Exponential tails, and the Gumbel law
Section of A first look at extremes, part of Stochastic processes
Exponential tails, and the Gumbel law
| Type | Section |
| Taught in | A first look at extremes |
| Part | Stochastic processes |
Let X_i be exponential with rate \lambda, so \mathrm{P}(X>x)=e^{-\lambda x} for x>0. Set b_n=(\log n)/\lambda and a_n=1/\lambda. Then
\mathrm{P}\Bigl(\frac{M_n-b_n}{a_n}\le x\Bigr) =\Bigl(1-e^{-\lambda(b_n+a_n x)}\Bigr)^n =\Bigl(1-\frac{e^{-x}}{n}\Bigr)^n \to e^{-e^{-x}}.
The limit is the Gumbel distribution function. The algebra uses \lambda b_n=\log n and \lambda a_n=1, so the exponential tail contributes exactly e^{-x}/n at the moving level b_n+a_n x. Light tails of exponential type fall into this family under suitable conditions on F.
Used in
See also
Power-law tails, and the Fréchet law · A finite endpoint, and a Weibull extreme-value law · Extreme-value classification · Pareto distribution