Fisher–Tippett–Gnedenko theorem

Let ${\displaystyle X_{1},X_{2},\ldots ,X_{n}}$ be a sequence of independent and identically-distributed random variables with cumulative distribution function ${\displaystyle F}$. Suppose that there exist two sequences of real numbers ${\displaystyle a_{n}>0}$ and ${\displaystyle b_{n}\in \mathbb {R} }$ such that the following limits converge to a non-degenerate distribution function:

${\displaystyle \lim _{n\to \infty }P\left({\frac {\max\{X_{1},\dots ,X_{n}\}-b_{n}}{a_{n}}}\leq x\right)=G(x)}$,

or equivalently:

${\displaystyle \lim _{n\to \infty }F^{n}\left(a_{n}x+b_{n}\right)=G(x)}$.

In such circumstances, the limit distribution ${\displaystyle G}$ belongs to either the Gumbel, the Fréchet or the Weibull family.[6]

In other words, if the limit above converges, then up to a linear change of coordinates ${\displaystyle G(x)}$ will assume the form:[7]

${\displaystyle G_{\gamma }(x)=\exp \left(-(1+\gamma \,x)^{-1/\gamma }\right),\;\;1+\gamma \,x>0}$

or else

${\displaystyle G_{0}(x)=\exp \left(-\exp(-x)\right)}$

for some parameter ${\displaystyle \gamma .}$ This is the cumulative distribution function of the generalized extreme value distribution (GEV) with extreme value index ${\displaystyle \gamma }$. The GEV distribution groups the Gumbel, Fréchet and Weibull distributions into a single one. Note that the second formula (the Gumbel distribution) is the limit of the first as ${\displaystyle \gamma }$ goes to zero.

The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution ${\displaystyle G(x)}$ above. The study of conditions for convergence of ${\displaystyle G}$ to particular cases of the generalized extreme value distribution began with Mises (1936)[3][5][4] and was further developed by Gnedenko (1943).[5]

Let ${\displaystyle F}$ be the distribution function of ${\displaystyle X}$, and ${\displaystyle X_{1},\dots ,X_{n}}$ an i.i.d. sample thereof. Also let ${\displaystyle x^{*}}$ be the populational maximum, i.e. ${\displaystyle x^{*}=\sup\{x\mid F(x)<1\}}$. The limiting distribution of the normalized sample maximum, given by ${\displaystyle G}$ above, will then be:[7]

• A Fréchet distribution (${\displaystyle \gamma >0}$) if and only if ${\displaystyle x^{*}=\infty }$ and ${\displaystyle \lim _{t\rightarrow \infty }{\frac {1-F(ut)}{1-F(t)}}=u^{-1/|\gamma |}}$ for all ${\displaystyle u>0}$.

This corresponds to what is called a heavy tail. In this case, possible sequences that will satisfy the theorem conditions are ${\displaystyle b_{n}=0}$ and ${\displaystyle a_{n}=F^{-1}\left(1-{\frac {1}{n}}\right)}$.

• A Gumbel distribution (${\displaystyle \gamma =0}$), with ${\displaystyle x^{*}}$ finite or infinite, if and only if ${\displaystyle \lim _{t\rightarrow x^{*}}{\frac {1-F(t+uf(t))}{1-F(t)}}=e^{-u}}$ for all ${\displaystyle u>0}$ with ${\displaystyle f(t):={\frac {\int _{t}^{x^{*}}1-F(s)ds}{1-F(t)}}}$.

Possible sequences here are ${\displaystyle b_{n}=F^{-1}\left(1-{\frac {1}{n}}\right)}$ and ${\displaystyle a_{n}=f\left(F^{-1}\left(1-{\frac {1}{n}}\right)\right)}$.

• A Weibull distribution (${\displaystyle \gamma <0}$) if and only if ${\displaystyle x^{*}}$ is finite and ${\displaystyle \lim _{t\rightarrow 0^{+}}{\frac {1-F(x^{*}-ut)}{1-F(x^{*}-t)}}=u^{1/|\gamma |}}$ for all ${\displaystyle u>0}$.

Possible sequences here are ${\displaystyle b_{n}=x^{*}}$ and ${\displaystyle a_{n}=x^{*}-F^{-1}\left(1-{\frac {1}{n}}\right)}$.

• Extreme value theory
• Gumbel distribution
• Generalized extreme value distribution
• Pickands–Balkema–de Haan theorem
• Generalized Pareto distribution
• Exponentiated generalized Pareto distribution