Unlock: Pareto Distribution

The Pareto distribution is the canonical power-law on a half-line. The Type I parameterization has survival function (x_m/x)^alpha for x at least x_m. The shape parameter alpha is the tail index. Three regimes of alpha matter for the law of large numbers and the central limit theorem: alpha at most 1 has no finite mean and breaks the LLN; 1 < alpha at most 2 has finite mean but infinite variance so the standard CLT fails (generalized CLT to a stable law); alpha greater than 2 admits both LLN and CLT in the usual form. Applications: wealth, city sizes, file sizes, network degree, insurance severity. The 80/20 'Pareto principle' is a specific case requiring alpha approximately 1.16.