A remarkable phenomenon in probability theory is that of *universality* – that many seemingly unrelated probability distributions, which ostensibly involve large numbers of unknown parameters, can end up converging to a universal law that may only depend on a small handful of parameters. One of the most famous examples of the universality phenomenon is the central limit theorem; another rich source of examples comes from random matrix theory, which is one of the areas of my own research.

Analogous universality phenomena also show up in *empirical* distributions – the distributions of a statistic $latex {X}&fg=000000$ from a large population of “real-world” objects. Examples include Benford’s law, Zipf’s law, and the Pareto distribution (of which the Pareto principle or *80-20 law* is a special case). These laws govern the asymptotic distribution of many statistics $latex {X}&fg=000000$ which

- (i) take values as positive numbers;
- (ii) range over many…

View original post 4,602 more words