---
tag: Y3Z99461HJ
kind: statement
role: theorem
name: Fisher–Tippett–Gnedenko theorem
msc: [60G70, 60F05]
uses: [00MRN0FARV, 0H8H7ZTHW2, 0SRYJGXKR4, 0E99F51GVB]
about: [0H8H7ZTHW2, 0SRYJGXKR4]
sources:
  - url: https://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem
same_as:
  wikidata: Q5454965
  wikipedia: Fisher–Tippett–Gnedenko theorem
---

## Statement

Let $X_1, X_2, \dots$ be [[i.i.d.|00MRN0FARV]] real random variables, and let $M_n = \max(X_1, \dots, X_n)$.
Assume that there are constants $a_n > 0$ and $b_n \in \R$ such that $(M_n - b_n) / a_n$ [[converges in distribution|0SRYJGXKR4]] to a random variable $Y$.
Assume that the law of $Y$ is not concentrated at one point.
Let $G$ be the [[cumulative distribution function|0H8H7ZTHW2]] of $Y$.
Then there are $a > 0$ and $b \in \R$ such that $x \mapsto G(ax + b)$ is one of the following three functions:

1. the Gumbel distribution function $\Lambda(x) = \exp(-e^{-x})$ for $x \in \R$;
2. a Fréchet distribution function $\Phi_\alpha(x) = \exp(-x^{-\alpha})$ for $x > 0$ and $\Phi_\alpha(x) = 0$ for $x \le 0$, for some $\alpha > 0$;
3. a reversed Weibull distribution function $\Psi_\alpha(x) = \exp(-(-x)^{\alpha})$ for $x \le 0$ and $\Psi_\alpha(x) = 1$ for $x > 0$, for some $\alpha > 0$.

## Remarks

The three families combine into the generalised extreme value distributions $G_\xi(x) = \exp\left(-(1 + \xi x)^{-1/\xi}\right)$ for $1 + \xi x > 0$, with $G_0 = \Lambda$ as the limit case $\xi \to 0$.
The sign of $\xi$ gives the type: $\xi > 0$ is Fréchet, $\xi = 0$ is Gumbel, and $\xi < 0$ is reversed Weibull.

Idea of the proof.
If $F$ is the distribution function of $X_1$, then $P(M_n \le a_n x + b_n) = F(a_n x + b_n)^n$.
Comparing the limits along $n$ and $nk$ shows that $G$ is max-stable: for each $k \ge 1$ there are $\alpha_k > 0$ and $\beta_k$ with $G(x)^k = G(\alpha_k x + \beta_k)$.
Solving this functional equation gives the three families.

Example: let the $X_k$ be [[exponential|0E99F51GVB]] with rate $1$, so $F(x) = 1 - e^{-x}$ for $x \ge 0$.
Then $P(M_n - \log n \le x) = (1 - e^{-x}/n)^n \to \exp(-e^{-x})$ for every $x$, so the Gumbel law appears with $a_n = 1$ and $b_n = \log n$.

The hypothesis that a non-degenerate limit exists is needed.
For some laws, such as the geometric and the Poisson distributions, no choice of $a_n$ and $b_n$ gives a non-degenerate limit.

## Explanations

### L1 human

The largest of many independent measurements has, after a shift and a rescaling, only three possible limit shapes.
Which shape appears depends on the tail of the distribution.
Heavy polynomial tails give the Fréchet law, and light tails such as the exponential or normal give the Gumbel law.
A finite upper end point can give the Weibull law.
This is the basis for modelling floods, record temperatures and other extremes.

### L2 human

The theorem plays for maxima the role that the [[central limit theorem|0EPZC4Z1FC]] plays for sums.
The set of laws whose maxima converge to a given type is its domain of attraction, and Gnedenko characterised these domains through conditions on the tail $1 - F$.
In statistics one fits the generalised extreme value family to block maxima of data.