# Fisher–Tippett–Gnedenko theorem

In statistics, the Fisher–Tippett–Gnedenko theorem (also the Fisher–Tippett theorem or the extreme value theorem) is a general result in extreme value theory regarding asymptotic distribution of extreme order statistics. The maximum of a sample of iid random variables after proper renormalization can only converge in distribution to one of 3 possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution. Credit for the extreme value theorem (or convergence to types theorem) is given to Gnedenko (1948)[1], previous versions were stated by Ronald Fisher and Leonard Henry Caleb Tippett (1928)[2] and Fréchet (1927).[3]

The role of the extremal types theorem for maxima is similar to that of central limit theorem for averages, except that the central limit theorem applies to the average of a sample from any distribution with finite variance, while the Fisher–Tippet–Gnedenko theorem only states that if the distribution of a normalized maximum converges, then the limit has to be one of a particular class of distributions. It does not state that the distribution of the normalized maximum does converge.

## Statement

Let ${\displaystyle X_{1},X_{2}\ldots ,X_{n}\ldots }$ be a sequence of independent and identically-distributed random variables, and ${\displaystyle M_{n}=\max\{X_{1},\ldots ,X_{n}\}}$. If a sequence of pairs of real numbers ${\displaystyle (a_{n},b_{n})}$ exists such that each ${\displaystyle a_{n}>0}$ and ${\displaystyle \lim _{n\to \infty }P\left({\frac {M_{n}-b_{n}}{a_{n}}}\leq x\right)=F(x)}$, where ${\displaystyle F}$ is a non-degenerate distribution function, then the limit distribution ${\displaystyle F}$ belongs to either the Gumbel, the Fréchet or the Weibull family.[4] These can be grouped into the generalized extreme value distribution.

## Conditions of convergence

If G is the distribution function of X, then Mn can be rescaled to converge in distribution to

• a Fréchet if and only if G (x) < 1 for all real x and ${\displaystyle {\frac {1-G(tx)}{1-G(t)}}\,{\xrightarrow[{t\to +\infty }]{\,}}x^{-\theta },\quad x>0}$. In this case, possible sequences are
bn = 0 and ${\displaystyle a_{n}=G^{-1}\left(1-{\frac {1}{n}}\right).}$
• a Weibull if and only if ${\displaystyle \omega =\sup\{G<1\}<+\infty }$ and ${\displaystyle {\frac {1-G(\omega +tx)}{1-G(\omega -t)}}\,{\xrightarrow[{t\to 0^{+}}]{\,}}(-x)^{\theta },\quad x<0}$. In this case possible sequences are
bn = ω and ${\displaystyle a_{n}=\omega -G^{-1}\left(1-{\frac {1}{n}}\right).}$

Convergence conditions for the Gumbel distribution are more involved.