Power-law tails, and the Fréchet law
Section of A first look at extremes, part of Stochastic processes
Power-law tails, and the Fréchet law
| Type | Section |
| Taught in | A first look at extremes |
| Part | Stochastic processes |
Suppose \mathrm{P}(X>x)=x^{-\alpha} for x\ge 1 and some \alpha>0, a Pareto tail. Set a_n=n^{1/\alpha} and b_n=0. For x>0,
\mathrm{P}(M_n/a_n\le x)=\bigl(1-(n^{1/\alpha}x)^{-\alpha}\bigr)^n=\Bigl(1-\frac{x^{-\alpha}}{n}\Bigr)^n\to e^{-x^{-\alpha}}.
The limit is the Fréchet distribution function. The heavy tail changes the scale from \log n to a power of n. Multiplying the sample size by a constant multiplies this scale by that constant to the power 1/\alpha.
Used in
See also
Exponential tails, and the Gumbel law · A finite endpoint, and a Weibull extreme-value law · Extreme-value classification · Pareto distribution