opposed to working specifically with Pareto distributions, comes without too much added complexity.