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), previous versions were stated by Ronald Fisher and Leonard Henry Caleb Tippett in 1928 and Fréchet in 1927.

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.

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. These can be grouped into the generalized extreme value distribution.

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