The Three Types
Gumbel, Fréchet, and Weibull — the only possible limiting distributions for maxima.
The Fisher-Tippett-Gnedenko theorem is one of the most elegant results in probability theory. It states that the normalized maximum of i.i.d. random variables can only converge to one of exactly three types of distributions. This parallels how the CLT says sums converge to the Gaussian — but here, there are three possibilities, not one.
The Fisher-Tippett-Gnedenko Theorem
Fisher-Tippett-Gnedenko Theorem
If there exist sequences and such that the normalized maximum converges in distribution:
where is a non-degenerate distribution, then must be one of the following three types.
Type I: The Gumbel Distribution
Gumbel Distribution (Type I)
The Gumbel distribution has CDF:
This is also called the double exponential or log-Weibull distribution.
Read: “G of x equals e to the minus e to the minus x”
The probability that the normalized maximum is at most x
The Gumbel distribution applies to distributions with exponentially decaying tails:
- Gaussian (Normal) distribution
- Exponential distribution
- Gamma distribution
- Lognormal distribution
Maximum of Gaussian Samples
For i.i.d. standard normal variables, the maximum satisfies:
where and .
The maximum grows extremely slowly — like . This is why "5-sigma events" are so rare for Gaussians.
Light Tails Lead to Gumbel
The Gumbel domain includes all distributions whose tails decay at least as fast as exponential. For these distributions, extremes are relatively well-behaved and predictable. There's a "typical" maximum that doesn't vary too wildly.
Type II: The Frechet Distribution
Frechet Distribution (Type II)
The Frechet distribution with shape parameter α has CDF:
The parameter is the tail exponent.
Read: “G of x equals e to the minus x to the minus alpha”
The Frechet CDF for positive x, with tail index alpha
This is the extreme value distribution for fat tails. If your data comes from a power-law distribution, maxima will follow Frechet.
The Frechet distribution applies to distributions with power-law (polynomial) decay:
- Pareto distribution
- Student's t distribution
- Cauchy distribution ()
- Any distribution with
Maximum of Pareto Samples
For i.i.d. Pareto(, ) variables, the maximum grows as:
The maximum grows as — a power of ! For , doubling the sample size multiplies the expected maximum by .
Frechet and Wild Extremes
Under Frechet, there is no "typical" maximum. The distribution of extremes is itself fat-tailed. This means:
- The next maximum could vastly exceed all previous ones
- Historical maximums are poor predictors of future maximums
- Risk estimates based on past extremes will consistently underestimate
Type III: The Weibull Distribution
Weibull Distribution (Type III)
The reversed Weibull distribution with shape parameter has CDF:
This applies to distributions with a finite upper endpoint.
The Weibull type applies to distributions that are bounded above:
- Uniform distribution
- Beta distribution
- Any distribution with a finite right endpoint
Maximum of Uniform Samples
For i.i.d. Uniform(0, 1) variables:
The maximum approaches the upper bound 1, with the gap shrinking as . The behavior is predictable and bounded.
The Generalized Extreme Value Distribution
All three types can be unified into a single family:
Generalized Extreme Value (GEV) Distribution
The GEV distribution has CDF:
for , where:
- : location parameter
- : scale parameter
- : shape parameter (tail index)
The shape parameter ξ (xi) determines the type:
| Shape Parameter | Type | Tail Behavior |
|---|---|---|
| Gumbel (Type I) | Exponential decay | |
| Frechet (Type II) | Power-law decay (fat tails) | |
| Weibull (Type III) | Bounded above |
The Sign of Xi Tells the Story
When fitting extreme value models to data, the estimated immediately tells you the tail type:
- : You're in fat-tail territory (Frechet domain)
- : Light tails (Gumbel domain)
- : Bounded distribution (Weibull domain)
For most financial and natural disaster data, — confirming fat tails.
Key Takeaways
- The Fisher-Tippett-Gnedenko theorem states that normalized maxima can only converge to one of three distributions
- Type I (Gumbel): For light-tailed distributions like Gaussian; extremes grow slowly (logarithmically)
- Type II (Frechet): For fat-tailed distributions like Pareto; extremes grow as powers of
- Type III (Weibull): For bounded distributions; maximum approaches the upper limit
- The GEV distribution unifies all three types with a single shape parameter
- The sign of tells you whether you're dealing with fat tails () or light tails ()